Question

An industry consists of two firms. Firm 1 has a total cost function given by
??1(?1)=?₁+?₁² ,
while firm 2 has a total cost function given by
??2(?2)=3?₂+12?₂²
for ?>0.
(a) Let ? denote the (exogenous) price at which each firm can sell its output. Write down each firm’s profit-maximization problem and the associated first-order conditions (FOCs).
(b) Derive the firms’ supply functions ?₁∗(?) and ?₂∗(?) and verify that these functions are linearly increasing in ?.
(c) Derive the industry supply curve ?(?). [Hint: Draw a picture and remember the notion of horizontal summation.

(d) Again assuming that the firms act as price takers, find the industry equilibrium when the industry demand curve is given by ??(?)=9/2−1/2? .

Answer 1

Solution:

a)

Give that:

TC₁(q₁)=q₁+q₁²

MC₁=d(TC₁)/dq₁=1+2q₁

Competitive firm sets its output level such that MC=P to maximize profit.So,

1+2q₁=p

2q₁=-1+p

q₁=-0.50+0.50p

Clearly function q1 should be zero or above zero i.e.

Supply function of firm 1 is given by

q*₁(p)=-0.50+0.50p where

q*1(p)=0 when p<1

TC₂(q₂)=3q₂+1/2q₂²

MC₂=d(TC₂)/dq₂=3+q₂

Competitive firm sets its output level such that MC=P to maximize profit.So,

3+q₂=p

q₂=-3+p

Clearly function q2 should be zero or above zero i.e.

Supply function of firm 2 is given by

q*₂(p)=-3+p where

q*₂(p)=0 when p<3

b)

Market supply is given by

Qs=q*₁+q*₂

Qs=-0.50+0.50p-3+p=-3.5+1.5p for

Qs=S(p)=q*₁ where

Qs=S(p)=-0.50+0.50p where

Qs=S(p)=0 for p<1

p	q1(p)=-0.5+0.5p	q2(p)=-3+p	S(p)
0
1	0		0
2	0.5		0.5
3	1	0	1
4	1.5	1	2.5
5	2	2	4
6	2.5	3	5.5
7	3	4	7
8	3.5	5	8.5

c)

Set S(p)=QD(p)

-3.5+1.5p=9/2-1/2p

2p=8

p=4

Clearly p is higher than 3, it means it lies in the domain of demand curve.

S(p)=-3.5+1.5*4=2.50

QD(p)=9/2-1/2*4=2.50

Equilibrium Price=$4

Equilibrium Quantity=2.50

***************************************Thank You************************************