Microeconomic Graphic Art
Microeconomic Graphic Art
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This website contains selected illustrations of microeconomics graphic art accompanied by descriptive matter. Instructors and students are welcome to use the illustrations and descriptive matter as they wish. Corrections and suggestions are welcome.
2. Marginal Utility and Risk
3. Derivatives
4. Two Independent Variables
5. Diminishing Marginal Utility
6. Demand Curve
7. Change of Demand
8. Demand Surface
9. Total Revenue
10. Marginal Revenue
11. Elasticity of Demand
12. Preference and Indifference
13. Production Vertical
14. Marginal Product of Labor
15. Capital Variable
16. Equamarginal
17. Production Horizontal
19. Isoquant Stages
20. Returns to Scale
21. Costs in the Short Run
22. Costs in the Long Run
23. Pure Competition
24. Entry and Exit
25. Competition in the Long Run
26. Pure Monopoly
27. Monopoly in the Long Run
28. Monopolistic Competition
29. Oligopolistic Competition
30. Price Discrimination
31. Jointly-Produced Products
32. Variable Proportions
33. Transfer Pricing
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Since a decision maker might be willing to trade off risk for return, he must be able to decide whether the higher expected value of a decision alternative is adequate compensation for the additional risk that must be assumed. Theoretically, a preference function relating satisfaction (or utility) to any pair of desirable goods (or phenomena) may be constructed for a decision maker. Practically, the specification of such a preference function is difficult if not impossible to achieve. A functional notation representation of such a preference function might appear as
U = f (V, (1-R) | ...),
where U is the amount of utility realized by the decision maker, V is the value of the return from a decision alternative, and R indicates the risk associated with the opportunity. The argument (1-R) is the complement of the amount of risk incurred, or the degree of certainty that a particular outcome within the range of all such possible outcomes will occur.
Graphically, such a preference function for a normal decision maker might appear as depicted in Figure A1-3. Risk (degree of uncertainty) and return occupy the floor axes of the three-dimensional graph; utility is measured in the vertical dimension. The right-hand side of the utility surface can be identified because the degree of certainty reaches a maximum, i.e., the degree of risk approaches zero. But the left-hand end of the risk-certainty axis cannot be specified since risk may increase without bound. The utility surface may be "sliced" parallel to the floor and at any altitude of utility. Four such slices are illustrated in Figure A1-3, and their vertical projections down into the floor of the surface have been drawn. These projections constitute so-called "indifference curves" for the two phenomena represented on the floor axes, i.e., return and degree of certainty. Any number of such indifference curves may be constructed by slicing the utility surface at any chosen utility altitudes.
Figure A1-4 depicts a map of selected indifference curves generated from the utility surface illustrated in Figure A1-3. In effect, the map depicted in Figure A1-4 is the sliced surface of Figure A1-3 viewed from above. In Figure A1-4, a movement from left to right along the horizontal axis represents an increase of certainty about which outcome within the range of all possible outcomes will occur (and implicitly a narrowing of the range of possible outcomes). At the right end of the horizontal axis, the decision maker may be certain that there is only one possible outcome, and thus that risk is minimal (or zero). Risk therefore increases from right to left along the horizontal axis, but the left end of the axis may not be specified in the sense that risk may increase without bound.
Any point in the coordinate space of the map represents some combination of risk and return, and each point lies on some indifference curve, such as I2, for a certain level of utility. Any movement along I2 would leave the decision maker in a state of indifference. For example, the movement from point A to point B would result in an increase of risk (a "decrease of certainty"), for which the decision maker would have to be compensated by an increase of return in order to remain at the same level of utility. Or, at the higher return associated with point B, the decision maker would tolerate more risk (less certainty). The indifference curves for a more risk-averse decision maker would be more steeply upward sloped (right to left) because such a decision maker would require even greater return for each level of risk.
The conditions under which a decision maker realizes satisfaction (i.e., utility) with respect to income (whether monetary or "in kind") are thought to influence his attitude toward risk. Three possible shapes of total utility functions are illustrated in Figure B1-2. For decision maker A, utility increases at a constant rate as income increases. Decision maker B's utility increases at a decreasing rate (the utility function, though positively sloped, is concave downward). This illustrates the so-called principle of diminishing marginal utility, and is thought to be descriptive of typical human behavior with respect to the receipt of income. Decision maker C realizes additional utility at an increasing rate as income increases (i.e., the curve is concave upward). Although this utility pattern is thought to be atypical of human behavior except in the very early stages of consumption, it would illustrate a phenomenon of increasing marginal utility. In all three cases, the utility function is extended into graphic quadrant III (negative values for both income and utility) for purpose of illustration. Although utility is hardly measurable, and certainly not comparable from one decision maker to another, we shall indulge in the heroic assumption of the same numeric scale on the vertical axis of all three graphs.
Figure B2-6 illustrates in its panel (a) the graphic plot of an idealized third order (cubic) function of form
y = a + bx1 + cx12 + dx13.
The cubic function has two directions of curvature separated by an inflection point at A. Panel (b) illustrates the associated first derivative or marginal function, of form
y' = b + 2cx1 + 3dx12.
The first derivative function is of one order lower than the original function, and has one fewer directions of curvature than does the original function. We note that the peak of the first derivative function occurs at the same x1 quantity at which the inflection point of the y function occurs. The second derivative function also attains zero y values at the levels of x1 at which the original y function reached both minimum and maximum.
Panel (c) illustrates the associated second derivative function of form
y" = 2c + 6dx1.
The second derivative function is one order lower than the first derivative function, and two orders lower than the original function. It has one fewer directions of curvature, in this case being linear, than does the first derivative function.
It is important to note that since the first derivative function reaches
a value of zero when the original function is at either minimum or maximum, it is not possible without plotting the graphs of the functions to know from computing the first derivative alone whether the original function has reached a maximum or a minimum. However, the second derivative provides a solution to this problem. Since, in Figure B2-6, the first derivative cuts the horizontal from below and is thus positively sloped at the x1 value for which the original function is at a minimum, the value of the second derivative function is positive at this level of x1. It is also true that the first derivative cuts its horizontal from above, and is thus negatively sloped at the x1 level at which the original function is at maximum. Hence, the second derivative exhibits a negative value when the original function is at a maximum.
A functional notation statement may be written to accommodate two independent variables, any others assumed constant:
y = f (x1, x2 | x3, ... xn).
An example of an equation containing two independent variables is
y = ax1 + bx12 + cx1x2 + dx2 + ex22.
This particular equation is of second order in both x1 and x2, but an equation could be specified to any order in as many independent variables as the analyst wishes to include.
The graph of an equation like this would be three dimensional with a parabolic appearance such as that in Figure B2-8.
Here u and v are the x1 and x2 coordinates of the point directly below the peak of the surface. Supposing for the moment that u and v are known coordinates, if we were to move a point such as r along the path from v toward t, we could observe that the slope of the surface, as measured by a tangent to the surface at r, becomes ever shallower until it is zero at the peak. Likewise, if a point such as s is moved along the path from u toward t, the slope of the surface as measured by a tangent at s would also become ever shallower, and also reach zero at the peak. It is knowledge of this behavior that allows us to find the values of u and v when they are unknown.
Economists have deduced from a great deal of personal and collective experience that consumers, in acquiring successive additional units of most goods or services, tend to realize declining amounts of additional value (i.e., utility or satisfaction). This phenomenon is referred to the in the economics literature as the "principle of diminishing marginal utility."
Economists recognize that the consumer may experience an initial surge of realized value from consuming the first few units of the good or service, but they have also become convinced of the certainty of eventual diminishing marginal utility for most goods. The qualification is in regard to goods (alcohol, drugs) or activities (hobbies, sex) that may be addictive or compulsive. Although there is much that is yet unknown in regard to addictive behavior, it may be hypothesized that the consumer realizes increasing marginal utility when consuming successive units of goods that are objects of addiction.
Figure C1-1 illustrates what economists think that a so-called total utility (TU) function and its derived marginal utility (MU) function might look like for a normal good, assuming all other factors constant. The underlying functional relationship can be given by
(1) TUx = f (Qx | ... ),
i.e., the total utility realized in consuming good x is determined by the quantity of x consumed, assuming all other factors constant. The graph of the TU function can be perceived as a two-dimensional section through a three-dimensional utility surface. As it is illustrated in panel (a) of Figure C1-1, the TU curve is concave upward initially, over the range from the origin to Q1. This is the initial consumption range over which the consumer may experience the surge of utility rising at an increasing rate. But beyond Q1, and up to quantity Q2, total utility increases at a decreasing rate. The key concept here is the decreasing rate of increase of total utility. It is apparent that the total amount of utility realized in the consumption of commodity x reaches a maximum at quantity Q2. Successive units consumed beyond Q2 actually yield negative satisfactions, so the total amount of utility decreases. We can observe that over the quantity range for which TU is increasing at an increasing rate, from the origin to Q1, MU (marginal utility) rises, reaching a peak at Q1. Over the quantity range for which TU is increasing at a decreasing rate, MU falls, reaching a value of zero at Q2 the quantity at which TU is maximum. The quantity range between Q1 and Q2 is described as the range of diminishing marginal utility. And it is this range that economists think represents the usual circumstances under which consumers make most of their choices. The reader is now invited to speculate on the likely appearances of the TU and MU curves for a commodity that is an object of addiction or compulsive consumption.
In the case of a non-addictive good or activity, the rational decision criterion is to continue to consume more of the good, even while realizing declining additional utility, until the marginal value realized in consumption is no longer greater than the marginal cost of the acquisition. In order to make such a comparison, the marginal cost of
acquisition must be perceived in units comparable to those in which satisfactions are measured. One way to do this is to regard the acquisition cost in terms of dissatisfaction or disutility at having to part with purchasing power to make the acquisition. If the marginal utility does indeed decline, a point at which additional consumption should cease will be reached. In Figure C1-1, curve MD (marginal disutility) represents the marginal cost of acquiring additional units of the commodity (constant as illustrated). The consumer should push consumption to Q3, beyond which the marginal utility falls below the marginal disutility realized in acquisition.
The functional notation representation of a demand relationship may be given by
(2) Qx = f ( Px | ... ),
i.e., the quantity demanded of a good or service is determined by the price of the good or service, assuming all other determinants to be constant. The functional relationship, f, is presumed to be inverse for the relationship between quantity and price for any normal good. This inverse relationship, i.e., the Law of Demand, can be illustrated by drawing a demand curve on a set of coordinate axes for price and quantity as in Figure C1-2. The downward (left to right) slope of the demand curve is a manifestation of the principle of diminishing marginal utility.
The demand curve in Figure C1-2 is drawn as a straight line with a negative slope. The equation for such a linear demand curve can be given in slope-intercept form as
(3) Qx = c + d(Px),
where c is the quantity-axis intercept, and d is the (assumed negative) slope of the demand curve. The linearity of this demand curve is assumed only for purposes of simplicity. In reality, a demand curve may exhibit any degree of curvature, and it may be concave upward or downward. Even if a straight line can approximate the price-quantity relationship, the linear demand curve may exhibit a range of slopes, from nearly horizontal at one extreme, to almost vertical at the other. And even these extremes are not effective limits on the possible slopes that demand curves may take. If the income effect of a price change of an inferior good were great enough to outweigh the substitution effect, the demand curve would slope upward from left to right in apparent contradiction to the law of demand.
If the demand curve illustrated in Figure C1-2 can be presumed to be a realistic representation of a real demand relationship for good x, then a decrease of the price from P1 to P2 can be expected to lead to an increase in the consumer's purchases of x from Q1 to Q2. Economists refer to such a movement from one point to another along a fixed-locus demand curve as a "change of quantity demanded." Such a change of quantity demanded is attributable exclusively to a change in the price of the good, given all other determinants.
The demand for any good or service is actually determined by many factors in addition to the price of the good or service. In fact, for some items the price may be one of the lesser-significant determinants of its demand. A more general specification of a demand curve may be given by
Qx = f ( Px, I, T, B, ... , Py, Pz, ... ),
where I is the income of the consumer, T stands for "tastes and preferences," B is the consumer's current level of indebtedness, Py is the price of a relevant substitute good, and Pz is the price of a related complement good. The ellipsis symbols ( ... ) between B and Py suggest that there are other non-price demand determinants that have not yet been specified (or even identified). The ellipsis symbols following Pz allow for prices of yet other substitute and complement goods.
There is nothing particularly significant about the order in which the determinants of demand are listed on the right side of the equation. The order of the listing can be changed at will, and any one of them can be moved to the head of the queue as required. The price of the good itself (i.e., the good's "own price") is typically listed in the first position because, historically, the attention of economists turned to this determinant first. Also, in the cases of most goods and services, the own price may indeed be the most important or significant (in a statistical sense) determinant of the quantity demanded. Yet any such hierarchy of determinants is something to be discovered by analysis, rather than assumed at the outset.
In order to draw the two-dimensional representation of the demand curve illustrated in Figure C1-2, it was necessary to treat all of the non-own price demand determinants as if they were constant, even if they in fact were not constant (more about this below). A revision of equation above to represent this specification is given by
Qx = f ( Px | I, T, D, ... , Py, Pz, ... ),
where the vertical bar ( | ) is used to separate the single demand determinant that is presumed to be variable (Px) from all the rest, that are assumed not to change. Indeed, if any of the other determinants are variable, it is technically not even possible to draw a discrete locus for the demand curve in the two-dimensional space of the P-Q coordinate axes.
