Economics 401
An example of a profit-maximizing firm and an example of a
cost-minimizing firm
1. Consider a price-taking, profit-maximizing firm in a one-output, two-input world.
Denote the price of output by p and input prices by w1 , w2 , and suppose its production
function is
1/2
q = 2z1
1/2
+ 4z2 .
(a) Check that this production function is strictly concave in its inputs and then derive
the firm’s input demand functions, ziπ (p, w1 , w2 ), i = 1, 2, output supply q(p, w1 , w2 ), its
profit function π(p, w1 , w2 ).
(b) Show that input demands and output supply are homogeneous of degree zero in
(p, w1 , w2 ) and π(p, w1 , w2 ) is homogeneous of degree one in the same variables.
(c) Verify that input demands slope downwards in their own prices and output supply
slopes upward in the price of output.
ANSWER
(a) If the Hessian of the production function is negative definite, then it is strictly concave.
Calculate
1 −1/2
−1/2
= z1
f1 (z1 , z2 ) = 2 z1
2
1 −1/2
−1/2
f2 (z1 , z2 ) = 4 z2
= 2z2
2
1 −3/2
f11 (z1 , z2 ) = − z1
2
f12 (z1 , z2 ) = 0
1 −3/2
−3/2
f22 (z1 , z2 ) = − 2z2
= −z2 .
2
1
The Hessian is:
f11 (z1 , z2 ) f12 (z1 , z2 )
f12 (z1 , z2 ) f22 (z1 , z2 )
=
−3/2
− 21 z1
0
0
−3/2
−z2
!
2
For z1 , z2 positive f11 < 0 and f11 f22 − f12
> 0. The conditions for a negative definite matrix
are satisfied, so the production function is strictly concave in z1 , z2 .
The firm’s profits are:
pf (z1 , z2 ) − w1 z1 − w2 z2
Note that the Hessian of profits is p times the Hessian of the production function. We can
conclude that profits are strictly concave in z1 , z2 and thus we are dealing with a well-behaved
maximization problem.
The first-order conditions for profit maximization are:
∂Profits
−1/2
= pf1 (z1 , z2 ) − w1 = pz1
− w1 = 0
∂z1
∂Profits
−1/2
− w2 = 0
= pf2 (z1 , z2 ) − w2 = 2pz2
∂z2
Solving these equations yields the input demand functions.
p2
w12
p2
z2π (p, w1 , w2 ) = 4 2
w2
z1π (p, w1 , w2 ) =
Substitute the input demand functions into the production function in order to get output
supply.
2
q (p, w1 , w2 ) = 2z1π (p, w1 , w2 )1/2 + 4z2π (p, w1 , w2 )1/2
2 1/2
2 1/2
p
p
q (p, w1 , w2 ) = 2
+4 4 2
2
w1
w2
p
p
q (p, w1 , w2 ) = 2
+4 2
w1
w2
p
p
q (p, w1 , w2 ) = 2
+8
w1
w2
4
1
+
q (p, w1 , w2 ) = 2p
w1 w2
w2 + 4w1
q (p, w1 , w2 ) = 2p
w1 w2
Substitute the input demand functions and the production function into profits in order to
get the profit function.
π (p, w1 , w2 ) = max pf (z1 , z2 ) − w1 z1 − w2 z2
z1 ,z2
π (p, w1 , w2 ) = pq (p, w1 , w2 ) − w1 z1π (p, w1 , w2 ) − w2 z2π (p, w1 , w2 )
2
2
p
p
w2 + 4w1
− w1
− w2 4 2
π (p, w1 , w2 ) = p 2p
2
w1 w2
w2
w2
2
2
p
p
w2 + 4w1
−4
−
π (p, w1 , w2 ) = 2p2
w1 w2
w1
w2
w2 + 4w1
1
1
2
π (p, w1 , w2 ) = p 2
−
−4
w1 w2
w1
w2
2w2 + 8w1 − w2 − 4w1
2
π (p, w1 , w2 ) = p
w1 w2
w2 + 4w1
1
4
2
2
π (p, w1 , w2 ) = p
=p
+
w1 w2
w1 w2
Checking the envelope theorem
∂π (p, w1 , w2 )
1
4
= 2p
+
= q (p, w1 , w2 )
∂p
w1 w2
∂π (p, w1 , w2 )
p2
= − 2 = −z1π (p, w1 , w2 )
∂w1
w1
∂π (p, w1 , w2 )
p2
= −4 2 = −z2π (p, w1 , w2 )
∂w2
w2
(b) A function g : <n → < is said to be homogeneous of degree γ if for all α > 0
3
g (αx1 , αx2 , . . . , αxn ) = αγ g (x, x2 , . . . , xn ) .
