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Exam 2
EconS 526
1. A firms has a production function 𝑦𝑦 = 𝑏𝑏𝑏𝑏𝑏𝑏 where b>0 is a parameter, y is output, L is labor and
K is capital. Assume that the price of labor and capital are both equal to w.
a. Derive the cost function and the conditional factor demand function for labor.
b. How does an increase in output affect labor use? Show a comparative static to prove your
answer.
a.
Objective of the firm: min 𝑤𝑤(𝐿𝐿 + 𝐾𝐾) 𝑠𝑠. 𝑡𝑡. 𝑦𝑦 = 𝑏𝑏𝑏𝑏𝑏𝑏
𝐿𝐿,𝐾𝐾
FOCs: 𝑤𝑤 − 𝜇𝜇𝜇𝜇𝜇𝜇 = 0, 𝑤𝑤 − 𝜇𝜇𝜇𝜇𝜇𝜇 = 0, 𝑦𝑦 − 𝑏𝑏𝑏𝑏𝑏𝑏 = 0 . From the first two FOCs, it is clear that L=K.
𝑦𝑦 0.5
Substituting this into the constraint yields, 𝐾𝐾 = 𝐿𝐿 = �𝑏𝑏 �
.
𝑦𝑦 0.5
.
𝑏𝑏
The corresponding cost function is then: 𝑐𝑐 = 2𝑤𝑤 � �
b.
More output increases labor. Using conditional factor demand, we have,
𝜕𝜕𝐿𝐿
𝜕𝜕𝜕𝜕
1 0.5
𝑏𝑏𝑏𝑏
= 0.5 � �
> 0.
𝛽𝛽 1−𝛽𝛽
2. The cost function for a firm is given by 𝑐𝑐 = 𝛼𝛼𝛼𝛼𝑝𝑝𝐿𝐿 𝑝𝑝𝐾𝐾
where y is output, 𝑝𝑝𝐿𝐿 is the price of
labor, 𝑝𝑝𝐾𝐾 is the price of capital and 𝛼𝛼, 𝛽𝛽 are parameters.
a. Derive the conditional factor demand for capital.
b. Derive an expression for the production function.
a.
b.
𝐾𝐾 =
𝜕𝜕𝜕𝜕
𝜕𝜕𝑝𝑝𝐾𝐾
𝛽𝛽 −𝛽𝛽
= 𝛼𝛼(1 − 𝛽𝛽)𝑦𝑦𝑝𝑝𝐿𝐿 𝑝𝑝𝐾𝐾
Get demand for labor:
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Re-arrange both equations:
𝐿𝐿 =
𝜕𝜕𝜕𝜕
𝜕𝜕𝑝𝑝𝐿𝐿
𝛽𝛽−1 1−𝛽𝛽
𝑝𝑝𝐾𝐾
= 𝛼𝛼𝛼𝛼𝛼𝛼𝑝𝑝𝐿𝐿
𝐿𝐿 1/(𝛽𝛽−1)
�
�
= 𝑝𝑝𝐿𝐿 /𝑝𝑝𝐾𝐾
𝛼𝛼𝛼𝛼𝛼𝛼
Re-arrange both equations:
�
1/𝛽𝛽
𝐾𝐾
�
= 𝑝𝑝𝐿𝐿 /𝑝𝑝𝐾𝐾
𝛼𝛼(1 − 𝛽𝛽)𝑦𝑦
𝐿𝐿 1/(𝛽𝛽−1)
�
= 𝑝𝑝𝐿𝐿 /𝑝𝑝𝐾𝐾
�
𝛼𝛼𝛼𝛼𝛼𝛼
Equating the two above equations and solving for y yields,
1−𝛽𝛽
1 𝛽𝛽
1
𝑦𝑦 = � � �
�
𝐾𝐾 1−𝛽𝛽 𝐿𝐿𝛽𝛽
𝛼𝛼𝛼𝛼
𝛼𝛼(1 − 𝛽𝛽)
3. There are 50 firms that behave in a competitive manner and have identical cost functions given
by 𝑐𝑐(𝑦𝑦) = 0.5𝑤𝑤𝑦𝑦 2 where w is input price and y is output. The demand curve for the product is
given by 𝐷𝐷(𝑝𝑝) = 1000 − 50𝑝𝑝 + 50𝐼𝐼 where p is price and I is income.
a. Derive the equilibrium output and price when average income is 10 and the input price is 1.
(Hint: first derive the supply function).
b. Assume that the firm is in long run equilibrium at the prices and quantities derived in part
(a). Assume further that there is a sudden increase in income and when more firms enter
into the market, the input price decreases. Given this scenario, describe (in detail) the
adjustment towards the new long run equilibrium. Use graph(s) to support your discussion
and identify the shape of the long run supply curve.
a.
individual firm max profit when: 𝑝𝑝 = 𝑤𝑤𝑤𝑤 so individual supply is 𝑦𝑦 = 𝑝𝑝/𝑤𝑤. Market supply is 𝑌𝑌 = 50𝑝𝑝/𝑤𝑤
Market supply is equated with market demand:
50𝑝𝑝
𝑤𝑤
= 1000 − 50𝑝𝑝 + 50𝐼𝐼 For I=10 and w=1, we have,
50𝑝𝑝 = 1000 − 50𝑝𝑝 + 500 . Therefore, p=1500/100=15 and Y = 50 x 15 = 750. Also, y=15.
b.