Economists employ the term "change of demand" to refer to the circumstance where some determinant of demand other than the item's own price has changed. The effect of such a change is to shift the own-price demand curve from its former locus to some position, as illustrated in Figure C1-3. Here, D1 is the original locus of the demand curve, and D2 is the new locus after something other than the price of the good has changed. For example, improving tastes and preferences for the good or service, or a decrease in consumer indebtedness, could possibly explain the illustrated right-ward shift of the demand curve.
A demand-shift phenomenon may be illustrated by a three-dimensional graphic representation of the demand relationship. In this representation, the vertical bar is moved to between the second and third determinants in the demand equation,
Qx = f ( Px, I | T, B, ... , Py, Pz, ... ).
In this relationship, two demand determinants, P and I, are presumed variable, while all other possible determinants are treated as if they were constant. A graphic representation of this relationship is given in Figure C1-4, where the third dimension (depth) is occupied by the income determinant. A number of "slices" (or vertical sections) have been made through the three-dimensional demand surface at different income levels.
If the three-dimensional surface were viewed from a perspective opposite the price-quantity plane, in effect collapsing the surface into two dimensions, the viewer would see something like that represented in Figure C1-5. Here, the several vertical slices through the three-dimensional surface appear as a demand curve that shifts in two dimensions.
In Figure C1-4, the intersection of the demand surface with the floor (the income-quantity plane) traces out path RSTUV, which when viewed from above yields the two dimensional graph illustrated in Figure C1-6.
In this figure, income is measured on the vertical axis, and is thus taken to be the demand determinant relative to quantity demanded on the horizontal axis. The path, RSTUV, thus traces out an income-demand curve (or Engel Curve, as it is referred to in the literature), for which the functional notation relationship would be given as
Qx = f( I | Px, T, D, ... , Py, Pz, ... ).
In this equation, the only variable determinant of demand is presumed to be income, while all other demand determinants, including price, are taken to be constant. The upward slope of RSTUV implies that the good being analyzed is a "normal good," i.e., with a direct relationship between income and quantity demanded. As noted above, price may not be the most significant demand determinant, and it is legitimate to move any of the demand determinants to the head of the list of determinants so that it may be analyzed, assuming all other determinants as givens.
The three-dimensional surface represented in Figure C1-4 is for two demand determinants, own-price and income, relative to quantity demanded. It is unfortunate that we can have access to no more than three graphic dimensions, because this necessarily limits our analysis to no more than two demand determinants at one time as long as we wish to stay with the graphic analysis. We can treat more than two determinants at one time only by exiting the graphics and entering the realm of multivariate algebra. However, within the realm of three-dimensional graphics, we can move any two determinants to the head of the determinant queue in order to construct a three-dimensional surface showing the relationship between quantity demanded and the two selected determinants. And by judiciously slicing the three-dimensional surface, we can extract two-dimensional demand curves showing the relationship between quantity demanded and any single demand determinant.
The price of a product can be understood to be its average revenue (AR), or the revenue per unit of the product sold by the enterprise. Thus, the total revenue (TR) that the enterprise will realize on the sale of Q units of its product can be computed by the formula
TR = P x Q,
or if total revenue is known, the average revenue, or price, can be computed by solving the equation for P, or
AR = P = TR/Q.
Knowledge of these relationships enables us to derive an equation for a total revenue function from the equation for the demand function, or an equation for the demand function if that for the total revenue function is known. Suppose that the demand equation has been specified with parameter values c=20 and d=-4, resulting in this equation,
Q = 20 + (-4)P.
In order to derive the total revenue equation, we must first solve the demand equation for P,
P = 5 - .25Q.
Since we know that TR = P x Q, we may derive the equation for total revenue by multiplying this equation through by Q,
P x Q = 5Q - .25Q2,
TR = 5Q - .25Q2.
Alternately, had the TR equation been specified first, since AR = TR/Q, the AR equation could be derived by dividing the TR equation through by Q,
TR/Q = 5Q/Q - .25Q2/Q,
AR = 5 - .25Q.
The derivation of these equations by simple algebraic manipulation enables us to illustrate in two dimensions the graphic relationship between AR and TR. The TR curve is shown in panel (a) of Figure C1-7; its associated demand curve (AR) is shown in panel (b).
Because the demand curve is linear with a negative slope, its associated total revenue curve is a second-order (or quadratic) equation that graphs as a parabola that opens downward and spans the positive-price range of the demand curve on the quantity axis. In Figure C1-7 we have also drawn a box in panel (b) below the demand curve formed by a horizontal at price P1 and a vertical at quantity Q1, the quantity that will be sold at price P1. We have also drawn a vertical in panel (a) below the TR curve a quantity Q1. By the formula for the area of a rectangle (area = length x width), we can assert that the area of the box in panel (b) measures the total revenue resulting from selling quantity Q1 at price P1. This same area is also represented by the altitude of the vertical at Q1 up to the TR curve in panel (a).
Figure C1-8 is a reproduction of Figure C1-7, but with several additional price-quantity boxes drawn below the D=AR curve, and corresponding verticals drawn below the TR curve.
The reader should verify by inspection of the boxes that as price falls toward P3
and quantity increases accordingly, the boxed areas increase to a maximum corresponding to the tallest vertical below the vertex of the TR parabola. If the demand curve is indeed linear, the maximum total revenue will occur at a quantity that is half the horizontal axis intercept of the demand curve. In Figure C1-8, prices successively lower than P3
yield revenue rectangles of progressively smaller area The graphic approach illustrated in Figure C1-8 provides a means of identifying the price-quantity combination that yields the maximum total revenue.
Economists refer the addition to total revenue consequent upon selling one more unit of an item as "marginal revenue." Letting the symbol "R" represent revenue and the symbol "Q" represent quantity, marginal revenue can be represented by the equation
MR = ΔR/ΔQ
for the smallest possible change of ΔQ (practically, one unit).
The graph of a marginal revenue curve shares a common price-axis intercept with the D = AR curve, but the slope of the downward-sloping MR curve is steeper than that of the downward-sloping AR curve.
Figure C1-9 shows the MR curve drawn in below the D = AR curve in panel (b). The most important observation to make in regard to Figure C1-9 is that the MR curve reaches zero at the quantity level for which the TR parabola attains its maximum value.
These relationships provide a useful managerial decision criterion. If the objective of the enterprise is to produce a quantity of an item and sell it at a price that yields the maximum possible revenue, it can do so by finding the quantity for which marginal revenue is zero.
A useful revenue-oriented decision criterion can be constructed by computing the ratio of the percentage change of quantity demanded to the percentage change in the price which resulted in the quantity change, or
%ΔQ / %ΔP,
letting the letter "D" stand for the Greek symbol for Δ (meaning "change of").
Economists refer to this ratio as the own-price elasticity of demand, and they interpret it as a measure of the sensitivity (or responsiveness) of quantity demanded to a change in the item's own price. The revenue-related importance of the own-price elasticity can be illustrated by demand equation Qx = 80 - 4Px, which is graphed in Figure C2-1.
The linear demand curve has been divided into ranges indicated by brackets. The upper portion of the demand curve, from price of $20 down to $10, is labeled the elastic range. It is characterized by positive marginal revenues and elasticity ratios (absolute values) greater than unity. The lower portion of the demand curve, from price $10 down to $0, is labeled the inelastic range. It is characterized by negative marginal revenues and fractional elasticity ratios (again, absolute values). The midpoint of the linear demand curve (at price $10) is labeled the unitarily elastic point because the absolute value of the demand elasticity ratio is precisely 1.0 at this point. Demand at the unitarily elastic point is also characterized by zero marginal revenue.
In the elastic range of the demand curve, any particular percentage decrease of price will result in a larger percentage increase of quantity demanded. Thus, what is lost to revenue by cutting price is more than made up for in increased quantity sold, so total revenue increases. For example, if price is lowered from $18 to $16, quantity demanded will increase from 8 units to 16 units, and total revenue will increase from $144 to $256. Price fell by 11 percent, but this was more than made up for by a 100 percent increase in quantity sold.
However, in the elastic range of the demand curve, any particular percentage increase of price will result in a larger-percentage decrease of quantity demanded, thus causing a decrease of total revenue. In this case, what is gained in raising the price is more than offset by the loss in quantity sold. Thus, if the manager can verify that the enterprise is presently selling the item at a point in the elastic range of the demand curve, the appropriate direction in which to change price in order to increase revenue is down.
The opposite conclusions emerge for the inelastic range of the demand curve. A price cut in the inelastic range will result in an increase of quantity demanded, albeit one of a smaller percentage magnitude, so that total revenue can be expected to decrease. If the objective is to increase revenue, price should be raised because a smaller-percentage decrease in quantity demanded will result. In the example illustrated in Figure C2-1, a price cut from $9 to $8 will result in an increase of quantity sold from 44 to 48 units, but a decrease of total revenue from $396 to $384.
Small percentage changes of price in the near neighborhood of the unitarily elastic point of the demand curve will be offset by the same percentage changes of quantity demanded (but in the opposite direction), thereby leaving revenue unchanged. In this example, if price is raised from $9 to $11, quantity demanded will fall from 44 to 36 units, leaving total revenue unchanged at $396.
If the enterprise were to progressively lower price, moving down the demand curve from its intercept with the price axis toward its intercept with the quantity axis, total revenue would increase to a maximum (at the unitarily elastic point), and then decrease; concurrently, marginal revenue would decrease from positive values, through zero (at the unitarily elastic point), to negative values. And, elasticity would fall from (absolute) values greater than unity to (absolute) values less than unity.
Figure C1A-1 illustrates a consumer's three-dimensional utility surface with utility measured vertically against the quantities of two goods, X and Y, represented on the floor axes. The functional notation representation of the utility surface is
U = f (X, Y / ...),
or "utility depends upon the quantities of goods X and Y consumed, given all else."
As illustrated in Figure C1A-1, this analytical approach to consumer behavior is to pass a plane, abcd, parallel to the floor of the utility surface at some utility level altitude, say I4, tracing out the horizontal slice, efgh. The path, e'f'g'h', in the floor of the surface is the vertical projection of the slice through the surface. Slice efgh is only one of an infinite number of slices which could be taken through the surface at different utility altitudes.
Figure C1A-2 shows a two-dimensional view from a perspective above the surface illustrated in Figure C1A-1. In effect, the three-dimensional surface appears to be collapsed into the floor when viewed from above, but the path e'f'g'h' represents the slice through the surface at utility altitude I4. Economists refer to the e'f'g'h' path as an iso-utility curve (meaning same utility) because it represents a sequence of points, the (K,L) coordinates of which are the combinations of the two goods which can yield the I4 level of utility. For example, utility I4 can be realized with the X1 quantity of good X combined with the Y1 quantity of good Y. Utility I4 can also be realized with Y2 if a larger, X2, is consumed. This suggests that good X and good Y are to some extent substitutable for one another in the consumer's preferences. Since path e'f'g'h' is drawn as a continuous curve, the implication is that there are an infinite number (or as many as there are points along the path) of combinations of X and Y which can yield the I4 utility level.
Economists also refer to path e'f'g'h' as an "indifference curve." The reason is that the consumer should be indifferent between the combinations of goods represented by different points along the same iso-utility curve because they all yield the same level of utility. Combination (X1, Y1) at point f' yields the I4 level of utility, but so also does combination (X2, Y2) at point g'. However, the consumer would prefer combinations of X and Y at utility level I4 to any combination of X and Y which yields a lower level of utility. But the consumer would give preference to any combination of X and Y which yields a higher level of utility than I4. We will refer to such curves as indifference curves rather than iso-utility curves in subsequent discussion.
Suppose that instead of selecting an (X,Y) combination represented by a point along the indifference curve, the consumer chooses the Y2 quantity of good Y with the smaller X2 quantity of good X, reaching point j on the utility surface in Figure C1A-1, and j' in Figure C1A-2. It should be apparent that the smaller quantity of good x combined with less of good y will yield some utility level lower than I4.
Figure C1A-3 is an elaboration of Figure C1A-2 to show the paths of several other "representative" indifference curves which could be generated by horizontally slicing the surface at altitudes other than I4. It is now apparent that the X1 quantity of good X along with the Y2 amount of good Y can yield only the I3 level of utility, which is less than I4 level. Theoretically, any number of such indifference curves could be generated by slicing the utility surface at different altitudes so that the floor of the surface might appear "dense" with concentric indifference curves. The collection of representative indifference curves may be referred to as an indifference curve map. The indifference curve map may be likened to the collection of isotemp or isobar lines on a weather map, or to the contour lines on a geological or military map.
Suppose that the horizontal slices taken through the surface to generate the indifference curve map illustrated in Figure C1A-2 were taken at successively higher utility altitudes which are equal utility increments apart. In this case, (I2- I1) would be equal to (I3- I2), and so on. When viewed from above the surface in Figure C1A-1, the indifference curve map illustrated in Figure C1A-2 betrays the likely shape of the surface which cannot be seen explicitly in Figure C1A-2. From indifference curve I1 up to indifference curve I4 the indifference curves appear to be getting closer together, implying that the surface is increasing at an increasing rate in the utility dimension. This corresponds to the range from the origin to Q1 along the total utility curve illustrated in panel (a) of Figure C1-1. From indifference curve I4 through indifference curve I7 the indifference curves appear to become farther apart, suggesting that utility is increasing at a decreasing rate. This corresponds to the range from the origin to Q1 along the total utility curve illustrated in panel (a) of Figure C1-1. The reader is invited to analyze the likely shapes of the indifference curve maps which could be generated from the surfaces linear and second-order (quadratic) shapes.
