Question

2. In the competitve market for widgets there are 50 identical consumers and 200 iden- tical firms. Each individual consumer has the following demand function for widgets qP(P) = 100 – 2P where qp is the quantity an individual consumes and P is the widget's price. Each firm has the following cost function: C(q) = 100+ 2q +q2. (a) (3 points) Find the market demand function for widgets QP(P). Find the industry supply function for widgets Qʻ(P), make sure to find each firm's supply function before you find the supply function for the industry as a whole. (b) (4 points) What is the equilibrium price in this industry? How many widgets does each consumer consume? How many widgets does each firm produce? How much of a profit or loss does each firm make? (c) (5 points) Suppose there is a per unit tax of 1 = $3 put in place on the firms. What are the prices in the market (reminder: tax wedge here)? What is the government revenue? What is the deadweight loss? What is the consumer's tax incidence (in terms of the percentage of the tax)? (d) (4 points) Ignoring the tax from the previous section, suppose the firm's long-run 92 cost function is C(q) = 25+ What is the long run competitive equilibrium 4 price P* and quantity Q* (hint: find the minimum long-run average cost)? How many firms are in the market?

Answer 1

a)

q^D = 100 - 2P

since there are 50 idetical consumers therefore market demand is

Q^D = 50(100 - 2P)

Q^D = 5000 - 100P  

c(q) = 100 + 2q + q²

MC = 2 + 2q

supply curve of a firm is given as

P = MC

P = 2 + 2q

P - 2 = 2q

q = (P - 2)/2

supply curve of a firm is q = (P - 2)/2

There are 200 identical firms thus industry supply curve is

Qs = 200(P - 2)/2

Qs = 100(P - 2)

Qs = 100P - 200

b)  

Industry equilibriun is given by the equality of market demand and market supply

Qd = Qs  

5000 - 100P = 100P - 200  

5000 + 200 = 100P + 100P

5200 = 200P  

P = 26  

Equilibrium market price is P = 26

Q = 100(26) - 200 = 2400

Equilibrium market quantity is Q = 2400  

We have q = (P - 2)/2

q = (26 - 2)/2

q = 24/2

q = 12

so each firm will produce 12 widgets

Individual cinsumer demand is

q = 100 - 2P

q = 100 - 2(26)

q = 100 - 52  

q = 48

So each consumer consumes 48 widgets

Profit of a firm = Pq - c(q)  

= Pq - 100 - 2q - q²

= 26$\times$12 - 100 - 2$\times$12 - 12²

= 312 - 100 - 24 - 144

= 312 - 268  

= 44

Each firm makes profit of $44

c)

Now a per unit tax of $ 3 is placed on the firms as a result cost function after tax  is

c(q) = 100 + 2q + q² + 3q  

c(q) = 100 + 5q + q²

MC = 5 + 2q

supply curve of a firm after tax is

P = MC

P = 5 + 2q  

P - 5 = 2q

q = (P - 5)/2

Insustry supply curve or function after tax

Qs = 200(P - 5)/2

Qs = 100(P - 5)

Qs = 100P - 500

Now Qd = Qs  

5000 - 100P = 100P - 500

5000 + 500 = 100P + 100P

5500 = 200P

P = 27.5

Price paid by consumer is 27.5

price received by a firm or seller is 27.5 - 3 = 24.5

Equilibrium market quantity after tax

Q = 100(27.5) - 500

Q = 2250

Govt. revenue = tax per unit $\times$ Equilibrium market quantity after tax

= 3$\times$2250

= 6750

DWL = 1/2(27.5 - 24.5)(2400 - 2250)

   = 1/2$\times$3$\times$150

= 225

Total per unit tax = 3

per unit tax paid by consumer is 27.5 - 26 = 1.5  

so incidence of tax on consumer in percentage term = (1.5/3)$\times$100 = 50%

d)

LR cost function

c(q) = 25 + q²/4

MC = 2q/4

MC = q/2  

AC = c(q)/q  

AC = (25 + q²/4 )/q

AC = 25/q + q/4

AC is minimum when MC = AC

q/2 = 25/q + q/4

q/2 - q/4 = 25/q

q/4 = 25/q

q² = 100  

q = 10  

In long run equilibrium P = MC = AC

P = MC  

P = q/2

P = 10/2  

P = 5

Thus long run equilibrium price P* = 5

we have calculated market demand curve or function in part (a)

Qd = 5000 - 100P  

Q = 5000 - 100(5)

Q = 5000 - 500

Q = 4500

Hence long run equilibrium market quantity is Q* = 4500

Number of firms n = Q*/q = 4500/10 = 450