Econ 121: Intermediate Microeconomics

Homework 6 Answer Key

Question 1

a.

Cost minimizing problem:

minx1,x2

p1x1 + p2x2

sub. to xα1x1−α2 ≥ y

F.O.C.s are therefore given by:

p1 − λαxα−11 x1−α2 = 0

p2 − λ(1− α)xα1x−α2 = 0

y − xα1x1−α2 = 0

b.

Solving the F.O.C.s give us the solution for input demands:

x∗1 = y

((1− α)p1αp2

)α−1

x∗2 = y

((1− α)p1αp2

)αc.

f(tx1, tx2) = (tx1)α(tx2)

1−α

= t.(xα1x1−α2 )

= tf(x1, x2)

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d.

The cost function is given by,

C(y) = p1y(

(1−α)p1αp2

)α−1+ p2y

((1−α)p1αp2

)α= y

[p1

((1−α)p1αp2

)α−1+ p2

((1−α)p1αp2

)α]Therefore the profit is given by,

π = y

[p− p1

((1− α)p1αp2

)α−1− p2

((1− α)p1αp2

)α]

Hence for positive and finite amount of production it must be the case that the coefficient is zero

or,

p = p1

((1− α)p1αp2

)α−1+ p2

((1− α)p1αp2

)αThis is a result of the CRS assumption because this makes the cost function linear in output. Since

revenue is linear in output as well, the profit also becomes linear in output which is why we need

this relation to hold, or otherwise there would be either zero or infinite production depending on

whether the coefficient in the profit equation is negative or positive.

Question 2.

a.

maxyπ = (A−By)y − 1

2y2

Therefore the F.O.C. implies:

A− 2By − y = 0

⇒ y∗ =A

2B + 1

⇒ p∗ =AB + A

2B + 1

Therefore the profit is given by,

π∗ = A

[A

2B + 1

]− (B − 1

2)

[A

2B + 1

]2

2

b.∂π

∂y(p∗, y) = p∗ − y

Therefore the derivative calculated at y = y∗ is given by,

∂π

∂y(p∗, y∗) =

AB + A

2B + 1− A

2B + 1=

AB

2B + 1> 0

c.

We know that the F.O.C. can be rewritten as,

p∗(

1− 1

ε∗

)= y∗

where

ε∗(p∗, y∗) = −dydp

(p∗, y∗)p∗

y∗=A−By∗

By∗

Now after the change of the demand to A′ and B′ we have a new elasticity given by,

ε′(p∗, y∗) =A′ −B′y∗

B′y∗=A−By∗

B′y∗<A−By∗

By∗= ε∗(p∗, y∗)

since,

A′ −B′y∗ = p∗ = A−By∗

So at (p∗, y∗) the F.O.C. equation would look like,

p∗(

1− 1

ε′

)< y∗

Hence to restore the equality the price must go up and output must fall.

Question 3.

a.

maxyπ = 0.9(A−By)y − 1

2y2

⇒ yM =0.9A

1.8B + 1<

A

2B + 1= y∗

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Price received by the consumers,

pC =0.9AB + A

1.8B + 1>AB + A

2B + 1= p∗

Price received by the firm,

pM = 0.9pC =0.81AB + 0.9A

1.8B + 1<AB + A

2B + 1= p∗

Profit of the firm,

πM =0.41A2

1.8B + 1< π∗

b.

In presence of the profit tax the optimization problem is

maxy

(1− t)[(A−By)y − 1

2y2]

which is the same problem as the original one in 2.a and hence the output and price decision is

same as in 2.a and the profit is just t proportion lower than the original one. Therefore given this

result the consumers will always prefer profit tax over sales tax as profit tax doesn’t distort the

pricing and supply decision of the firm while sales tax does,

c.

The fixed cost again doesn’t change the profit maximizing price-output choice of the firm, but lowers

profit by the amount of the fixed cost. Therefore the argument that selling band of air-wave lengths

to the highest bidder through auctions would rive up the price is not well founded as the cost paid

to government to get this right is like a fixed cost to the monopolist.

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