The slope of an indifference curve at any point may be measured by the slope of a tangent drawn to the curve at the point, or it may be approximated by the slope of a chord connecting two points near to each other along the indifference. For example, the slope of the I4 indifference curve over the f' to g' range in Figure C1A-3 can be approximated by the slope of the chord from f' to g', or
ΔY/ΔX = (Y2 - Y1) / (X2 - X1) = f'j' / j'g'
The technical name for this slope is the Marginal Rate of Substitution (MRS). The MRS is interpreted as the rate at which the consumer can substitute good X for good Y while remaining at the same level of utility. Over the f'g' arc of the I4 indifference curve, the consumer can maintain utility at the I4 level as the good Y amount is reduced from Y1 to Y2 only by increasing the quantity of good X consumed from X1 to X2. It should be obvious that between f' and g' the MRS is negative since Y2 is less than Y1. The MRS will normally be negative over the economic range of consumption.
It will be instructive to examine the right triangle formed by the points f', g', and j'. The ratio of the sides of this triangle, f'j'/j'g', measures the slope of the third side (or hypotenuse), which forms the f'g' chord. The diagonal movement from f' to g' may be regarded as two separate adjustments represented by the vertical and horizontal sides of the triangle. Specifically, the movement from f' to j' is a decrease in the consumption of the Y good (-ΔY) which, other things remaining the same, would decrease the consumer's utility from I4 to I3, or -ΔI1. The movement from j' to g' is an increase in the consumption of X (ΔX), which by itself would cause output to increase from I3 to I4, or ΔI2. ΔI2 is of the same magnitude, but opposite sign, as ΔI1, so we shall use ΔI without sign or subscript to refer to both quantities. Identification of the utility changes allows us to specify the marginal utilities of good X and good Y as
MUY = ΔI/ΔY and MUX = ΔI/ΔX.
Further, we may note that the ratio of the marginal utility of good X to the marginal utility of good Y also measures the MRS:
MUX/MUY = (ΔI/ΔX)/(ΔI/ΔY) = ΔY/ΔX = MRS.
The indifference curve map provides information about consumption possibilities, but by itself cannot provide the consumer with a criterion for selecting an optimal combination of two goods. Additional information is needed in the consumer's budget (or that portion which is reserved for the two goods) and the prices of the two goods, X and Y. Suppose that the prices of good X and good Y are PX and PY, and the consumer has determined that the budgeted outlay must be limited to the dollar sum B. This information can be brought together in the following equation:
X*PX + Y*PY <= B.
The sense of this equation is that the product of the number of units of good X consumed (X) times its price (PX) plus the number of units of good Y consumed (Y) times its price (PY) cannot exceed the budgeted outlay (B). The consumer could underexpend the budget, but suppose that the consumer typically commits the entire budget to the purchase of quantities of good X and good Y so that the inequality symbol may be ignored. What quantities of each good should the consumer purchase in order to maximize the consumer's utility?
This equation can be usefully rearranged in the following format:
Y = ( B / PY ) - ( PX | PY ) * X.
Equation (5) is linear and in the so-called "slope-intercept" format, y = a + bx, where the slope of the line which represents the equation is the ratio of the price of labor (PX) to the price of capital (PY). It should be noted that this ratio is negative. The conceptual interpretation of this slope is the rate at which the consumer can substitute good X for good Y while remaining within the budget The vertical axis intercept is the ratio of the budgeted outlay (B) to the price of good Y (PY), which determines the maximum amount of good Y which could be purchased by the consumer if he or she purchased none of good X. Equation (5), henceforth referred to as the budget line, can be plotted on the same set of coordinate axes containing the indifference curve map. We may suppose that the values of B, PX, and PY are such that the budget line is plotted as in Figure C1A-4.
At what point along the budget line should the consumer select a combination of good X and good Y? As the reader has likely already guessed, the relevant decision criterion is a comparison of the slopes of the indifference curve and the budget line, i.e., whether
MRS >=< ( PX / PY ).
Point m' is one option involving a large amount of good Y with a small amount of good X. At point m' the slope of the indifference curve, though negative, is very steep, implying that if the consumer were to give up the (Y3- Y4) quantity of good Y, he or she could purchase the (X4- X3) quantity of good X while remaining within the budget. This additional quantity of good X is a great deal more than the consumer would have to have to remain at the I2 level of utility. In fact, to exchange the (Y3- Y4) quantity of good Y for the (X4- X3) quantity of good X would enable the consumer to move to a higher level of utility on indifference curve I3 at point n'. This is obviously a good move which the consumer should undertake. The reader is now invited to repeat this analysis using point n' as the departure point.
Suppose that the consumer maker had first determined to try the input combination represented by point s'. At this point the consumer purchases a relatively large quantity of good X along with substantially less of good Y. The MRS at point s' is negative, but quite small (i.e., the slope of the indifference curve is shallow). The consumer is considering replacing the (X5 - X6) quantity of good X with additional good Y. Only a small amount of additional good Y will be needed to allow the consumer to remain at the same level of utility, I3. However, by giving up the (X5- X4) quantity of good X the consumer can still remain within the budget by purchasing the (Y6- Y5) quantity of good Y, which will enable the consumer to increase utility almost to the I4 level. This is a good move which the consumer should undertake.
As may be readily deduced, the consumer will find a utility-increasing incentive to substitute good X and good Y by moving downward along the budget line from points such as m' and n' until the incentive disappears. Likewise, a similar incentive will be found to move upward along the budget line from points such as r' and s' until the incentive is eliminated.
When does the substitution incentive expire? This occurs when a point is reached along the budget line where the slope of the budget line is just equal to the slope of the indifference curve. At such a point, e.g., f' in Figure C1A-4, the conceptual interpretation is that the rate at which the consumer can substitute good X for good Y while remaining at the same level of utility (the MRS) is just equal to the rate at which the consumer can substitute good X for good Y while remaining within its outlay budget (PX/PY). This is the point of tangency of the budget line with an indifference curve. It is also the highest-utility indifference curve which the budget line can reach. At this point the consumer will have found the utility-maximizing combination of good X and good Y, given the limitation of the budgeted outlay.
In Figure C1A-5, budget line B1 represents the initial conditions under which the consumer can purchase quantities of X and Y, given their respective prices, while not exceeding the budget. The slope of the budget line is measured by the ratio, PX/PY. The consumer thus can maximize utility at point J, the tangency of the budget line with indifference curve I4, purchasing the quantities X1 and Y1.
However, this increase of utility consequent upon the price change is composed of two effects, a substitution effect as the consumer shifts purchases away from other goods to X, and an income effect due to the fact that with a lower price of something in the consumer's budget, the consumer can now purchase more of all goods. These two effects may be illustrated by hypothetically removing enough from the consumer's budget at the new PX/PY price ratio to return the consumer to the former level of utility, I4. This would be represented by the budget line B3, which reaches tangency with indifference curve I4 at point L where the consumer would purchase quantities X3 and Y3.
The substitution effect now can be illustrated in Figure C3A-5 as the movement along the original indifference curve, I4, from point J to point L. In order to remain at the same level of utility as the price of X falls, the consumer would purchase a larger quantity, X3, but a smaller quantity, Y3. However, with the implicit increase in purchasing power of the consumer's income consequent upon the fall in the price of X, the budget line implicitly shifts from B3 to B2, allowing the consumer to purchase the larger quantities, X2 and Y2.
We may note that in the case of a normal good as illustrated, the income effect is in the same direction as the substitution effect, and thereby reinforces the substitution effect. With the fall in the price of X the consumer purchases more of X due to the substitution effect, any yet more of X due to the income effect. In the case of a normal good, the demand curve slopes downward from left to right for two reasons, the substitution effect and the income effect. The reader is invited to construct a graphic analysis to illustrate the substitution and income effects consequent upon a price increase.
In the case of an inferior good, the income effect is in the opposite direction to that of the substitution effect, and therefore offsets (at least partially) the substitution effect. In such a case, with a fall in the price of X, more of good X is purchased due to the substitution effect, but not as much more as might have been purchased had there been no offsetting income effect. In the case of an inferior good, the demand curve still slopes downward from left to right, but only because the substitution effect is strong enough to be only partially offset by the income effect, but not overwhelmed by it. The reader is invited to imagine the shape of the indifference curve map for two goods, X and Y, to illustrate the case of an inferior good consequent upon price changes in either direction.
The analysis of production requires an examination of how inputs are combined to produce output. In a functional notation representation of a production function, letting broad categories of physical inputs be represented by the symbols,
L = labor
R = resources
K = capital
a generalized production function can be represented as,
Q = f ( L, R, K, ... )
where Q is the volume of output.
This equation is an incomplete specification of the production function. Technology, entrepreneurship, and managerial capacity, represented by the symbols T, E, and M, respectively, both enable and constrain the combination of the physical inputs in the production of outputs.
In recognition that T, E, and M are not physical inputs, a production function can be represented in two stages as:
Q = f ( L, R, K )
f = g ( T, E, M ).
This representation indicates the relationship between output and the physical inputs, and it signifies that the physical inputs production function is itself a function of other conditions, i.e., technology, entrepreneurship, and managerial capacity.
In the long run, all inputs are presumed to be variable. The analysis of returns to scale thus belongs to the long-run. A change of a single input, given fixed quantities of other inputs, is an analysis of the short run. It is tempting to identify capital as the input class that typically is fixed in the short run, but we must recognize that this concept is not descriptive of all real-world situations.
A short-run production perspective may be analyzed in Figure C3-2 by taking a vertical slice through the surface, parallel to either floor axis. Assuming that some quantity of capital, K1, is available from an already-constructed plant, a vertical slice is cut parallel to the labor axis, and emanating from the point K1 on the capital axis.
The problem for the production decision maker is to choose an appropriate amount of labor to employ with capital input (plant size) K1. The analysis may be conducted by assuming alternate labor-employment decisions that follow the path across the floor of the diagram from L1 through L2 and L3. Theoretically, any other quantities of labor along this path might have been chosen; these are simply a few representative quantities. The real-world production process might be characterized by a few, discrete labor-quantity choices, such as L2 or L3.
As the labor employed with capital K1 is increased from L1 toward L4, output increases along the path on the surface from Q1 to Q2, Q3, and Q4. It is apparent that over the labor input range from L1 to L2, output increases at an increasing rate (the surface is concave upward) from Q1 to Q2. Point Q2 in the surface path is near what mathematicians would call the inflection point, i.e., where a curve changes concavity, in this case from being concave upward to being concave downward.
As the labor input is further increased from L2 to L3, output continues to increase to Q3, but at a decreasing rate of increase. Further increases of the labor input from L3 to L4 yield additional output, also at a decreasing rate over the Q3 to Q4 range. It should be clear that the ΔQ3 output increment is smaller than the ΔQ2 output increment. This phenomenon of output increasing at a decreasing rate continues some beyond Q4, and until the output path peaks around Q5 and turns downward.
The labor input range from L1 to L2 is described by economists as the range of increasing returns. It is thought to be an early or temporary phenomenon in the production process, and may not be observable in most real-world production situations.
The labor input range from L2 to L5 is described by economists as the range of diminishing returns. Its essential characteristic is that output increases at a decreasing rate as the labor input increases. Its graphic illustration is the downward concavity of the production surface, and the output path formed by the vertical slice through the surface. Note that the range of diminishing returns to the variable input ends at the peak of the output path. Beyond L5 and its associated Q5 in Figure C3-2, output can be expected to decrease in absolute terms as the then-excessive quantities of labor greater than L5 are employed.
The principle of diminishing returns is thought to govern all real-world production processes. Diminishing returns may not be evident in the very early stages of production characterized by low levels of labor employment, but it becomes obvious as progressively more labor is employed. It is simply implausible to believe and unreasonable to expect that output can continue to increase at increasing or even constant rates forever as the labor input is progressively increased vis-a-vis a given plant size. This physical relationship has been adopted by economists as the fundamental behavioral premise in the explanation of short-run input-output relationships.
Theoretically, any number of different vertical sections, parallel to the labor axis, could be cut through a production surface. Each one would differ from the others by the amount of capital (i.e., the size of plant) in use. Practically, the number of discrete plant sizes that can be built is likely to be rather small. For the time being we shall continue to analyze the representative section illustrated in Figure C3-2 by extracting it from the surface and laying it out on a set of two-dimensional coordinate axes in panel (a) of Figure C3-3. Here the production function section traces out a path that economists refer to as a total product (TP) curve. This TP curve is specific to a given technology, entrepreneurial ability, managerial capacity, and plant size; variation along it is accounted for solely by variation in the labor input.
We can now trace out in panel (b) of Figure C3-3 the average product (AP) and marginal product (MP) curves that correspond to the TP curve in panel (a). The average product of labor may be defined and computed as the amount of output, Q, divided by the quantity of labor, L, employed in its production, given all other inputs, i.e.,
APi = Qi / Li,
for the ith amount of labor employed. For example, the average product of the L2 volume of labor employed is Q2/L2. The behavior of the labor AP curve may be discerned by observing the slopes of rays drawn from the origin to successive points on the TP curve. This is so because the slope of the ray drawn to a point on the TP curve is the hypotenuse of a right triangle formed by the horizontal axis and a vertical erected from the labor quantity point on the axis to the TP curve. Then, the trigonometric tangent of the angle so formed is the ratio of the opposite to the adjacent sides of the triangle, e.g., Q2/L2, which have already been defined as the average product of the L2 quantity of labor.
In panel (a) of Figure C3-3, rays to the successive points along the TP curve have progressively steeper slopes until Q3 is reached, beyond which the rays become shallower of slope. Thus we can draw the AP curve in panel (b) as rising from the origin to a peak at the L3 quantity of labor, beyond which it falls back toward the horizontal axis. The vertical axis units in panel (b) have been expanded relative to those in panel (a) so that the behavior of AP can be made quite obvious. The relationships among the computed average products for the points along the TP curve illustrated in Figure C3-3 are:
Q1/L1 < Q2/L2 < Q3/L3 > Q4/L4 > Q5/L5.