Consider prices (αp, αw1 , αw2 ). The firm’s profits are:
αpf (z1 , z2 ) − αw1 z1 − αw2 z2
The first-order conditions when maximizing profits with respect to (z1 , z2 ) are:
∂Profits
−1/2
= αpf1 (z1 , z2 ) − αw1 = αpz1
− αw1 = 0
∂z1
∂Profits
−1/2
= αpf2 (z1 , z2 ) − αw2 = 2αpz2
− αw2 = 0
∂z2
The α’s will cancel out, so that the first order conditions and the solution are the same as
they were under prices (p, w1 , w2 ). Therefore, the input demand functions are homogenous
of degree zero in prices.
ziπ (αp, αw1 , αw2 ) = ziπ (p, w1 , w2 ) , i = 1, 2
As a result, the output supply is also homogeneous of degree zero.
q (αp, αw1 , αw2 ) = 2z1π (αp, αw1 , αw2 )1/2 + 4z2π (αp, αw1 , αw2 )1/2
= 2z1π (p, w1 , w2 )1/2 + 4z2π (p, w1 , w2 )1/2
= q (p, w1 , w2 )
Substituting (p, w1 , w2 ) with (αp, αw1 , αw2 ) in the profit function shows it to be homogeneous
of degree one in prices.
π (αp, αw1 , αw2 ) =
=
=
=
αw2 + 4(αw1 )
(αp)
(αw1 )(αw2 )
α(w2 + 4w1 )
2 2
α p
α 2 w1 w2
w2 + 4w1
2
αp
w1 w2
απ (p, w1 , w2 )
2
(c) The derivatives of input demands and output supply with respect to own prices are:
∂z1π (p, w1 , w2 )
p2
= −2 3 < 0
∂w1
w1
π
∂z2 (p, w1 , w2 )
p2
= −8 3 < 0
∂w2
w2
w2 + 4w1
∂q (p, w1 , w2 )
= 2
>0
∂p
w1 w2
4
Given positive prices, input demands are downward sloping and output supply is upward
sloping in own price.
2. Consider a price-taking, cost-minimizing firm in a one-output, two-input world. Denote input prices by w1 , w2 , and suppose its production function is
1/2
q = 2z1
1/2
+ 4z2
(a) Given your answer to 1(a) explain why this firm’s technology has strictly convex upper
contour sets and then use Lagrange’s method to derive the firm’s input demand functions,
zic (w1 , w2 , q), i = 1, 2, and the cost function c(w1 , w2 , q).
(b) Show that input demands are homogeneous of degree zero in (w1 , w2 ) and c(w1 , w2 , q)
is homogeneous of degree one in w1 , w2 .
(c) Verify that input demands slope downwards in their own prices.
ANSWER
(a) Since the production function is strictly concave, it is also strictly quasiconcave and
therefore has strictly convex upper contour sets (see Assignment 2 question 1(d)). The
method of Lagrange multipliers can then be used to solve the cost minimization problem.
