Problem Set 3. Profit Maximization and Profit Functions
EconS 526
1. The production function for good z is 𝑓(𝑥) = ln𝑥 where x is an input. The price of good z is p
and the input price for x is w.
a. Set up the problem for a profit maximizing firm and solve for the demand function for x.
Do not forget to show the first order condition and show if the second order condition
satisfies the condition for a maximum.
max 𝑝ln𝑥 − 𝑤𝑤
𝑥
First order condition,
Therefore, 𝑥 ∗ =
𝑝
−𝑤 = 0
𝑥
𝑝
𝑤
Second order condition,
−𝑝/𝑥 2 < 0
b. Derive the profit function. What is the derivative of the profit function with respect to w?
Show the expression and interpret it.
Substitute 𝑥 ∗ =
𝑝
𝑤
into profit equation so,
Take derivative with respect to w,
𝑝
𝜋 = 𝑝ln � � − 𝑝
𝑤
𝑝
𝜋𝑤 = − 𝑤 = −𝑥 ∗ . We get back the (negative of ) input demand.
c. The firm decides to use an alternative technology to produce good z. The production
function with this new technology is 𝑔(𝑥) = 20x − x 2 . Output and input prices remain the
same. What condition is required for x>0? When will the firm opt to produce x=0?
max 𝑝(20x − x 2 ) − 𝑤𝑤
𝑥
First order condition,
Therefore, 𝑥 ∗ = 10 −
𝑤
.
2𝑝
𝑝(20 − 2𝑥) − 𝑤 = 0
Thus, we need 20p>w for x>0. If 20p<=w then x=0.
2. A competitive profit maximizing firm has the following profit function: 𝜋(𝑝, 𝑤1 , 𝑤2 ) =
𝑝(𝑔(𝑤1 ) + ℎ(𝑤2 )).
a. Characterize the functions 𝑔(𝑤1 ) and ℎ(𝑤2 ). (Describe their first and second order
conditions for the profit function to be well behaved).
Since the profit function is convex and decreasing in input prices, we would need 𝑔′ < 0, 𝑔′′ >
0, ℎ′ < 0, and ℎ′′ > 0.
b. How does the price of input 2, 𝑤2 , affect factor demand for input 1? Prove your answer.
It has no effect. First, we can get factor demand for input 1 using envelope theorem,
𝜕𝜕
−
= −𝑝𝑔′ (𝑤1 ) ≡ 𝑥1
𝜕𝑤1
Take the cross derivative with respect to w2,
𝜕𝜕
𝜕2𝜋
=
=0
−
𝜕𝑤1 𝜕𝑤2 𝜕𝑤2
3. A Cobb-Douglas production function characterizes production of good y, 𝑓(𝑥, 𝑧) = x 𝑎 z 𝑏 . The
output price is p, the input prices for x and z are px and pz, respectively.
a. Calculate the input demand functions, the supply function and the profit function.
max 𝑝x 𝑎 z 𝑏 − 𝑤𝑥 𝑥 − 𝑤𝑧 𝑧
𝑥,𝑧
FOCs:
𝑎𝑎x 𝑎−1 z 𝑏 −𝑤𝑥 = 0
1/𝑏
𝑤𝑥 x1−𝑎
� .