The average product of the variable input is relatively easy to measure; the only information required is the amount of output and the corresponding quantity of the input required to produce the output. Because it is easy to measure, the AP of the variable input is a tempting criterion for production decision making. However, economists usually reject it in favor of the marginal product of the variable input. The average product of labor does find usefulness in aggregate production settings where it is commonly referred to as the output per capita of the labor force.
The marginal product of labor (MPL) may be defined as the ratio of an increment of output (ΔQ) divided by the smallest possible increment of labor (ΔL), i.e.,
MPL = limit (ΔQ/ΔL), as ΔL approaches zero.
The MPL measures the rate of change of TP, and can be illustrated graphically as the slope of a tangent to the TP curve at a selected point. This concept provides the means for discerning the behavior of the MPL. Tangents have been drawn to each of the representative points on the TP curve illustrated in panel (a) of Figure C3-3. As the labor input is increased, and output consequently increased, the slopes of the tangents become progressively steeper. The maximum steepness is reached at point Q2, beyond which they become shallower until a zero slope is reached at Q5. Beyond Q5 the slopes of tangents to points on the TP curve are negative. Thus, we draw the MP of labor curve as rising from the origin to a peak at the L2 level of labor input, then falling until it is zero at the L5 level of labor input, beyond which it is below the horizontal axis.
Since the APL and the MPL curves are superimposed in panel (b) of Figure C3-3, we may note the corresponding behaviors of the two curves. Over the initial range of labor input, MPL rises much faster than does APL. For example, at Q1 the slope of the tangent is steeper than the slope of the ray to Q1. MPL remains greater than APL even after the peak of the MPL curve is reached. MPL decreases and passes through the peak of the APL curve at the labor input level L3. Here, the tangent to the TP curve at Q3 is coincidental with a ray from the origin to Q3. For all labor input levels greater than L3, MPL is both decreasing and less than APL. The slope of the tangent at Q4 is shallower than the slope of the ray drawn to Q4.
Thus far we have illustrated a vertical slice through the production surface, parallel to the labor axis, on the premises that capital is the fixed input and labor is the variable input in the short run. But we have also noted the possibility of the opposite identities of the fixed and variable inputs in a short-run setting. Any number of vertical slices may be taken through the production surface, parallel to the capital axis, thus simulating fixed quantities of labor, with variable quantities of capital. We could also repeat all of the discussion on the last several pages, but in reference to capital as the variable input. Figure C3-4, depicts a labor-constant, capital-variable vertical slice through the production surface.
Now that we have demonstrated the applicability of the production principles to either labor or capital as the variable input, the stage is set to identify the so-called relevant range of production. To this end we reproduce in Figure C3-5 the surface and vertical slice through it illustrated in Figure C3-2, but with obvious alterations. We retain the assumption that capital is the fixed input with plant K1 in place; the slice parallel to the labor axis implies that labor is the variable input in the short run.
Any point like Q2 on the production surface lies on both a TP curve for labor as variable input and a TP curve for capital as variable input. Thus we have drawn line segments cross-wise to the path of the slice at each of the selected points along the labor-variable TP curve. These cross-wise line segments represent tangents to the surface, the slopes of which measure the marginal products of capital. Even though the quantity of capital does not change, we should be able to compute for every different amount of labor employed both the MP of labor and the MP of capital.
How do the marginal products of labor and capital vary with respect to each other? Figure C3-6 illustrates a TP curve for labor as a variable input, assuming one unit of installed capital.
In this illustration, the TP curve is carried to the point where so much labor has been employed that output has fallen to zero. For purpose of illustration, we assume that the horizontal axis between the origin and the point at which TP returns to zero can be divided into ten equal labor units. In panel (b) of Figure C3-6 we represent the explicit labor-variable MP and AP curves that correspond to the TP curve, and further plot selected points along implicit MP and AP curves for capital as if it were the variable input. Along the horizontal scale we note that even though the labor input increases explicitly from zero to ten units, the capital input remains constant at one unit.
In Figure C3-6 we may examine the results of explicitly decreasing the labor input from ten toward zero units. When this happens, the quantity of capital employed per unit of labor implicitly increases from 1/10 of a unit to 1/9 of a unit, then to 1/8, 1/7, and so on. Conversely, when the quantity of labor is explicitly increased from one unit to two units, the quantity of capital per unit of labor is implicitly decreased from 1/1 unit to 1/2 unit. Recognition that the relative quantity of each fixed input does implicitly change consequent upon an explicit change of a variable input is essential to an understanding of how the marginal products change vis-a-vis each other.
As may be seen in panel (b) of Figure C3-6, over the initial range of labor input, the MP of labor increases to a peak, then begins to decrease; but the MP of capital is negative. Economists identify this range as the Stage I of production for labor, but the Stage III of production for capital. Likewise, moving from right-to-left as the quantity of capital is implicitly increased, the MP of capital rises to a peak and begins to decrease; but the MP of labor is negative. Economists identify this range as the Stage I of production for capital, but the Stage III of production for labor. A rational and perceptive production decision maker would not choose to operate in either Stage I or Stage III. In labor's Stage I (capital's Stage III), labor is underutilized while capital is overutilized (implied by the negative marginal productivity of capital). In labor's Stage III (capital's Stage I), labor is overutilized (implied by the negative marginal productivity of capital) while capital is underutilized.
We may thus identify Stage II (common to both inputs) as the relevant range of production. Within Stage II the marginal products of both inputs are positive, although they vary in opposite directions. The boundaries for Stage II for each input are found at the point of zero MP for the other input, which coincidentally correspond to the intersections of the MP and AP curves for each of the inputs.
Some very important behavioral relationships may now be noted. Within production Stage II, as either input is increased in quantity, its MP decreases, but the MP of the other input (which explicitly does not vary) increases. Conversely, if the quantity of either input is decreased, its MP will increase, but the MP of its complement(s) may be expected to decrease. Knowledge of this relationship enables us to answer a very important question for the production manager: how can the (marginal) productivity of labor be increased? Two answers are possible: either by decreasing the labor input, or by providing labor with more capital equipment. The former response is typically regarded as a short-run adjustment, the latter as a long-run change.
Where within Stage II should the production manager choose to operate? It is tempting to pick the intersection of the two MP curves, but this would be the appropriate point only in a very special circumstance. To answer the question, we need two additional pieces of information, the prices per unit of the two inputs. Then, whatever happens to be the ratio of the unit prices of the two inputs, it follows logically that a corresponding ratio between their marginal products would be warranted, i.e.,
MPL/MPK = PL/PK.
An alternate version of this relationship may be expressed as:
MPL/PL = MPK/PK.
This very important relationship is known as the Equamarginal Principle. Stated as an operational decision criterion, the optimal combination of labor and capital is the one for which the marginal product per dollar's worth of labor is equal to the marginal product per dollar's worth of capital. But this is really not a decision criterion; rather, it is an equilibrium condition. The practical decision criterion might better be stated:
MPL/PL >=< MPK/PK.
Following this decision criterion, if the MP per dollar's worth of capital is greater than the MP per dollar's worth of labor, the production decision maker should employ more capital and/or less labor. By so doing, the MP of capital will fall and the MP of labor will rise, thereby tending to bring about an equality of the marginal product per dollar's worth of the two inputs. We leave it to the reader to explore alternate possibilities. In Figure C3-6, if the price of a unit of capital is twice the price of a unit of labor, then approximately 5.2 units of labor should be employed to operate the one unit of capital since at 5.2 units of labor the MP of capital is approximately twice the MP
Figure C3A-1 illustrates a production surface where a plane, abcd, parallel to the floor at quantity altitude Q4, has been passed through the production surface tracing out the horizontal slice, efgh. The path, e'f'g'h', in the floor of the surface is the vertical projection of the slice through the surface. Slice efgh is only one of an infinite number of slices that could be taken through the surface at different quantity altitudes.
Figure C3A-2 shows a two-dimensional view from a perspective above the surface illustrated in Figure C3A-1. In effect, the three-dimensional surface appears to be collapsed into the floor when viewed from above, but the path e'f'g'h' represents the slice through the surface at quantity altitude Q4. Economists refer to the e'f'g'h' path as an isoquant (meaning same quantity) because it represents a sequence of points, the (K,L) coordinates of which are the input combinations that can produce the Q4 level of output. For example, output Q4 can be produced with the L1 quantity of labor employed with the K1 quantity of capital. Output Q4 can also be produced with less capital, K2, if a larger labor quantity, L2, is utilized. This suggests that labor and capital are to some extent substitutable for one another. Since path
e'f'g'h' is drawn as a continuous curve, the implication is that there are an infinite number (or as many as there are points along the path) of combinations of K and L that can produce the Q4 output. In reality, technological considerations may limit the effective number of such combinations to a relatively small number.
Suppose that instead of selecting a (K,L) com-bination represented by a point along the isoquant the production decision maker chooses to use the K2 quantity of capital with the smaller L2 quantity of labor, reaching point j on the production surface in Figure C3A-1, and j' in Figure C3A-2. It should be apparent that the smaller quantity of labor working with less capital will produce some output smaller than Q4.
Figure C3A-3 is an elaboration of Figure C3A-2 to show the paths of several other "representative" isoquants that could be generated by horizontally slicing the surface at altitudes other than Q4. It is now apparent that the L1 quantity of labor using the K2 capital input can produce only the Q3 output, which is less than Q4. Theoretically, any number of such isoquants could be generated by slicing the surface at different altitudes so that the floor of the surface might appear "dense" with concentric isoquants. The collection of representative isoquants may be referred to as an isoquant map. The isoquant map may be likened to the collection of isotemp or isobar lines on a weather map, or to the contour lines on a geological or military map.
Suppose that the horizontal slices taken through the surface to generate the isoquant map illustrated in Figure C3A-2 were taken at successively higher quantity altitudes that are equal quantity increments apart. In this case, (Q2- Q1) would be equal to (Q3- Q2), and so on. When viewed from above the surface in Figure C3A-1, the isoquant map illustrated in Figure C3A-2 betrays the likely shape of the surface that cannot be seen explicitly in Figure C3A-2. From isoquant Q1 up to isoquant Q4 the isoquants appear to be getting closer together, implying that the surface is increasing at an increasing rate in the quantity dimension. From isoquant Q4 through isoquant Q7 the isoquants appear to become farther apart, suggesting that output is increasing at a decreasing rate.
The isoquant map provides information about production possibilities, but by itself cannot provide the production decision maker with a criterion for selecting an optimal input combination. Additional information is needed in the firm's production budget and the prices of the two inputs, K and L. Suppose that the prices of Labor and Capital are PL and PK, and the firm's management has determined that the production cost outlay must be limited to the dollar sum C. This information can be brought together in the following equation:
L*PL + K*PK <= C.
The sense of this equation is that the product of the number of units of labor utilized (L) times its price (PL) plus the number of units of capital consumed (K) times its price (PK) cannot exceed the budgeted outlay (C). The firm could underexpend its budget, but suppose that the management has decided to commit the entire budget to the purchase of quantities of labor and capital so that the inequality symbol may be ignored. What quantities of each input should the firm employ?
The equation can be rearranged in the following format:
K = C/PK - PL/PK . L.
This equation is linear and in the so-called "slope-intercept" format, y = a + bx, where the slope of the line that represents the equation is the ratio of the price of labor (PL) to the price of capital (PK). It should be noted that this ratio is negative. The conceptual interpretation of this slope is the rate at which the firm can substitute labor for capital while remaining within the budget. The vertical axis intercept is the ratio of the budgeted outlay (C) to the price of capital (PK), which determines the maximum amount of capital that could be purchased by the firm if it purchased no labor at all. The equation of the budget line can be plotted on the same set of coordinate axes containing the isoquant map. We may suppose that the values of C, PL, and PK are such that the budget line is plotted as in Figure C3A-4.
At what point along the budget line should the firm choose to employ a combination of labor and capital? As the reader has likely already guessed, the relevant decision criterion is a comparison of the slopes of the isoquant and the budget line, i.e., whether
MRTS >=< PL/PK.
Point m' is one option involving a large capital input used with a small amount of labor. At point m' the slope of the isoquant, though negative, is very steep, implying that if the firm were to give up the (K3- K4) quantity of capital, it could purchase the (L4- L3) quantity of labor while remaining within its budget. This additional quantity of labor is a great deal more than the firm would have to have to remain at the Q2 level of output. An exchange of the (K3- K4) quantity of capital for the (L4- L3) quantity of labor would enable the firm to move to a higher level of output on isoquant Q3 at point n'. This is obviously a good move which the firm should undertake.
Suppose that the production decision maker had first determined to try the input combination represented by point s'. At this point the firm employs a relatively large quantity of labor with substantially less capital. The MRTS at point s' is negative, but quite small (i.e., the slope of the isoquant is shallow). The production decision maker is considering replacing the (L5- L6) quantity of labor with additional capital. Only a small amount of additional capital will be needed to allow the firm to remain at the same level of output, Q3. However, by giving up the (L5- L4) quantity of labor the firm can still remain within its budget by purchasing the (K6- K5) quantity of capital, which will enable it to increase output almost to the Q4 level. This is a good move that the firm should undertake.
As may be readily deduced, the production manager will find an output-increasing incentive to reallocate labor and capital by moving downward along the budget line from points such a m' and n' until the incentive disappears. Likewise, a similar incentive will be found to move upward along the budget line from points such as r' and s' until the incentive is eliminated.
