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Microeconomic Theory
Profit maximization
Profit equation is:
Π=𝑝∙𝑦−𝒘∙𝒙
max Π(L) = 𝑝 ∙ 𝑦(𝒙) − 𝒘 ∙ 𝒙
Solution is 𝒙∗ (𝑝, 𝒘), and by putting it in Π(𝒙) one gets the maximum profit equation 𝜋(𝑝, 𝒘):
𝜋(𝑝, 𝒘) = Π[𝒙∗ (𝑝, 𝒘)] = 𝑝 ∙ 𝑦[𝒙∗ (𝑝, 𝒘)] − 𝒘 ∙ 𝒙∗ (𝑝, 𝒘)
or
𝜋(𝑝, 𝒘) = 𝑝 ∙ 𝑦(𝑝, 𝒘) − 𝑤 ∙ 𝒙(𝑝, 𝒘)
Hotelling lemma
𝜕𝜋
𝜕𝑝
=
𝜕𝜋
𝜕𝑤𝑖
𝜕Π
𝜕𝑝
=𝑦
𝜕Π
(Supply of product)
= 𝜕𝑤 = −𝑥𝑖
𝑖
or
𝜕𝜋
𝜕𝑝
= 𝑦 𝑆 (𝑝, 𝒘)
𝜕𝜋
𝜕𝑤𝑖
= −𝒙𝑖 𝐷 (𝑝, 𝒘)
(unconditional demand for inputs)
Problem 1. Company produces a good which quantity is q and its price p. Production
function is 𝒒(𝑲) = √𝑲. The price of input K is r.
a) Find profit function 𝚷(𝑲)
Π(𝐾) = 𝑝 ∙ 𝑞 − 𝑟 ∙ 𝐾 = 𝑝√𝐾 − 𝑟𝐾
b) Find profit maximizing quantity of capital.
𝑑Π
p
p
p
=
−𝑟 =0⇒
= 𝑟 ⇒ √K = | ↑2
dK 2√K
2r
2√K
𝐾∗ =
𝑝2
4𝑟 2
c)Find maximum profit function.
𝑝2
𝑝2
𝑝2
𝜋(𝑝, 𝑟) = 𝛱(𝐾 ∗ ) = 𝑝√ 2 − 𝑟 2 =
4𝑟
4𝑟
4𝑟
d) Deduct supply of product and demand for capital using Hotelling lemma.
𝑑𝜋
𝑑𝑝
= 2𝑟 = 𝑞 𝑆
𝑝
𝑑𝜋
𝑑𝑟
= − 4𝑟2 = −𝐾 𝐷 ⇒ 𝐾 𝐷 = (2𝑟)
𝑝2
𝑝 2
e) Find product supply if r = 0,25?
𝑞 𝑆 (𝑝, 𝑟 = 0,25) =
𝑝
= 2𝑝
2𝑟
f) Find capital demand if p = 2?
2 2
1
𝐾 𝐷 (𝑝 = 2, 𝑟) = ( ) = 2
2𝑟
𝑟
g) What is the relation between marginal revenue product and unconditional demand for
inputs?
𝑀𝑅𝑃𝐾 =
𝜕𝑇𝑅𝑃
.
𝜕𝐾
𝑇𝑅𝑃 = 𝑝 ∙ 𝑞(𝐾) ⟹ 𝑀𝑅𝑃 = 𝑝 ∙
𝑟=
𝑝
2√𝐾
⟹ √𝐾 =
𝑑𝑞
𝑝
=
𝑑𝐾 2√𝐾
𝑝
𝑝 2
⟹𝐾=( )
2𝐾
2𝐾
MRP is the inverse demand for input.
h) Find MC(q). What is the relation between MC and results of the Hotelling lemma?
𝑇𝐶(𝑞) = 𝑇𝐸[𝐾 −1 (𝑞)] = 𝑟𝐾 −1 (𝑞)
𝑀𝐶(𝑞) = 𝑟
𝑑𝐾
𝑟
= 𝑑𝑞 = 2𝑟√𝐾 = 2𝑟𝑞
𝑑𝑞
𝑑𝐾
𝑝(𝑞) = 2𝑟𝑞 ⟹ 𝑞(𝑝) =
𝑝
2𝑟
MC is the inverse supply of products.
Problem 2. Price of a good is p, price of an input x is w, and production function is y =
f(x). Express maximum profit function, demand for input, supply of product and check
Hotteling lemma if production function is:
a) y = x ¼
b) y = x2
c) y = xn
d) y = x
Solutions:
𝑝2
𝑝2
𝑝
𝑤2
𝑤2
1
𝑤
𝑤 𝑎 1−𝑎 1
(𝑎
a) 𝜋 = 2𝑤 , 𝑥 = 4𝑤 2 , 𝑦 = 2𝑤, b) 𝜋 = − 4𝑝 , 𝑥 = 4𝑝2 , 𝑦 = 2𝑝, c) 𝜋 = ( 𝑎𝑝 )
1
𝑤 𝑎−1
( )
𝑎𝑝
, d) 𝜋 = 0
− 1) , 𝑥 =