1/2
1/2
L = w1 z1 + w2 z2 + λ q0 − 2z1 + 4z2
5
First-order conditions:
∂L
−1/2
= w1 − λ z1
=0
∂z1
∂L
−1/2
= w2 − λ 2z2
=0
∂z2
∂L
1/2
1/2
= q0 − 2z1 + 4z2
=0
∂λ
Take the ratio of equations (1) and (2) and re-arrange:
−1/2
−1/2
(1)
(2)
(3)
1/2
w1
λz1
z1
z2
=
=
=
−1/2
−1/2
1/2
w2
λ2z2
2z2
2z1
1/2
1/2 w1
z2 = 2z1
w2
Substitute this result into equation (3) and solve for one of the input demands:
1/2
2z1
1/2
4z2
+
q0 −
1/2
1/2 w1
q0 − 2z1 + 4 2z1
w2
w1
1/2
z1
2+8
w2
2w2 + 8w1
1/2
z1
w2
= 0
= 0
= q0
= q0
1/2
z1
q0
=
2
w2
w2 + 4w1
z1c (w1 , w2 , q0 )
q2
= 0
4
w2
w2 + 4w1
2
Substitute the input demand for z1 back into the result from the first-order conditions in
order to obtain the other input demand
1/2
z2
q02
4
= 2
w2
w2 + 4w1
2 !1/2
q0
w2
w1
= 2
2 w2 + 4w1
w2
w1
1/2
z2
= q0
w2 + 4w1
2
w1
c
2
z2 (w1 , w2 , q0 ) = q0
w2 + 4w1
1/2
z2
6
w1
w2
Use both of these input demands to solve for the cost function
c(w1 , w2 , q) = w1
q02
4
w2
w2 + 4w1
2 !
+ w2
q02
w1
w2 + 4w1
2 !
q02 w1 w2 (w2 + 4w1 )
c(w1 , w2 , q) =
4
(w2 + 4w1 )2
2
q0
w1 w2
c(w1 , w2 , q) =
4 w2 + 4w1
(b) Consider input prices (αw1 , αw2 ). Input demands are now:
z1c (αw1 , αw2 , q0 )
z2c (αw1 , αw2 , q0 )
q2
= 0
4
2
2
αw2
αw2 + 4αw1
2
q02
w2
=
4 w2 + 4w1
= z1c (w1 , w2 , q0 )
αw1
=
αw2 + 4αw1
2
w1
2
= q0
w2 + 4w1
c
= z2 (w1 , w2 , q0 )
q02
Therefore, input demands are homogeneous of degree zero in input prices. The cost function
is now:
c(αw1 , αw2 , q) =
=
=
=
αw1 αw2
q02
4 αw2 + α4w1
q02
αw1 w2
4 w2 + 4w1
q02
w1 w2
α
4 w2 + 4w1
αc(w1 , w2 , q)
The cost function is homogeneous of degree one in input prices.
(c) The derivatives of the input demands with respect to own prices are:
∂z1c (w1 , w2 , q0 )
2q02
=
∂w1
4
w2
w2
−4
w2 + 4w1
(w2 + 4w1 )2
w22
= −2q02
<0
(w2 + 4w1 )3
7
∂z2c (w1 , w2 , q0 )
= 2q02
∂w2
w1
w1
−4
w2 + 4w1
(w2 + 4w1 )2
w12
<0
= −8q02
(w2 + 4w1 )3
For positive input prices and output supply, these derivatives are strictly less than zero;
therefore, input demands slope downwards in their own prices.
Note that when q0 = q(p, w1 , w2 ) (from the profit maximization problem), the input demands
from the profit maximization and cost minimization problem are the same.
z1c (w1 , w2 , q(p, w1 , w2 ))
z2c (w1 , w2 , q(p, w1 , w2 ))
2p
w2 +4w1
w1 w2
2
2
w2
=
4
w2 + 4w1
2 2
2
4p w2 + 4w1
w2
=
4
w1 w2
w2 + 4w1
p2
=
w12
= z1π (p, w1 , w2 )
=
=
=
=
2 2
w2 + 4w1
w1
2p
w1 w2
w2 + 4w1
2 2
w2 + 4w1
w1
2
4p
w1 w2
w2 + 4w1
2
p
4 2
w2
π
z2 (p, w1 , w2 )
8