When does the input-reallocation incentive expire? This occurs when a point is reached along the budget line where the slope of the budget line is just equal to the slope of the isoquant. At such a point, e.g., f' in Figure C3A-4, the conceptual interpretation is that the rate at which the firm can substitute labor for capital while remaining at the same level of output (the MRTS) is just equal to the rate at which the firm can substitute labor for capital while remaining within its capital outlay budget (PK/PL). This is the point of tangency of the budget line with an isoquant. It is also the highest output-level isoquant that the budget line can reach. At this point the production manager will have found the output-maximizing combination of labor and capital, given the limitation of the budgeted capital outlay. It is then also the cost-per-unit minimizing combination of labor and capital that can be accommodated by the capital-outlay budget.
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Cost minimization occurs at the point representing the input combination that meets the criterion,
MRTS = PL/PK.
Since MRTS also measures the ratio of MPL to MPK, this equation can also be expressed as
MPL/MPK = PL/PK.
which by algebraic rearrangement can be expressed as
MPL/PL = MPK/PK.
This is the same Equamarginal Principle that resulted in the optimal resource allocation criterion via the approach of horizontal slices through the production surface.
The stages of production can be identified on an isoquant map by specifying two ridge lines. One ridge line follows a path formed by the points on the isoquants where they are vertical, the path r1r in Figure C3A-5.
At any point below r1r an increase of the capital input will increase output because a higher isoquant will be reached. This implies that the MPK is positive. However, at any point above r1r an increase in the capital input will decrease output (lower quantity altitude horizontal slices will be reached), implying a negative MPK. The region above r1r is Stage III for capital, while that below r1r is capital's Stage II.
Any point to the left of r2r the MPL is positive, while at any point to the right of r2r the MPL is negative. This identifies Stages II and III for labor. Capital's Stage III is coincident with labor's Stage I, and labor's Stage III is coincident with capital's Stage I. The MRTS at any point along an isoquant in Stage II will be negative, and the marginal productivities of both inputs in Stage II will be positive. Outside of Stage II, the MRTS will be a positive slope, and the marginal productivity of one or the other of the inputs will be negative.
In the short run, some (at least one) inputs may not be changed. In the long run, all inputs are presumed to be variable. The analysis of returns to scale thus belongs to the long-run.
Returns to scale can be analyzed by cutting a vertical slice through a production surface, but along a linear path emanating from the origin at some angle with the labor and capital axes. The angle with respect to either axis is the 90-degree complement of the angle with respect to the other axis. In Figure C4-1 the path to be followed on the floor of the production surface (a ray from the origin) is 0ABCD. The corresponding path traced out on the production surface is 0Q1Q2Q3Q4.
The angle chosen for the ray from the origin implies a certain capital-labor ratio which remains fixed as long as the line input path is followed, that is,
K1/L1 = K2/L2 = K3/L3 = K4/L4.
If the firm has been producing Q1 by employing the L1 quantity of labor in a K1 size plant, the firm can increase its scale of operations by increasing both inputs (and all others not explicitly included in the three dimensional diagram) by the same proportion. For example, it might increase both the plant size and the employed labor force by the same proportion to reach point B where it will employ the L2 labor force in the K2 plant which will enable it to produce the Q2 output. Similar same-proportion increases of both inputs would enable it to reach points C and D.
Consequent upon the shift from point A to point B in the floor of the diagram (a less than doubling of both inputs), output increases to Q2, which is more than double the output at Q1. We may describe the A to B input change as an output range of increasing returns to scale. The B to C input shift, an increase of about 50 percent for both inputs, resulted in an output increase from Q2 to Q3, only about 25 percent. We can describe the B to C input change as an output range of decreasing returns to scale.
For any particular change of all inputs in a production process by the same proportion, if the output change is in the same direction and
a. of smaller proportion than the input change, this is a case of decreasing returns to scale.
b. of the same proportion as the input change, this is a case of constant returns to scale.
c. of larger proportion than the input change, this is a case of increasing returns to scale.
In the transition from the analysis of production to the analysis of costs, we note that the concavities of the total curves are reversed. In the production context, as the labor input is increased, output may initially increase at an increasing rate; but corresponding to this Stage I phenomenon, variable costs tend to increase at a decreasing rate. In the production context, beyond some point further increases of the variable input resulted in output increasing at a decreasing rate, i.e., the phenomenon of diminishing returns in Stage II. And corresponding to this Stage II phenomenon, costs will increase at an increasing rate. Economists refer to this phenomenon as a revelation of the law of increasing cost. The law of increasing cost is the cost analysis variant of the production principle of diminishing returns.
Diminishing returns and increasing costs can also be illustrated with marginal and average cost functions derived from a total variable cost (TVC) function. The average variable cost (AVC) curve, illustrated in panel (b) of Figure C5-4, may be derived from the TVC curve in the same fashion that the AP curve is derived from the TP curve.
Specifically, AVC at any level of output, Q1, may be measured as
AVC = TVC/Q
The ratio of TVC/Q may be measured graphically as the slope of a ray from the origin to the TVC curve at the selected Q. For successively larger outputs, it can be seen in panel (a) of Figure C5-4 that the rays from the origin at first decrease in slope, reach a minimum at Q3, and then increase in slope as output increases beyond Q2.
Correspondingly, the AVC curve illustrated in panel (b) of Figure C5-4 decreases to a minimum at Q3, and then increases beyond Q3. The increase of AVC beyond Q3 is attributable to the principle of diminishing returns, and is illustrative of the law of increasing costs.
Likewise, the marginal cost curve, MC, illustrated in panel (b) of Figure C5-4, may be derived from the TVC curve in the same fashion that the MP curve was derived from the TP curve. Marginal cost may be computed as
MC = ΔTVC / ΔQ
in the limit as ΔQ approaches zero. This may be illustrated graphically by the slopes of tangents to successive points along the TVC curve.
The average variable cost is easiest to compute (only two pieces of information are needed, the current total variable cost and the quantity being produced), and for this reason it is tempting to try to base output decisions upon it. Indeed it can be used as an output decision criterion if the goal of the enterprise is to minimize per-unit variable costs. But the circumstances of cost minimization, revenue maximization, and profit maximization are unlikely to coincide.
If the goal of the enterprise's management is profit maximization, then AVC is an inadequate criterion. The appropriate cost-related decision criterion for profit maximization is marginal cost.
We may now state unique relationships among cost curves:
a. Over the output range for which TVC is increasing at a decreasing rate, both AVC and MC decreases, but MC is less than AVC.
b. MC reaches its minimum point at the Q for which TVC reaches its inflection point; at its minimum, MC is less than AVC.
c. AVC reaches its minimum point at the Q for which a ray from the origin to the TVC is of minimum slope (point C in panel (a) of Figure C5-4). Coincidentally, the ray from the origin to this point is a tangent to the TVC, so MC and AVC are equal at this output level (Q3 in Figure C5-4). MC is less than AVC up to this point.
d. For all output levels beyond the minimum of the AVC, both AVC and MC increase, with MC rising at a faster rate (i.e., MC lies above AVC).
e. Neither AVC nor MC is ever negative.
We can reach some general conclusions on how costs are affected when the long-run change is to adjust the scale of the enterprise's operation by constructing a larger or smaller plant. A larger plant than K1in Figure C6-1 results in another slice through production function surface at, say K2. The total product curve drawn from K2 likely rises higher into the output dimension, and may reach its peak at a larger volume of labor utilization than does the K1 total product curve. Any number of successively larger plants could be represented by cutting total product slices through the production surface at quantities of capital greater than K2.
The total cost curves associated with the K2 plant likely lie farther to the right than those associated with plant K1, as illustrated by TC2 relative to TC1 in Figure C6-2. Correspondingly, the average total cost curve for plant K2, illustrated by ATC2 in Figure C6-2, likely lie slightly to the right of the position of ATC1 for plan K1, and ATC2 may reach minimum at a lower per-unit cost level than does ATC1.
Figure C6-3 illustrates an idealized representative set of such ATC curves which might result from building a sequence of ever larger plants, given the same technology and managerial capacity. This is an admittedly heroic representation since most firms would have the occasion to build only one plant for any particular technology. However, if we may indulge in the presumption that the management might at least contemplate the construction of any of a range of possible plant sizes, we may note that the successively larger plants initially achieve ever higher and wider output ranges accompanied by falling per-unit costs. This occurs up to point A in Figure C6-3.
Economists describe this phenomenon as the range of economies of scale and note that the decreasing costs correspond to the phenomenon of increasing returns to scale. Increasing returns to scale, with the consequent scale economies, are attributable to greater division and specialization in the use of labor, or (and perhaps to say the same thing in other terms) more efficient use of plant and equipment.
Beyond point A in Figure C6-3 the ATC curves, though they reach higher output ranges, incur ever increasing per-unit costs. Economists describe this as the range of diseconomies of scale which are associated with the phenomenon of decreasing returns to scale. Scale diseconomies are most often attributed to limitations on the abilities of management to coordinate and control the ever more complex productive relationships involved in the larger-scale operations.
The curve in Figure C6-3 which is tangent to the sequence of ATC curves is a long run average total cost curve, LATC. It is also referred to as an "envelope" curve because it envelopes the sequence of ATC curves.
Each of the ATC curves represents the cost conditions for a plant of particular capacity which the enterprise may construct. The LATC curve, however, represents no single such entity, but rather is a sequence of points each on a different ATC curve. Each of these points represents the single optimal plant size, rate of production, and cost level to meet a particular level of demand. The LATC curve may therefore be regarded as a long-run "planning horizon" concept which may assist the production manager in selecting the appropriate plant size, given the existing or anticipated level of demand for the product.
The revenue and cost curves for a firm in a purely-competitive market are illustrated in panel (a) of Figure D1-1. Market prices will adjust toward an equilibrium at price P1 and quantity Q1. Because all firms in the market are similar to one another and significant differences are unlikely to emerge, we can analyze the behavior of a "representative firm" in panels (b) and (c).
The locus of the total cost curve is not shown in panel (c) or the average total cost curve in panel (b) of Figure D1-1 because fixed costs are not relevant to short-run decision making. All that the manager should attempt to discern in the short run is whether, after the fact of the decision, the operating costs are covered. An appropriate long-run consideration is whether revenues also cover the overhead costs and make a contribution to profit, hence the concept of a "contribution margin."
No single firm in the competitive market will find incentive to charge any price but the market price, P1; at any higher price no one will buy from the firm; the firm can sell all that it can produce at the market price. Therefore, the firm is what we call a "price taker;" it exercises no discretion concerning price except to get into line with the market price when it changes. For this reason, the representative firm's demand curve is drawn as a horizontal line at price P1. When demand is linear and horizontal, total revenue, TR, is a straight line emanating from the origin of its axes, and the marginal revenue curve, MR, is coincident with the demand curve.
Output quantities q1 and q2 are variable-cost break-even levels. At outputs smaller than q1 or larger than q2, total variable cost is greater than total revenue; likewise average variable cost exceeds average revenue. At any output between q1 and q2 the firm covers all operating costs and makes some contribution to covering overhead costs and profit.
Output q3 is one of the output levels between q1 and q2 for which revenue exceeds variable costs, TR > TVC, and makes a contribution to overhead costs and profit. At q3, a small increase of output will increase total revenue by more than it will increase total cost. We can discern this relationship graphically in panel (c) because the slope of the TVC curve is shallower than is the slope of the TR curve at output q3. This relationship can be verified in panel (b) of Figure D1-1 where at q3 the marginal revenue curve (coincident with the demand curve) lies above the marginal cost curve. If the decision maker's goal is profit maximization, it is clear that an increase of output from q3 will add more to total revenue than to total cost, and thereby either increase profit or diminish loss. The appropriate decision criterion is a comparison of marginal revenue and marginal cost. If marginal revenue is greater than marginal cost, MR > MC, then output should be increased.
Output q4 is also in the range of q1 to q3, but at q4 marginal cost is greater than marginal revenue. An increase of output from q4 will have the effect of adding more to total cost than to total revenue, thereby diminishing profit or increasing loss. A decrease of output will decrease both total revenue and total cost, but total cost will decrease by more than total cost decreases, thereby decreasing loss or increasing profit. Again, if the goal of the manager is to maximize profit, the firm should decrease output when marginal cost is greater than marginal revenue, i.e., MR < MC.
We could examine other output levels between q3 and q6 to draw inferences similar to those drawn for q3. Likewise, we could examine other output quantities between q4 and q6 in order to draw inferences similar to those drawn for q4. The general principle which should govern output decision making when the goal is to maximize profit is to increase output if marginal revenue is greater than marginal cost, MR > MC, but to decrease output if marginal cost is greater than marginal revenue, MR < MC.
Marginal revenue is easy to identify in the purely competitive market because it is equal to price, and the MR curve is coincident with the demand curve. Marginal cost is not directly observable, but it can be computed for any output level if the equation of the TC or MC curve is known. However, knowing either of these equations probably requires the exercise of a costly data collection and model specification and estimation process.
It is tempting then to look at average revenue and average cost as possible decision criteria because it is so easy to compute both at any level of output. Average revenue at any output level such as q2 can be computed as TR2/q2, but in the purely competitive market, average revenue is also equal to the price of the product. Average variable cost is easily computed as TVC2/q2 at output level, q2. If the magnitudes of AR2 and AVC2 are so easy and cheap to compute, what would be the problem in using them as the output decision criteria?
