Unconstrained Optimization: The Proﬁt Function 287
Equation (7.21) expresses proﬁt not directly as a function of output, but as a function of the inputs employed in the production process, in this case capital and labor. Equation (7.21) allows us to examine the proﬁt-maximizing conditions from the perspective of input usage rather than output levels. Taking partial derivatives of Equation (7.21) with respect to capital and labor, the ﬁrst-order conditions for proﬁt maximiza-tion are
∂π = P Ê ∂Qˆ −P = 0
∂π = P Ê ∂Qˆ −P = 0
The second-order condition for a proﬁt maximum is
∂2 π < 0; ∂2 π < 0;Ë ∂2 π¯Ë ∂2 π¯ −Ë ∂∂2 π ¯ > 0 Equations (7.22) may be rewritten as
P ¥ MP = P
P ¥ MP = P
(7.22a)
(7.22b)
(7.23)
(7.24a)
(7.24b)
The term on the left-hand side of Equations (7.24) is called the marginal revenue product of the input while the term on the right, which is the rental price of the input, is called the marginal resource cost of the input. Equa-tions (7.24) may be expressed as
MRP = MRCK
MRP = MRCL
(7.25a)
(7.25b)
Equations (7.24) are easily interpreted. Equation (7.24a), for example, says that a ﬁrm will hire additional incremental units of capital to the point at which the additional revenues brought into the ﬁrm are precisely equal to the cost of hiring an incremental unit of capital. Since the marginal product of capital (and labor) falls as additional units of capital are hired because of the law of diminishing marginal product, and since MRPK < MRCK, hiring one more unit of capital will result in the ﬁrm losing money on the last unit of capital hired. Hiring one unit less than the amount of capital required to satisfy Equation (7.24a) means that the ﬁrm is for going proﬁt that could have been earned by hiring additional units of capital,since MRPK > MRCK.
Problem 7.9. The production function facing a ﬁrm is Q = K.5 .5
288 Proﬁt and Revenue Maximization
The ﬁrm can sell all of its output for $4. The price of labor and capital are $5 and $10, respectively.
a. Determine the optimal levels of capital and labor usage if the ﬁrm’s operating budget is $1,000.
b. At the optimal levels of capital and labor usage,calculate the ﬁrm’s total proﬁt.
Solution
a. The optimal input combination is given by the expression
MP = MP L K
Substituting into this expression we get (0.5K0.5L−0.5 ) (0.5K−0.5 0.5 )
5 10 K = 0.5L
Substituting this value into the budget constraint we get 1,000 = 5L + 10K
1,000 = 5L + 10(0.5L)
L* = 80 1,000 = 5(80)+ 10K
K*= 40
b. π = TR − TC = P(K0.5L0.5) − TC = 4[(40)0.6(80)0.4] − 1,000 = −$821.11
MONOPOLY
We continue to assume that total cost is an increasing function of output [i.e.,TC =TC(Q)].Now,however,we assume that the selling price is a func-tion of Q, that is,
P = P(Q) (7.26)
where dP/dQ < 0. This is simply the demand function after applying the inverse-function rule (see Chapter 2). Substituting Equation (7.26) into Equation (7.10) yields
π(Q) = P(Q)Q−TC(Q) (7.27)
For a proﬁt maximum, the ﬁrst- and second-order conditions for Equa-tion (7.16) are, respectively,
dπ dQ = P +QË dP¯ − dTC = 0 (7.28)
Unconstrained Optimization: The Proﬁt Function 289
or
P +QÊ dPˆ = MC (7.29)
The term on the left-hand side of Equation (7.29) is the expression for marginal revenue.The second-order condition for a proﬁt maximum is
d2π =QË d2P ¯ + 2 dP − d2TC < 0 (7.30)
If we assume that the demand equation is linear, then Equation (7.26) may be written as
P = a+bQ (7.31) where b < 0. Substituting Equation (7.31) into (7.27) yields
π(Q) = (a+bQ)Q−TC(Q) = aQ+bQ2 −TC(Q) (7.32) The ﬁrst- and second-order conditions become
dπ = a+ 2bQ− dTC = 0
where
a+ 2bQ = MR (7.33)
Note that the marginal revenue equation is similar to the demand equation in that it has the same vertical intercept but twice the (negative) slope.Note also that, by deﬁnition, Equation (7.31) is the average revenue equation, that is,
AR = TR = aQ+bQ2 = a+bQ = P
The second-order condition for a proﬁt maximum is
d2π = 2b− d2TC < 0 (7.34)
The conditions for proﬁt maximization assuming a linear demand curve are shown in Figure 7.9. The cost functions displayed in Figure 7.9a are essentially the same as those depicted in Figure 7.9. All of the cost func-tions represent the short run in production in which 7.8, the prices of the factors of production are assumed to be ﬁxed.The fundamental difference between the two sets of ﬁgures is that in the case of perfect competition the ﬁrm is assumed to be a “price taker,” in the sense that the ﬁrm owner can sell as much product as required to maximize proﬁt without affecting the market price of the product.The conditions under which this occurs will be
290 Proﬁt and Revenue Maximization
b
D
0 Q1 Q2 Q* C
Q4
Q
c
MR, MC
C MC P*
P5 D AR
0 Q1 Q3 Q* Q5 MR FIGURE 7.9 Proﬁt maximization: monopoly.
Unconstrained Optimization: The Proﬁt Function 291
discussed in Chapter 8.In the case of monopoly,on the other hand,the ﬁrm is a “price maker,” since increasing or decreasing output will raise or lower the market price.The simple explanation of this is that because the monop-olist is the only ﬁrm in the industry, increasing or decreasing output will result in a right- or a left-shift in the market supply curve.
As always, the ﬁrm maximizes proﬁt by producing at an output level where MR = MC, which in Figure 7.9 occurs at an output level of Q*. As before, total proﬁt is optimized at output levels Q1 and Q*, where dπ/dQ = 0.At both Q1 and Q* the ﬁrst-order conditions for proﬁt maximization are satisﬁed, however only at Q* is the second-order condition for proﬁt max-imization satisﬁed. In the neighborhood around point D in Figure 7.9b, while the slope of the proﬁt function is positive, it is falling (i.e., d2π/dQ2 < 0).At point C d2π/dQ2 > 0, which is the second-order condition for a local minimum.Again,note that at D¢ the proﬁt-maximizing condition MC = MR, with MC intersecting the MR curve from below, is satisﬁed. At C¢, MC = MR but, MC intersects MR from above, indicating that this point corre-sponds to a minimum proﬁt level.
Note that the marginal cost curve in Figure 7.9c reaches its minimum value at output level Q3, which corresponds to the inﬂection point on the total cost function in Figure 7.9a. Unlike the case of perfect competition, however, while the selling price is by deﬁnition equal to average revenue, in the case of monopoly the price is greater than marginal revenue. Once output has been determined by the ﬁrm,the selling price of the product will be deﬁned along the demand curve. In fact, any market structure in which the ﬁrm faces a downward-sloping demand curve for its product will exhibit this characteristic. Only in the case of perfect competition, where the demand curve for the product is perfectly elastic,will the condition P = MR be satisﬁed for a proﬁt-maximizing ﬁrm.
Problem 7.10. The demand and total cost equations for the output of a monopolist are
Q = 90− 2P
TC =Q3 −8Q2 +57Q+ 2
a. Find the ﬁrm’s proﬁt-maximizing output level. b. What is the proﬁt at this output level?
c. Determine the price per unit output at which the proﬁt-maximizing output is sold.
Solution
a. Deﬁne total proﬁt as
π =TR−TC
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