7/29/2019 max-econ

1/4

A FIRM MAXIMIZING PROFIT

MATH 232

1. Two Products

1 Assume a firm makes two products with output levels ofQ1 and Q2, prices P1 and P2, and revenue isR = P1Q1 + P2Q2. Assume the cost of production is C= Q

21

+ Q1Q2 + Q22

, so the profit is

= R C= P1Q1 + P2Q2 Q2

1 Q1Q2 Q2

2.

We want to maximize the profit under two different assumptions.

First Case: In this case, we assume there is pure competition and the prices are determined externally tothe firm and are considered fixed by the firm. Taking the partial derivatives with respect toQ1 and Q2 and

setting them equal to zero, we get the two equations

0 =

Q1= P1 2Q1 Q2 and

0 =

Q2= P2 Q1 2Q2.

Solving forQ1 and Q2, we get the critical values

Q1 =2P1 P2

3and Q2 =

2P2 P13

.

The second partial derivatives are

2

Q21

= 22

Q2Q1= 1

2

Q1Q2= 1

2

Q21

= 2,

so

2

Q21

= 2 < 0 and

2

Q2

1

2

Q2

1

2

Q2Q12

= (2)(2) (1)2 = 3 > 0,

and the quantities (Q1, Q

2) maximize profit.

Second Case: In this case, we assume that the firm has a monopoly and can set the prices of the two

products. However, once the prices are fixed, the quantities purchased of the two products is determined by

1Based on the treatment in Fundamental Methods of Mathematical Economics by Alpha Chiang, McGraw Hill, Inc., 1984

1

7/29/2019 max-econ

2/4

2 MATH 232

a demand function set by the consumers. We assume that the products are interchangeable, and the demand

of each product depends on the prices of both products by the rules

Q1 = 40 2P1 + P2

Q2 = 15 + P1 P2.

Substituting in the quantity demanded for the prices, we get the expression for the revenue in terms of the

prices,

= P1(40 2P1 + P2) + P2(15 + P1 P2) (40 2P1 + P2)2

(40 2P1 + P2)(15 + P1 P2) (15 + P1 P2)2.

We want to maximize the profit as a function of the prices.Taking the partial derivatives with respect toP1 and P2, we get

0 =

P1= (40 2P1 + P2) 2P1 + P2 + 4(40 2P1 + P2)

+ 2(15 + P1 P2) (40 2P1 + P2) 2(15 + P1 P2)

= 160 10P1 + 5P2, and

0 =

P2= P1 + (15 + P1 P2) P2 2(40 2P1 + P2)

(15 + P1 P2) + (40 2P1 + P2) + 2(15 + P1 P2)

= 10 + 5P1 4P2.

Thus, we have the two equations

0 = 160 10P1 + 5P2

0 = 10 + 5P1 4P2.

Multiplying the second equation by 2 and adding, we get 0 = 140 3P2 or P2

= 140/3 = 462/3. Then10P1 = 160 + 5(140/3) and P

1

= 118/3 = 391/3.The second partial derivatives are

2

P21

= 102

P2P1= 5

2

P1P2= 5

2

P21

= 4,

so

2

P21

= 10 < 0 and

2

P21

2

P21

2

P2P1

2= (10)(4) (5)2 = 15 > 0,

and the prices (P1, P

2) maximize profit.

7/29/2019 max-econ

3/4

A FIRM MAXIMIZING PROFIT 3

2. Multiple inputs and one product

Assume a firm makes one product from n inputs. Let p be the price of the output, xj be the amount ofthe jth input used, and pj be the price of the j

th input. Let G(x1, . . . , xn) be the production function, whichgives the amount of output in terms of the inputs. The profits is

= pG(x1, . . . , xn) p1 x1 pn xn.Assuming the prices are fixed, the inputs which maximize profit satisfy

0 =

xi= p

G

xipi or

G

xi=

pip.

Thus, at the critical point, the marginal product of each input equals the price of the input relative to the

price of the output.

The matrix of second partial derivatives is2

xixj

=

p

2G

xixj

.

Thus, for the critical point to be a maximum it is necessary that not only p2G

x2i< 0, but also that the

principal determinants have the correct signs,

(1)k det

p

2G

xixj

1i,jk

> 0

for each 1 k n. The requirement that p2G

x2i< 0 could be viewed as saying that a small change in

the input xi makes more difference for small values ofxi than for large input: diminishing returns. For twoinputs, we need

p2G

x21< 0 and

p22G

x21

2G

x22

p2

2G

x1x2

> 0.

7/29/2019 max-econ

4/4

4 MATH 232

3. RATE OF CHANGE OF MINIMAL COST OF PRODUCTION

Let Q = F(L,K) be the production function of a single output in terms of two inputs, laborL and capitalK. Let w be the price of labor (wages) and r the price of capital (interest rate). Thus the cost function isC = wL + rK. Assume the outputQ = Q0 is fixed and the cost is minimized. LetL = L(w, r) be the

amount of labor andK= K(w, r) be the amount of capital which realizes this minimum. Define

C(w, r) = w L(w, r) + r K(w, r)

be the minimal cost at these values. Shepherds Lemma says that

C

w(w0, r0) = L(w0, r0) and

C

r(w0, r0) = K(w0, r0).

This says that the rate of change of cost with respect to change in wages is equal to the size of the labor

force, and does not depend on the change of the size of the labor force or amount of capital. See [2] for a

discussion of the economic interpretation.

The derivation is not too difficult but does involve the introduction of a new function. LetL0 = L(w0, r0)and K0 = K(w0, r0) be the size of labor and capital at this minimum for wages s0 and interest rate r0. Weform the function

g(w, r) = C(w, r) wL0 rK0.

The terms subtracted involve the changing cost of labor and capital while holding the amounts fixed. The

value g(w0, r0) = C(w0, r0) w0L0 r0K0 = 0 by the definitions. For (w, r) = (w0, r0), C(w, r)

wL0 + rK0 because (L0,K0) satisfies the production constraint and the new values L(w, r) and K(w, r)minimize the cost. Therefore, g(w, r) 0 and g attains its maximum at (w0, r0), so

0 =g

w(w0, r0) =

C

w(w0, r0) L0 and

0 =

g

r (w0, r0) =

C

r (w0, r0) K0,so

C

w(w0, r0) = L0 = L(w0, r0) and

C

r(w0, r0) = K0 = K(w0, r0)

as claimed.

REFERENCES

[1] C. Simon and L. Blume, Mathematics for Economists, W. W. Norton & Company, New York, 1994

[2] D. Besanko and R. Braeutigam, Microeconomics: An Integrated Approach, John Wiley & Sons, Inc., New York, 2002.