It should be apparent in panel (b) of Figure D1-1 that the profit-maximizing output level cannot be found simply by comparing AR and AVC unless all possible outputs between q1 and q2 are examined. Another way to say this is that the comparison of AR and AVC can reveal whether or not the firm is covering its operating costs, but can yield no guidance about whether to increase or decrease output. The comparison between AR and AVC can serve as a useful decision criterion if the goal of the management of the firm is to minimize per-unit costs (this occurs at output level q5 in Figure D1-1), or if the goal is simply to operate at any output level for which revenues cover operating costs. The moral of this story is that MR and MC are the appropriate decision criteria if the goal of the firm is to maximize profit
The firms should shut-down operations if the revenue generated by selling the output cannot cover even the operating costs (or variable costs) of producing the output. The shut-down criterion can be illustrated in panel (b) of Figure D1-1 at any output for which AR < AVC, or in panel (c) at any output for which TR < TVC. Graphically, TR would lie completely below TVC, and P=AR=MR would be below MC at all output levels. In these circumstances, the firm should not operate because the revenue resulting from operation would not cover all of the operating costs, and could make no contribution at all to the overhead costs. In shut-down mode, the firm minimizes its losses by incurring only the fixed costs. The fixed costs, which continue in the short run whether the firm operates or not, can be saved (or avoided) only by exiting the industry, a long-run decision.
Exogenous phenomena that are not under the discretionary control of the firm's management may impinge upon the firm in the short run. For example, a change in the price of an input will have the effect of shifting the firm's short-run cost curves upward or downward and may thereby create a shut-down or restart situation. Such a short-run change may create the need or basis for a long-run decision to exit the industry if the problem cannot be satisfactorily remedied by short-run adjustment. In the long run, revenues must cover all costs of production because there are no fixed costs in the long run. Figure D1-2 includes the TC curve in panel (c) and the ATC curve in panel (b). TC is greater than TVC by the amount of the total fixed costs, and ATC is greater than AVC by the average fixed cost at each level of output. The long-run exit criterion applies to any output in panel (b) of Figure D1-2 for which AR < ATC, which also corresponds in panel (c) to any output for which TR < TC.
Under the circumstances illustrated by P1 and TR1 in Figure D1-2, firms presently operating in the market can cover all of their variable and fixed costs and enjoy a profit at the current market price. When these profits are perceived by outsiders, and if these profits are greater than can be earned in other markets, the outsiders may exercise their entrepreneurship to enter the market and try to share in the supernormal profits. This entry into the market will have the effect of increasing market supply, shifting it to the right to some position like S2 in panel (a) of Figure D1-2. As a result, market price will fall toward P2, which will become the new locus of the demand and marginal revenue curves as well. Correspondingly, the TR curve will rotate downward to its new position, TR2. As consequence, the firm's profit-maximizing output level will change to q7, and the profit earned by the firm will be smaller.
Theoretically, this adjustment process, driven by continuing entry into the market, could continue until the price falls to P2 and the total revenue curve rotates to position TR2; here, supernormal profits are eliminated and the market price just covers the firm's variable and fixed costs, allowing only a normal return to the firm's ownership interest. With no effective way for firms to prevent entry into the market, all super-normal profits will be competed away. But no firm in the market will be suffering because each will be paying or earning normal returns for all of the resources under its employ. Capital and entrepreneurship, having entered the market, will continue in their present occupation until the prospect of supernormal profits appears elsewhere.
Perfect market knowledge is never a fact in the purely competitive market. Suppose that there is a time lag in the dissemination of information so that potential entrants into the market fail to realize that enough new firms have already entered the market to cause price to fall to the level of ATC. Continued entry by new producers who are yet unaware that supernormal profits are no longer possible in the market will cause price to fall to below the minimum point of ATC, and the total revenue curve to rotate downward to TR3 which lies below TC. Now firms in the market are realizing insufficient revenue to cover all of their costs. At price P2, all of the direct or variable costs are covered, and a contribution is being made toward the overhead or fixed costs, so the firm should keep on operating in the short run. However, if due to excessive entry into the market price should happen to fall below the minimum point of AVC, the firm should then shut down and suffer only the fixed costs.
If a short-run loss situation persists, the management of the firm should give thought to making the long-run decision to exit the market. The managements of some firms will decide to do just this; their exit from the market will decrease market supply, causing market price to rise toward the level where all costs, direct and overhead, again can be covered, and the firms remaining in the market once again can earn normal profits. The competitive market is in equilibrium when all firms still in the market are realizing normal profits, and there are no supernormal profits to tempt entry or subnormal profits to induce exit.
The management decision options of the long-run extend to
(a) entering or exiting a market;
(b) adding to the firm's plant or equipment, disposing of capital assets, or letting them depreciate to non-functionality without replacement;
(c) building a larger or smaller plant of the same technology;
(d) choosing a different technology, and consequently an appropriate plant size to implement that technology; technologies new to the firm may be acquired externally by licensing, purchasing new equipment, or hiring new personnel, or they may be developed internally through research and development (R&D) efforts;
(e) gaining of managerial capacity internally by experience and training or externally by employment of new personnel; or losing managerial capacity through retirement, death, or movement of personnel to other firms; or
(f) gaining or losing entrepreneurial ability.
An alteration of any of the environmental parameters (technology, management, or entrepreneurship) means that the firm's production and cost functions, short and long-run, have shifted and should redefined by intuitive or empirical estimation means. How the functions shift, and whether they exhibit warps or twists relative to previous attitudes, depend entirely upon the particular change.
A change of plant size within the same technology is amenable to analysis using the long-run average total cost curve. Such a plant-size change may have scale economy or diseconomy implications if the LATC curve is U-shaped. If the management can discover enough about the shape of its LATC curve, it may devise a long-run (capital investment) strategy to change plant size in anticipation of exploiting scale economies or avoiding scale diseconomies. Also, as market demand changes, the managements which know enough about their costs in the long run may seek to construct plants of more optimal capacity to meet market demands.
Suppose, for whatever historical reason, the firm has constructed the plant represented by ATC1 and MC1 illustrated in Figure D1-3.
At price P1 the firm maximizes profits by producing the q1 output. But we there is a more optimal plant for producing the q1 output. This plant is ATC2, the one for which average total cost curve is tangent to the LATC curve at q1. The profit-maximizing output for plant ATC2 is not q1, but rather q2 where MC2 crosses the MR curve. However, there is an even lower-cost plant which could be constructed to produce the q2 output level, ATC3, but the profit maximizing output level for ATC3 is q3. Points a, b, and c are on the respective marginal cost curves and lie directly below the points of tangency of the respective ATC curves with the LATC curve. These points trace out a path which can be construed as a long-run marginal cost curve, LMC.
Unless the firm's managers have perfect knowledge of the firm's long-run cost conditions (i.e., they know the loci of all of the possible short-run cost curves), it will be through a process of discovery that the firm will eventually be led to build the ATCm plant and operate it at the qm level of output where the firm will maximize profits. Given market price P1, plant ATCm is the plant which will yield the largest mass of profit for the firm. At output level qm, the short-run ATCm curve is tangent to the LATC curve; there is no other plant which is more appropriate for producing the qm output. Theoretically, this plant can be found at the intersection of the LMC curve with the MR curve. The appropriate plant then is the one whose MC curve crosses MR at the intersection of the LMC curve with the MR curve.
Let us suppose for a moment that the manager of the competitive firm does have perfect knowledge of its production and cost functions, but completely lacks foresight of likely future market changes. Given price P1 and assuming that it will persist forever, the manager makes the long-run decision to build plant ATCm, intending to operate it at qm level of output. Managers of many other competitive firms do likewise. This phenomenon, plus the effect of entry of new firms into the market to try to capture a share of the super-normal profits can be expected to bid the market price down toward Pe in Figure D1-4, for which demand curve De and marginal revenue curve MRe happen to be tangent to the minimum point of the LATC curve. When prices reaches Pe, the appropriate plant size is described by ATCe, which is tangent to the LATC curve at its minimum point. And the manager who opted for plant ATCm when price was P1 can now be seen to have been both naive and short-sighted because plant ATCm incurs unit costs (even at its minimum point) which are greater than market price Pe.
Managers who committed to plants like ATCm or ATC1 may find their firms realizing losses relative to firms whose managers more astutely chose to build a least-cost plant. They are stuck with plants which are too small or too large for the duration of the lives of the plant and equipment. Their short-run options are to continue to operate while suffering losses if Pe happens to exceed their average variable cost, or to shut-down if Pe is less than their average variable cost. If price remains as low as Pe, the firms which are realizing losses have the option of exiting the market. Exit from the market will likely involve capital losses if the plant and equipment cannot be economically converted to other applications, or if they must be sold at prices below their depreciated book values.
In reality, managers of competitive firms have neither perfect foresight of market conditions nor perfect knowledge of their production and cost functions. Collectively they will be led through processes of discovery to the construction of optimal plants (ATCe in Figure D1-4) as market price falls toward Pe. Some will build suboptimal plants and others will build superoptimal plants. It is these who will prematurely exit the market. The ones who survive and remain in the market are those who through exceptional knowledge of their own long-run cost possibilities, astute understandings of future market dynamics, or sheer good luck, are induced to build the right-sized plants and operate them at the right levels of output. Competition smiles upon those who learn enough about their costs and are knowledgeable of market dynamics. Competition ruthlessly dismisses those who are ignorant of their costs or naive of market realities.
We have examined decision-making criteria from the perspective of the manager, but we should note that from the perspective of society as a whole, plant ATCe operated at output level qe when price is pe is also a socially-optimum condition. Competitive firms are induced to build least-cost plants and operate them at least-cost output levels while covering all economic costs but realizing no supernormal profits. The market price of a good can be interpreted as society's valuation of the good; the marginal cost of producing a good can be interpreted as the cost to society of using the resources to produce the good rather than other goods. Because in competitive equilibrium the firm's marginal cost of producing the output is just equal to market price, it may be said that the social cost of using resources to produce the output is just equal to the social valuation of the output produced. From this may be drawn the inference that the resources used in the production of the output are efficiently allocated. Enough, but not too
much, of the output is being produced.
Figure D2-1 illustrates the short-run revenue and cost curves for a pure monopolist. The market demand and supply curves illustrated in a pure competition graph are missing from Figure D2-1. They are in fact present in Figure D2-1, but a separate set of coordinate axes is not needed to house them because the monopolist's demand curve is the market demand curve when there is only one seller in the market. The market supply curve consists of one of the monopolist's short-run cost curves, depending upon the monopolist's behavioral objective.
The monopolist's total revenue curve, TRm, is drawn as a parabola opening downward, and its demand curve (which is also its average revenue curve), Dm, slopes downward from left to right. The marginal revenue curve, MRm, as was noted in Chapter C1, diverges from the demand curve and lies below it when the total revenue curve is a parabola.
The practical reason that D slopes downward is that the monopolist, in order to increase sales, finds it necessary to lower the product price (or do something to cause the demand curve to shift to the right). The conceptual reason that marginal revenue is always less than average revenue (i.e., MR lies below D) is that the lowered price applies to all units which are sold, not just to the additional unit sales resulting from the price decrease. Since marginal revenue is the addition to total revenue consequent upon selling more units, marginal revenue decreases because total revenue increases at a decreasing rate. This can also be stated in the opposite direction: when TR increases at a decreasing rate, MR decreases. Because marginal revenue eventually reaches zero and negative values, total revenue correspondingly reaches a peak and becomes negatively sloped beyond the peak.
The cost curves illustrated for the pure monopolist in Figure D2-1 have essentially the same appearances as those illustrated for the pure competitor. Short-run cost relationships are not the real differences between the pure competitor and the pure monopolist. In a local market, or in the case of a newly-invented product, the pure monopolist could be physically as small a firm as we assume a pure competitor to be, in which case the cost curves could be identical. We usually imagine the monopolist to be a gigantic firm, having gotten so by growth and exploiting economies of scale. In this case, the quantity units scale on the horizontal axis may cover a much larger range than would be expected of the small-firm pure competitor, but we can still expect the short-run cost curves to be of similar shape, only to spread horizontally across the larger range.
The thought process which the pure monopolist must use in selecting the right output level is virtually identical to that of the pure competitor. The same decision criteria are pertinent to both pure competition and pure monopoly. Output should be increased if marginal revenue exceeds marginal cost because more will be added to total revenue than to total cost, thereby increasing profit (or diminishing loss if the firm is operating unprofitably). Such is the case at output level Q1 in Figure D2-1. If marginal cost is greater than marginal revenue, as at output level Q2, output should be decreased; the resulting decrease in total cost will be less than the decrease in total revenue, so that profit will be increased (or loss diminished).
The major difference between the pure monopolist and the pure competitor lies in the fact that the pure monopolist has at least two realms of decision discretion whereas the pure competitor has only one. The pure competitor is a price taker; it has no alternative but to accept market price as a given. The pure monopolist not only can alter market price if it wishes; it must change market price if it wishes to change the quantity which it sells. For example, suppose that it has been selling the Q1 quantity at price P1 (point A in Figure D2-1). The manager realizes that since marginal revenue is greater than marginal cost output should be increased to Q2. If the manager now does not also lower price to P2, a larger quantity will be produced than can be sold at the unchanged price, resulting in an inventory accumulation of Q2-Q1, or the line segment AC in Figure D2-1. It would certainly be nice to be able to sell the larger quantity, Q2, at the unchanged price, P1, but point C does not happen to be on the firm's demand curve. In fact, to increase output without correspondingly decreasing price will result in increased total costs but no increase in total revenue. The impact on profit would be precisely the opposite of the desired effect.