A sudden increase in income will shift demand up if the good is normal from DO to D1. This will lead to a
higher price from P0 to P0’ and positive profit for existing firms. This will signal other firms to enter the
market. As more firms enter, supply shifts to the right from S0 to S1 which decreases price. Since input
price decreases with more firms, AC and MC decrease from AC0 and MC0 to AC1 to MC1. The resulting
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new equilibrium price is lower than the original equilibrium price (P0 to P1). Therefore the long run
supply curve is downward sloping (S_LR).
p
p
MC0
D1
MC1
S0
S1
D0
AC0
P0’
AC1
P0
S_LR
P1
Y
y
4. A monopolist has a convex cost
function 𝑐𝑐(𝑦𝑦) and is faced with an
inverse demand 𝑝𝑝(𝑦𝑦, 𝑝𝑝𝑟𝑟 ) where y is output level and pr is the price of a related good. The related
good is a complement to the output produced by the monopolist.
a. Set up and solve the monopolist’s problem. Given your solution, derive an expression
showing the effect of an increase the price of the related good on optimal output and price
of the monopolist. What assumption(s) do you need to make to sign the comparative static?
b. Assume that you found out that the inverse demand takes the following form: 𝑝𝑝(𝑦𝑦, 𝑝𝑝𝑟𝑟 ) =
𝑎𝑎(𝑦𝑦) + 𝑏𝑏(𝑝𝑝𝑟𝑟 ) where 𝑎𝑎(𝑦𝑦) and 𝑏𝑏(𝑝𝑝𝑟𝑟 ) are functions. How does this specific functional form
affect the comparative static that you calculate?
a.max 𝑝𝑝(𝑦𝑦, 𝑝𝑝𝑟𝑟 )𝑦𝑦 − 𝑐𝑐(𝑦𝑦)
𝑦𝑦
FOC: 𝑝𝑝𝑦𝑦 (𝑦𝑦, 𝑝𝑝𝑟𝑟 )𝑦𝑦 + 𝑝𝑝(𝑦𝑦, 𝑝𝑝𝑟𝑟 ) − 𝑐𝑐𝑦𝑦 (𝑦𝑦) = 0. Thus, y*(𝑝𝑝𝑟𝑟 ). SOC: 𝑝𝑝𝑦𝑦𝑦𝑦 𝑦𝑦 + 2𝑝𝑝𝑦𝑦 − 𝑐𝑐𝑦𝑦𝑦𝑦 < 0
𝑑𝑑𝑦𝑦 ∗
Get 𝑑𝑑𝑝𝑝 by substituting y*(𝑝𝑝𝑟𝑟 ) into the FOC and totally differentiating with respect to I to
𝑟𝑟
derive,
𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 𝑦𝑦 + 𝑝𝑝𝑝𝑝𝑟𝑟
𝑑𝑑𝑦𝑦 ∗
=−
𝑑𝑑𝑝𝑝𝑟𝑟
𝑝𝑝𝑦𝑦𝑦𝑦 𝑦𝑦 + 2𝑝𝑝𝑦𝑦 − 𝑐𝑐𝑦𝑦𝑦𝑦
The denominator is negative because that is the second order condition. The numerator is ambiguous
because we do not know the sign of 𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 . Here, 𝑝𝑝𝑝𝑝𝑟𝑟 < 0 because the goods are complements. If we
𝑑𝑑𝑦𝑦 ∗
𝑑𝑑𝑦𝑦 ∗
assume that if 𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 < 0 then
< 0. Note it can also be 𝑑𝑑𝑝𝑝 > 0 if 𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 > 0 and �𝑦𝑦𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 � > �𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 �.
𝑑𝑑𝑝𝑝
𝑟𝑟
The impact on price has a direct and indirect effect:
𝑟𝑟
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𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑑𝑑𝑑𝑑
𝑑𝑑𝑝𝑝∗
=
+
𝑑𝑑𝑝𝑝𝑟𝑟 𝜕𝜕𝑝𝑝𝑟𝑟 𝜕𝜕𝜕𝜕 𝑑𝑑𝑝𝑝𝑟𝑟
The direct effect is negative by assumption of complementarity,
𝜕𝜕𝜕𝜕
𝜕𝜕𝑝𝑝𝑟𝑟
< 0. The indirect effect is positive
because they are two negative numbers multiplied together. Therefore, the total effect is ambiguous.
One would need to assume something between the magnitude of the direct versus indirect effects.
Usually direct effects are larger (not always). If this is the case, then price declines.
b.
If we have that functional form then 𝑝𝑝𝑦𝑦𝑝𝑝𝑟𝑟 = 0. Therefore,we now have an unambiguous result that
𝑑𝑑𝑦𝑦 ∗
𝑑𝑑𝑝𝑝𝑟𝑟
< 0. No additional assumption is needed. The price effect is still the same and ambiguous.
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