250 Economics Some Applications of the Calculus of Functions of One Variable
Don Ferguson September 30, 2004
Reading: Parts of chapters 4 and 5. The following terms are used in the examples that follow. totals, marginals and averages Marginal constructs are pervasive in Economics (marginal revenue, marginal cost, marginal utility, marginal product, marginal tax rate, etc.). If y = f (x) denotes a total (total revenue, total cost and so on) as it depends on some variable x, then the derivative function f (x) denotes the corresponding marginal function of x and f (x) x is the corresponding average function. point elasticities Given a function y = f (x), the point elasticity of y with respect to x is dened as x dy xf (x) yx = = . y dx f (x) Note that, typically, the point elasticity is a function that will vary with x. If y is a decreasing function of x, then the absolute value of xf (x)/f (x) is often used for the elasticity.
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1. Example: the relation between averages and marginals Let T (x) denote a total function and let M (x) = T (x) and A(x) = T (x)/x denote the marginal and average functions. Applying the quotient rule we nd that A (x) = xT (x) T (x) x2 1 = (M (x) A(x)) . x
If, as is often the case with economic variables, x > 0, then this implies the following general rule about the relation between marginals and averages A (x) > (<)0 as M (x) > (<)A(x). In other words, the average is increasing or decreasing as the marginal is greater or less than the average. 2. Example: Price elasticity of demand If the demand function is D(p) = a bp, a > 0, b > 0, p a/b
then, using the absolute value convention, the price elasticity of demand is Dp (p) = pb . a bp
On the other hand, if the demand function is D(p) = apb , then a > 0, b > 0, p > 0
pbapb1 =b apb where, once more, the absolute value convention has been used. This last case is the only type of demand function for which the elasticity of demand does not vary with price. (Can you prove this?) Dp =
3. Example: Marginal revenue product of an input Consider an imperfectly competitive rm which has a revenue function R(y) which depends on how much it sells y. Suppose that the rm has a production function y = f (x) where x is the amount of an input employed by the rm. We can then express revenue as a function R(x) R(f (x)) 2
of x. This is the total revenue product of the input. Applying the chain rule we nd that the marginal revenue product of the input is R (x) = R (y)f (x) which is the product of the marginal revenue of the output and the marginal product of the input. 4. Marginal Revenue and the Price Elasticity of Demand Using the inverse demand function p = P (q), total revenue can be expressed as R(q) = qP (q). Applying the product rule, the marginal revenue is R (q) = P (q) + qP (q) qP (q) = P (q) 1 + p
.
If we let q = D(p) denote the demand function, then, using the rule for the derivative of an inverse function, we have P (q) = 1 D (p)
and, hence, the marginal revenue can be expressed as R (q) = P (q) 1 + = P (q) 1 where Dp = 1
pD (p) q
1 Dp
pD (p) D(p)
is the price elasticity of demand (assuming that demand is downward sloping and the absolute value convention is applied). 5. Example: Kinks It is often the case that total functions have kinks in them. For example, typically marginal tax rates are constant within an income bracket, but change between brackets. Similarly, marginal costs may change abruptly upon hitting a capacity constraint. In all such the cases, marginal functions (derivative 3
functions) are discontinuous at the kinks even though the total functions are continuous. Kinked Tax Function For example, suppose that taxes are a function of income y and that the marginal tax rates vary depending on the income bracket T (y) = where a2 = a1 + b1 y1 b2 y1 .1 Both the tax function T (y) and the average tax function A(y) = T (y)/y are continuous, but the marginal tax function M (y) = T (y) is discontinuous. 6. Example: Short-run Production with a Leontief Technology Leontief technologies are characterized by xed coecients of production. For example, suppose that the amount of labour and capital required per unit of output are constant. Let aL and aK denotes these xed input coecients. The total amounts of labour and capital used to produce an amount Y of output will be aL Y aK Y. Consider a rm which has a Leontief technology. In the short run the amount of productive capital available to the rm is xed and this imposes a capacity constraint on the rms output. Specically, if K is the amount of capital available, then it is possible that the rm may leave some of its capital idle, however it cannot use more than this amount. This means that aK Y K. The resulting short run production function Y = f (L) for the rm will be L a aL if aK L K L f (L) = K if aK L > K aK aL This is illustrated in Figure 6. The average and marginal functions are shown in Figure 6. Notice that the total and average functions are continuous, but the marginal function is discontinuous.
This insures that at the transition between brackets (y = y1 ), the same amount of tax is owing under both schedules.
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a1 + b1 y if y [0, y1 ) a2 + b2 y if y [y1 , )
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Y
K/aK
KaL /aK Figure 1: Short Run Leontief Production Function
L
AP(Y /L), MP(dY /dL)
1/aL
AP, MP AP MP L /aK Ka L
Figure 2: Short Run Leontief Marginal and Average Product
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p S(p)
p
D(p) D(p), S(p) Figure 3: Equilibrium with Demand Equal Supply
7. The Existence of Walrasian Equilibrium in a Single Market Walras focused on the role of prices in equilibrating market...