To accomplish the desired objective of increasing profits, the monopolist must move along the demand curve to some point like B. Such a movement is a "change of quantity demanded," which any student who has had a course in principles of economics knows to be caused by a change of price. In order to get to a point like C, the monopolist would have to effect a "change of demand," i.e., to shift the demand curve to the right. This may indeed be possible by mounting an effective promotional effort.
Even if the monopolist does not have perfect knowledge of the shapes and loci of his revenue and cost curves (management almost never will), an iterative process (trial and error) employing as decision criteria the comparison between marginal revenue and marginal cost can lead the monopolist toward the profit maximizing price and output levels, P3 and Q3 in Figure D2-1. Output level Q3 is that for which marginal revenue is equal to marginal cost. Given the locus of the demand curve at Dm, there is no other quantity sold at any alternative price which can yield any more profit than can the Q3 output sold at the P3 price.
In addition to Q3, there are other output levels which are also notable:
Q4, the output which maximizes total revenue;
Q5, the output which minimizes average variable cost;
Q6, the output which minimizes average total cost
Q7, the upper break-even output level
Q8, the maximum output at which all variable costs are covered; and
Q9, the lower break-even output level.
Each of these is notable because it could be a decision objective of the firm in lieu of profit maximization. For example, suppose that the objective of the firm is growth rather than profit. Output Q7 is the largest quantity that can be produced and sold without incurring loss. Output Q8 is the largest output that can be produced and sold while covering all variable costs, but sustaining a maximum loss equal to the average fixed costs. Output Q9, the lower break-even output, could be an objective if the firm is attempting to present a low profile to the antitrust authorities: it is the smallest output which can be produced and sold without incurring a loss.
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The monopolist's long-run average total cost curve is more likely to be flat-bottomed or to slope downward from left to right as depicted in Figure C6-4, rather than to have the classic U-shape depicted in Figure C6-3. The monopolist's demand and marginal revenue curves are downward-sloping as depicted by curves D1 and MR1 in Figure D2-3. If market demand, D1, is still small relative to scale economy possibilities, the monopolist should select plant size ATC1, for which its marginal cost curve crosses the long-run marginal cost curve at the latter's intersection with MR1.
Although it may be heroic for the monopolist to know so much about its revenue and cost possibilities as to be able to find such an intersection, there is no more efficient plant for meeting the D1 demand than the ATC1 plant. However, selection of plant size ATC1 could be short-sighted if there is prospect for market demand to shift to the right from position D1. It might do so if the economy is growing and the income elasticity of demand for the product is positive. The monopolist may also promote its rightward shift by engaging in effective marketing activity. If either of these prospects is significant, the monopolist might be wise to build a plant to meet the demand forecasted for some number of years into the future, even though it may yield suboptimal profits or even losses at present demand levels.
The right-ward shift of demand through time may enable the monopolist to exploit potential scale economies if LATC slopes downward. By the time that demand has shifted to position D2, plant ATC2 can be constructed and operated profitably. It will yield lower unit costs and extend to a larger output range than can be attained by plant ATC1. The exploitation of scale economies as demand increases can contribute to the profitability of the monopolist.
Some industries grow, but others contract with technological advances or the dynamics of comparative advantage. Sometimes a monopolist is faced with contracting demand conditions. In these circumstances, a firm which has built a plant like ATC2 in Figure D2-3 may find demand to have collapsed to some level like D1. Given demand D1, there is no level of output at which plant ATC2 can be operated profitably. The monopolist's options are to continue to operate plant ATC2 as long as output can be sold at a price above average variable cost, or to shut down when price falls below AVC. The manager can then given consideration to exiting the market or building a smaller plant, such as ATC1.
In the discussion of pure competition in Chapter D1, we spoke of a "representative firm in the market." This terminology may not be appropriate to monopolistic competition because products are not homogeneous, and because monopolistic competitors may develop managerial distinctives. The differentiated products, however, are enough alike to be construed as being within the same market group. We shall analyze the operation of a typical firm in the monopolistically-competitive market group.
Figure D3-1 represents the revenue and cost curves for such a typical monopolistic competitor. The cost and revenue curves for the typical monopolistic competitor illustrated in Figure D3-1 differ from those for a representative purely competitive firm in that the market is not separately represented, the demand curve slopes downward from left to right (even if only slightly), the MR curve diverges from the AR curve, and the TR curve has some downward concavity. The Figure D3-1 curves are actually quite similar to those for the pure monopolist with the exception of the relative shallowness of the slope of the AR and MR curves, and the fact that the TR curve is of such slight concavity that its peak and downward-sloping range lie well beyond the total cost curves.
The process that the manager of the monopolistically competitive firm might go through in finding the output Q1 and the corresponding price P1 is similar to that for a firm in pure competition. The comparison of marginal revenue with marginal cost serves just as well for the manager of the monopolistically competitive firm as it does for the pure competitor or pure monopolist in discovering whether to increase or decrease output.
Assuming that the monopolistically-competitive firm depicted in Figure D3-1 is typical of all such firms in the market group, the large amount of supernormal profit realized at the profit-maximizing price and output level will not go unnoticed by entrepreneurs presently outside the market. There will likely ensue a rush to enter the market in order to likewise reap such handsome above-normal returns, but they will surely be competed away just as in pure competition.
Suppose that the entire market demand for all of the close-substitute products which comprise the market group can be identified, and that there are n such typical firms in the market group. Unless any of them can distinguish itself and its product to capture a larger share of the market demand, each can expect to exploit a 1/nth share of the market demand. When, in response to the perception of the supernormal profits being realized in the market, k additional firms enter the market, each firm in the market (new as well as old) now can count on only a 1/(n+k) share of market demand, which is smaller than the former 1/n share.
Thus, the firm's demand curve shifts to the left consequent upon entry into the market. The leftward shift of each firm's demand curve consequent upon a diminishing share of the market induces each firm's management to accept a progressively lower price consistent with the goal of maximizing profits. The effect of the decreasing market share and falling price is to decrease the profits of each of the typical firms. Theoretically, enough additional firms will enter the market until all supernormal profits have been competed away, and each of the typical firms is left in a state similar to that depicted in Figure D3-3. This state may be described as market (-group) equilibrium.
This result is similar to the market equilibrium in pure competition in that the typical firm in the market is covering all of its economic costs, including normal returns to the entrepreneur and management, but is not realizing supernormal profit. It is different from the purely competitive conclusion because the typical firm operates at an output rate, Qm, which is below that at which the pure competitor would produce, Qc, and sells at a price which is slightly higher than that of the representative pure competitor in market equilibrium.
Carried into the long run, the manager of the monopolistically competitive firm will be led to build a (slightly) too-small plant, operate it at a (slightly) too-low rate of output, and charge a (slightly) too-high price, all while realizing no more profit than would have been realized by the representative pure competitor.
With the small amount of pricing discretion implicit in the shallowly-sloped demand curve, the manager of the monopolistically-competitive firm may be tempted to experiment with price. In the interest of increasing the market share, a price cut is much more likely to be attempted than a price increase. The highly elastic demand leads to the belief that a small price cut can vastly increase sales, thereby to increase both revenues and profits. And this strategy would work, too, if none of his competitors noticed the price cut.
Alas, some of the competitors at first will become aware that price has fallen somewhere in the market (but because there are so many sellers, they will be oblivious to the identity of the perpetrator) and cut their prices as a defensive measure. More and more firms will meet what is rapidly becoming the reality of a new market price. As this phenomenon ensues, each firm will find itself selling a few more units of the product (people do tend to buy more at lower prices), but if the movement along the demand curve is into the inelastic range, each firm's revenues will actually decrease and profit will fall. The perpetrator of the price cut, so full of hope that cutting price will get him a larger market share and more profit, will have gotten burned by "playing with the matches" of price experimentation. It is experiences like this that make managers of monopolistically-competitive firms much more amenable to non-price forms of competition and promotion.
For each of the other market structure types (pure competition, pure monopoly, and monopolistic competition), it was possible to describe one fairly standard model generally accepted by economists. Unfortunately, this is not possible for oligopolistic competition because the circumstances and the potential for competitor reaction render each oligopolistic situation unique. Since no two oligopolistically competitive situations are alike, it is necessary to model each one to fit the specifics. There is no single oligopoly model as there is a single monopoly model. This section will illustrate and describe a few oligopolistic models.
If the managers of oligopolistically competitive firms are so naive as to assume that competitors will continue to charge the same price forever more, and if they are incapable of learning from experience, they may get into price wars that leave them just breaking even or suffering losses. But real-world business firm managers are neither so naive nor incapable of learning from experience. As intelligent and perceptive decision makers, they are unlikely to lapse into such mindless competition. However, if the manager of one oligopolistically competitive firm has a cost advantage or a greater financial capacity relative to competitors, a price war may be initiated with the intention of driving price down below the AVC of the competitors, thereby to induce them to shut down in the short run, and to exit the market if the low price continues long enough.
In Figure D4-2, suppose that two duopolists equally share the market demand so that their demand curves are coincident and each equal to 1/2 of market demand, Dm. Firm B, however, has lower costs, represented by ATCb, AVCb, and MCb. Firm A's costs are higher at ATCa, AVCa, and MCa. Firm A would like to maximize profit at P2, while Firm B prefers the lower price, P1, for maximization of its profits. But if the manager of Firm B knows enough about both the firm's own costs and those of Firm A, he or she will realize that price can be taken as low as P4 before incurring a loss greater than average fixed costs. Price P3 is at the minimum point of Firm A's AVC curve. Firm A's options now are either to meet price P4 and go out of business in the long run(because P4 is less than its AVC), or to stay at Price P2 and lose market share as its demand curve shifts leftward far enough for it to go out of business because of declining market demand for its product. Thus, by pursuing a deliberate profit-nonmaximizing strategy in the short run, Firm B may be able to achieve monopoly position which will allow it to maximize profits in the long run. But this sort of behavior is likely to be regarded as predatory by the antitrust authority.
Dominant firm price leadership is where the market consists of a dominant firm surrounded by a competitive fringe of smaller firms. The dominant firm behaves as a benevolent monopolist, tolerating the existence of the smaller firms and allowing them to sell any amount of the product which they wish at the price that the dominant firm prefers. The dominant firm then takes its demand as the residual of the market demand not met by the competitive fringe firms, and proceeds to behave as a pure monopolist in maximizing profits. In Figure D4-5, the market demand is Dm. The competitive supply, Sc, is the sum of the marginal cost curves of the competitive fringe firms. The locus of the dominant firm's demand curve, Dd, is found by subtracting the competitive supply from Dm at each possible price. The dominant firm then setsprice at P1 to maximize its profits by selling output Q1, while the competitive firms behave as purely competitive price takers to sell quantity (Q2-Q1). This form of oligopolistic market organization is quite workable, and can persist as long as antitrust law is vigorously enforced and the dominant firm behaves itself. The dominance of the dominant firm may break down when one or more of the competitive firms begins price experimentation or product differentiation/promotion in the effort to capture a larger share of the market.
Differential characteristics price leadership may be based on three aspects of the constituent firms' characteristics:
(1) differences in per-unit costs;
(2) differences in sizes of plant; or
(3) differences in market shares.
Combinations of these differences may also be bases for price leadership.
Suppose, in Figure D4-6, that there are two firms in a market, and that, as illustrated in panel (a), they initially share the market demand equally, i.e., the demand curves are coincident at Df, each of which is one-half of market demand. Firm B has a cost advantage (it hires labor or buys material inputs, components, or energy in lower-cost resource markets) than does Firm A. In order to maximize its profits, Firm B would prefer the lower price, PB, at which it sells quantity QB, than that preferred by Firm A, PA, at which it sells the smaller quantity QA in order to maximize its profits. Which firm has the potential for exercising price leadership?
If Firm A chooses to charge its preferred price, PA, ignoring Firm B's preferred lower price, PB, some of Firm A's customers will defect to purchase from Firm B. This constitutes a change of a non-price determinant of demand for both firms, i.e., the population of consumers purchasing from each firm. Firm A's demand curve will shift to the left toward position DA as illustrated in panel (b), carrying with it its marginal revenue curve toward position MRA, with the consequence that Firm A will prefer an ever-lower price. Firm B's demand curve will shift to the right toward position DB in panel (b), carrying with it its marginal revenue curve toward position MRB, with the consequence that it will prefer an ever-higher price. Theoretically, these shifts will continue until the preferred prices converge to a common price, PC, but with a significant difference: the two firms now have divergent market shares, Firm B now with a larger share than Firm A. Firm A will sell an even smaller quantity, QA', and Firm B will sell an even larger quantity, QB'.
Alternately, had the manager of Firm A been willing to meet Firm B's preferred lower price (a deliberate profit sub-maximizing strategy in the short run), it could have preserved its share of the market. Thus, the firm naturally preferring the lower price (in this case, Firm B) has the potential to be the price leader when demand or cost circumstances change. The other firm(s) may choose to follow or not; they can either go ahead and meet the leader's preferred lower price (and thereby preserve market share), or they can loose market share and end up preferring the same price as the leader.
The same phenomenon can be seen where there is no cost or demand difference between firms, but there is a difference in plant sizes. In Figure D4-7, the two firms again have equal initial market shares. They also use the same technology and have access to the same labor and materials markets, or different markets with the same market prices. The evidence of this is that their ATC curves have the same shapes and reach bottom at the same per-unit cost levels. Firm B, however, has a slightly larger plant evidenced by the overlap of its ATC curve to the right of Firm A's ATC curve. Again, Firm B has the potential for price leadership because it naturally prefers a lower prices in order to maximize profits. But in this case, the basis for price leadership lies in Firm B's larger plant.