𝑎𝑎
Using the first FOC, we get 𝑧 = �
the following expression,
𝑏𝑏x 𝑎 z 𝑏−1 −𝑤𝑧 = 0
Substitute this into the second FOC and solve for x*. I get
𝑥∗ =
𝑤𝑧 𝑏/(𝑎+𝑏−1) 𝑤𝑥 (1−𝑏)/(𝑎+𝑏−1)
𝑏𝑏/(𝑎+𝑏−1) 𝑎(1−𝑏)/(𝑎+𝑏−1) 𝑝1/(𝑎+𝑏−1)
𝑧∗ =
𝑤𝑧 (1−𝑎)/(𝑎+𝑏−1) 𝑤𝑥 𝑎/(𝑎+𝑏−1)
𝑏 (1−𝑎)/(𝑎+𝑏−1) 𝑎𝑎/(𝑎+𝑏−1) 𝑝1/(𝑎+𝑏−1)
Substitute this into the expression for z, and we get optimal z*,
The supply function is,
𝑤𝑧 𝑎𝑎/(𝑎+𝑏−1) 𝑤𝑥 𝑎(1−𝑏)/(𝑎+𝑏−1)
𝑤𝑧 𝑏(1−𝑎)/(𝑎+𝑏−1) 𝑤𝑥 𝑏𝑏/(𝑎+𝑏−1)
𝑦 = 𝑎𝑎/(𝑎+𝑏−1) 𝑎(1−𝑏)/(𝑎+𝑏−1) 𝑎/(𝑎+𝑏−1) 𝑏(1−𝑎)/(𝑎+𝑏−1) 𝑎𝑎/(𝑎+𝑏−1) 𝑏/(𝑎+𝑏−1)
𝑏
𝑎
𝑝
𝑏
𝑎
𝑝
∗
Simplifies to,
𝑦∗ =
The profit function is,
π=𝑝
Where 𝜑 ≡
𝑤𝑧 𝑏/(𝑎+𝑏−1) 𝑤𝑥 𝑎/(𝑎+𝑏−1)
𝑏𝑏/(𝑎+𝑏−1) 𝑎𝑎/(𝑎+𝑏−1) 𝑝(𝑎+𝑏)/(𝑎+𝑏−1)
𝑤𝑧 𝑏/(𝑎+𝑏−1) 𝑤𝑥 (1−𝑏)/(𝑎+𝑏−1)
𝑤𝑧 𝑏/(𝑎+𝑏−1) 𝑤𝑥 𝑎/(𝑎+𝑏−1)
−
𝑤
𝑥 𝑏/(𝑎+𝑏−1) (1−𝑏)/(𝑎+𝑏−1) 1/(𝑎+𝑏−1)
𝑏𝑏/(𝑎+𝑏−1) 𝑎𝑎/(𝑎+𝑏−1) 𝑝(𝑎+𝑏)/(𝑎+𝑏−1)
𝑏
𝑎
𝑝
(1−𝑎)/(𝑎+𝑏−1)
𝑎/(𝑎+𝑏−1)
𝑤𝑧
𝑤𝑥
− 𝑤𝑧 (1−𝑎)/(𝑎+𝑏−1) 𝑎/(𝑎+𝑏−1) 1/(𝑎+𝑏−1)
𝑏
𝑎
𝑝
1
𝑏𝑏/(𝑎+𝑏−1) 𝑎 𝑎/(𝑎+𝑏−1)
−
π=
1
𝑤𝑧 𝑏/(𝑎+𝑏−1) 𝑤𝑥 𝑎/(𝑎+𝑏−1)
𝜑
𝑝1/(𝑎+𝑏−1)
𝑏
1−𝑏
𝑏𝑎+𝑏−1 𝑎 𝑎+𝑏−1
−
1
𝑏(1−𝑎)/(𝑎+𝑏−1) 𝑎 𝑎/(𝑎+𝑏−1)
b. Use the profit function and envelope theorem to derive the effect wz on input demand for
z and x. Are x and z complements or substitutes?
−π𝑤𝑧 ,𝑤𝑧
𝑏
𝜕𝑧 ∗
𝑏
𝑏
𝑤𝑧 𝑎+𝑏−1−2 𝑤𝑥 𝑎/(𝑎+𝑏−1)
=
=−
�
− 1�
𝜑
𝜕𝑤𝑧
𝑎+𝑏−1 𝑎+𝑏−1
𝑝1/(𝑎+𝑏−1)
−π𝑤𝑧 ,𝑤𝑧
𝑏
𝜕𝑧 ∗
1
𝑤𝑧 𝑎+𝑏−1−2 𝑤𝑥 𝑎/(𝑎+𝑏−1)
=
=�
�
𝜑
𝜕𝑤𝑧
𝑎+𝑏−1
𝑝1/(𝑎+𝑏−1)
This is negative only if a+b<1.
−π𝑤𝑥 ,𝑤𝑧
So if a+b<1, then
𝜕𝑥 ∗
𝜕𝑤𝑧
𝑏
𝑎
𝜕𝑥 ∗
𝑎
𝑏
𝑤𝑧 𝑎+𝑏−1−1 𝑤𝑥 𝑎+𝑏−1−1
=
=−
�
�
𝜑
𝜕𝑤𝑧
𝑎+𝑏−1 𝑎+𝑏−1
𝑝1/(𝑎+𝑏−1)
< 0. Therefore x and z are complements.