Finally, we suppose in Figure D4-8 that the two firms have identical plants and hire labor and purchase materials from the same resource markets. These conditions are evidenced by the fact their ATC curves are coincident. Firm B, however, initially has a slightly larger market share than does Firm A. In this case, Firm A prefers the lower price in order to maximize its profits, and thereby has the potential for price leadership. In this example, the basis for price leadership lies in the smaller initial market share (we can only imagine that Firm A's sales force is pushed to be more aggressive, "We're Number 2, we try harder!"). If Firm B now does not meet the lower price preferred by Firm A, it (Firm B) will lose market share to Firm A. This process theoretically could continue until they have the same market shares and thus prefer the same market price.
While the potential for price leadership can readily be discerned in
each of these models, it does not follow that the actual price leader will
coincide with the potential price leader. The theoretical follower could
"take the bull by the horns" and undercut the price preferred by the theoretical leader in each example. But by so-doing he risks the initiation of a price war as described ea
Price discrimination is possible only when demand elasticity varies among groups or from time to time, and it is not feasible for a prospective client to jump from one group or time frame to another. If clients can jump market segments, the basis for price discrimination is destroyed. If two conditions can be met, the company can increase its revenue by price discriminating across its markets: (a) demands are of different elasticities in the different markets; and (b) there is some means segmenting markets and keeping customers in the different market segments from jumping segments or from buying for one another.
Figure D5-1 illustrates the possibility of price discrimination across two separable markets, A and B. Demand in Market A is somewhat more inelastic than is demand in Market B. When the demands are summed (horizontally), Dc (the combined demand) has the appearance of a bend where Db is joined to Da, so that the marginal revenue curve, MRc is as drawn in panel (c). The firm has a single plant for which its marginal cost curve is MC. The intersection of MC with SMR identifies the quantity Qc and price Pc which would maximize profits without price discrimination. The total revenue will be the area 0PcTQc.
Suppose now that the manager of the company identifies the quantities and prices in the two markets separately for which MR in each is equal to MC, the common marginal cost. On this criterion, Q1 can be sold at Pa in market A, and Qb can be sold at Pb in Market B. Pa is higher and Pb is lower than Pc. A careful examination of total revenue rectangles 0PaRQa and 0PbSQb should reveal that the sum of their areas is greater than that of total revenue rectangle 0PcTQc. Thus, whatever the firms costs happen to be, its revenues with price discrimination will be greater than its revenues without price discrimination, so price discrimination will yield more profit (or less loss) than can be realized without price discrimination
The manager of an imperfectly competitive company may by price discrimination increase the company's revenues, but only by incurring the costs of establishing and enforcing market separation, and often by risking antitrust prosecution. It may be very troublesome (and trouble translates into costs) to seal off the markets from one another. The costs of enforcing market separation may be greater than the additional revenue realized from discrimination.
Jointly-produced products are those which result from a common production process. Classic examples are beef and hides, gasoline and fuel oil, mutton and wool. Even where the objective is to produce one primary product, e.g., metal stampings for auto body parts, there are likely to be marketable by-products such as the metal scrap. In any short-run situation, such joint products are produced in fixed proportions. The relevant questions are what quantity of the output mix is to be produced and at what prices are the individual items in the mix to be sold. In the long run, the management often can vary the output proportions, so that the relevant question for the long run is the profit-maximizing output combination.
The short-run decision problem can be analyzed with a variant on the multimarket price discrimination model. In Figure D5-2, the marginal revenue curves for the jointly produced products are summed vertically (they were summed horizontally in the price discrimination model) to construct the joint marginal revenue curve, MRJ. We note that for all outputs larger than Q2, MR1 is negative so that the MRj curve is coincident with the path of MR2. The relevant short-run decision criterion is the comparison of marginal cost with joint marginal revenue. The manager should increase output as long as joint marginal revenue exceeds marginal cost, or decrease output if joint marginal revenue is less than marginal cost. In panel (a) of Figure D5-2, if marginal cost is given by MCA, the product 1 profit-maximizing price is P1 at which output Q1 of product 1 should be sold. Price P2 should be charged for product 2, and all units of both products should be sold.
If marginal cost should fall to MCB in panel (b) of Figure D5-2, it intersects the joint marginal revenue curve to the right of where MR1 has become negative. Since it would be irrational to sell so large a quantity of any product as to reduce total revenue (i.e, where MR is negative), output Q3 of both products is produced, but only Q2 of product 1 should be sold at price P4. The rest of product 1 (Q3-Q2) should be withheld from the market and possibly destroyed or "dumped" in another market (dumping is then a special case of price discrimination). All of product 2 produced, Q3, should be sold at price P3.
The problem of joint products produced in varying proportions can be illustrated by constructing a three-dimensional cost function where the dependent variable, TC, is a function of two outputs, Qa and Qb, for products A and B. The cost surface has the appearance of that in Figure D5B-1.
Figure D5B-1 illustrates horizontal slices taken at successively higher altitudes (i.e., cost levels or amounts budgeted for expenditure on production of the two items). The projections of the slices through the cost surface into the floor generate a set of concentric isocost curves, which, when looked at from above, have the appearance of the isocost map illustrated in Figure D5B-2. Along any single isocost, e.g., TC3, are points the coordinates of which represent the combinations of quantities of A and B which can be produced at the same level of expenditure on production, TC3. The isocost curves in Figure D5B-2 appear to get closer together to the northeast because the surface rises more steeply in this range.
A three-dimensional total revenue function where TR depends on the prices of A and B and the quantities of each sold can be represented as in Figure D5B-3, assuming pure competition in both product markets (the reader might speculate on how the shape of the surface would be changed in any form of imperfect competition).
When this TR surface is sliced at different altitudes (i.e., levels of revenue), and the slices are projected down into the floor, this gives the appearance when viewed from above of the isorevenue map illustrated in Figure D5B-4. All of the isorevenue curves have the same slopes, determined by the prices of items A and B.
The isorevenue map may be superimposed over the isocost map as depicted in Figure D5B-5 (or in three-dimensions the revenue surface may be shown cutting the cost surface). Because the isorevenue map is dense (i.e., the revenue surface can be sliced at any and all altitudes), isorevenue curves of the same slope are drawn where needed tangent to the isocost curves. The profit can be computed at the tangency of each isocost curve with an isorevenue curve by subtracting the total cost from the total revenue. The largest profit, in this case profit = 100, occurs at the tangency of TC200 with TR300, and the coordinates of the point of tangency represent the quantities of A and B which will maximize the profits from the jointly produced products. No point along an isocost that is not at a tangency with an isorevenue curve can yield as much profit as that realized at a tangency point.
Suppose that the company is currently producing at point R, the coordinates of which represent the quantities of items A and B being produced. Although the company can reallocate its production in any direction from point R, let us suppose that it limits its changes to moving along either an isorevenue curve or an isocost curve. If the company moves downward along the isorevenue curve toward point T, its revenue will remain the same as it produces more of item A and less of item B, but its costs will fall (it reaches ever lower isocost curves). If the company moves downward along the isocost curve from point R to point S, its costs will remain the same while again it produces more of item A and less of item B, but its revenue will increase (it reaches higher isorevenue curves). In either case, profits will increase. We leave it to the reader to deduce what would happen to profits if the company should move upward along either the isocost or the isorevenue curve from point R.
The slope of the isorevenue curve, Pb/Pa, measures the rate at which item A can be substituted for item B within the company's product mix while remaining at the same level of revenue. This rate will be referred to as the marginal rate of revenue substitution (MRRS) of A for B.
The slope of the isocost curve, ΔQb/ΔQa, measures the rate at which item A can be substituted for item B within the company's product mix while remaining at the same level of cost. This rate will be referred to as the marginal rate of cost substitution (MRCS) of A for B.
Now a decision criterion can be specified. If the MRRS is greater than the MRCS, i.e.,
MRRS > MRCS,
the company can increase its profits by producing more of item A and less of item B. Graphically, if the slope of the isorevenue curve (the MRRS) is steeper than the slope of the isocost curve (the MRCS), the company should produce more A and less B to increase profits (both the isocost and isorevenue curves have negative signs). The sense of this is that if the rate at which costs can be reduced while revenue remains constant (i.e., by moving along the isorevenue curve) is greater than the rate at which revenue can be increased while costs remain constant (i.e., by moving along the isocost curve), profits can be increased by producing more of item A and less of item B. This paragraph may be reread, making appropriate alterations on the assumption of a reversal in the direction of the inequality.
A perfectly vertically integrated firm performs all operations in proper sequence to convert a batch of raw materials into a final product. The alternative to vertical integration is vertical segmentation (or disintegration) where each identifiable productive operation is performed by a separate company. Each company in performing its operation adds value to the intermediate product. The partially-processed product is then sold to another company which adds more value by performing the next operation in sequence, and so on until the state of "final product" is attained. If a production process were perfectly vertically segmented, each of the companies in the vertical sequence would maximize profits by finding the price and quantity for which marginal revenue is equal to marginal cost. The price which each company would charge would be equal to its marginal revenue if market conditions were purely competitive, but price would exceed marginal revenue in imperfectly competitive markets. In a vertically-segmented production process, each company's price would become part of the next company's per-unit costs.
A vertically integrated company may be organized into separate, semiautonomous divisions within which managers are given discretion for determining output and responsibility for controlling costs and earning profits. It then becomes necessary to determine the prices of the intermediate goods as they are transferred from each division to the division which will undertake the next stage of processing. Realistic transfer prices are as important to the allocation of resources between divisions within the vertically integrated company as realistic intermediate goods prices are to the allocation of resources between companies in a vertically segmented production process. A too-low transfer price will result in a sub-normal profitability of the division, and resources will tend to be underallocated to the division as division output is decreased. Since the too-low transfer price becomes a too-low per-unit cost to the next division, profits there will be supernormal, and resources will tend to be over-allocated to that division.
Figure D5-3 can be used to illustrate the determination of the transfer price at any stage of production when the manager of the division is given no discretion to source the intermediate product from the market or sell product to the market after the division's processing of it. The product demand and marginal revenue curves are represented as D and MR. The product is assumed to go through j stages of processing where each stage is accomplished in a semi-autonomous division of the company. The marginal cost curve after the final, or jth, stage of processing is MCj. The company will maximize profits at output level Q1, the output level at which MR is equal to MCj.
As processing ensues through the stages of production from the first to the jth, the marginal cost curves (as well as the average variable and average total cost curves) for the respective stages can be imagined to step upward, approaching MCj as the limiting and final-product position. The rationale for the upward stepping of the MC curves is that the transfer price set in any stage becomes the per-unit cost of the intermediate product at the next stage, to which the marginal costs at the next stage are added to find the next transfer price. Thus the transfer price at each stage is higher than the transfer price at the previous stage of processing. On analogy, if the unit prices of the materials inputs in any production process were to increase, the whole set of per unit cost curves (average variable, average total, and marginal cost) would shift upward. In the divisionalized, vertically integrated company, the price of the intermediate product rises as more value is added to it at each stage of processing.
Given that the executive decision has been made to produce Q1 units of output to be sold at the P1 price, and assuming that the required cost functions can be specified and statistically estimated at each stage of production (i.e., by each division), the transfer price for the ith division can be found at the intersection of the vertical from Q1 with the division's marginal cost curve, MCi. In a competitive industry, this marginal cost curve above its AVC minimum would constitute the ith division's intermediate product supply curve.
The demand by division i+1 for the output of division i is in effect perfectly elastic at the marginal cost of producing the value added by processing at the ith stage of production, but only up to output level Q1. Beyond Q1, division i+1 has no demand at all (nor does anyone else) for the output of the ith division. Hence, since MRi = Di when demand is perfectly elastic, MRi is equal to MCi at output Q1, and the appropriate transfer price for the ith division is Pi, which is just equal to the marginal cost of the value added to the intermediate product by processing it in the ith division. Whether division i is functioning profitably at transfer price Pi then depends on the level of its average variable costs, the magnitude of the overhead costs, and how the overhead costs are allocated to the various divisions.
Suppose that division i+1 can purchase an equivalent intermediate product from a competitive external market at a market price below Pi. In this case, disintegration is likely to ensue since none of the output of the intermediate product should be purchased by division i+1 from division i. All of the required intermediate product should be purchased from the external market, and the company should dispose of division i. We can imagine that the intermediate product after processing by division i-1 might be sold to another company to perform the ith stage of value-added processing, then be purchased by division i+1 for further processing, thereby completely skipping the ith stage of processing within in company. This phenomenon occurs in effect when a company contracts with another company to perform a certain stage of processing, then proceeds with further processing in-house. An executive decision might be made to mandate the purchase of the intermediate product by division i+1 from division i even at a transfer cost above the external market price. In this case, other divisions of the company subsidize the higher-cost division. The likely consequence is distortion of both the allocation of resources within the company and the attribution of returns among the divisions.
Suppose that division i finds that it can sell the intermediate product after its processing on a competitive external market at a price above Pi, say Pt in Figure D5-4. The manager of division i should take the demand and marginal revenue curves to be Dt and MRt, respectively, and will find incentive to push his division's output to Q2. With this larger output being
produced, Q1 will be transferred to division i+1, while quantity (Q2- Q1) can be sold on the external market. However, since the marginal cost of producing the larger quantity is higher, the transfer price to division i+1 will be the higher price Pt rather than Pi. This will have the effect of shifting upward all of the marginal cost curves of the subsequent processing divisions, including that of the final product, MCj. This will likely reduce the maximum profit output for the firm below Q1. This can be imagined to set in motion a series of iterative readjustments by processing divisions all along the vertical sequence until a new equilibrium solution is found.
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