Question

Consider a perfectly competitive market in good x consisting of 250 consumers with a utility function: Denote Px to be the price for good x and suppose Py = 1. Each consumer has income equal u(x, y) = xy to 10. There are 100 firms producing good x according to the cost function c(x) = x^2 + 1.

(a) Derive the demand curve for good x for a consumer in the market.

(b) Derive the market demand curve for good x.

(c) Derive the individual firm’s supply curve for good x.

(d) Derive the market supply curve for good x.

(c) Derive the individual firm’s supply curve for good x.

(e) Determine the equilibrium price and quantity in the market for good x. (f) Is the market currently in the long-run equilibrium? Why or why not?

(d) Derive the market supply curve for good x.

Answer 1

1) utility Maximizing condition:

MUx/px= MUy/py

Y/px=x/1

Y=px*x

Budget constraint,

M=py*y+px*x

M=1*(px*x)+px*x=2*px*x

X=M/(2*px)

M=10

X=5/px

2) Market demand is sum of individual demand

X=250*5/px=1250/px

3) individual firm supply function is nothing marginal cost function.

MC=2x

P=2x( inverse supply)

X=0.5p( supply function)

4) Market supply is sum of individual supply,

X=100*0.5p=50p

5) Equilibrium at market demand= market supply

1250/p=50p

P^2=25

P*=5

X*=50*5=250

The long run Equilibrium price is equal to Minimum average cost.

The average is Minimum at , where it is equal to marginal cost.

ATC=x+1/x

ATC= MC

X+1/x=2x

1/x=x

X^2=1

X=1

Min ATC=1+1/1=2

So price is not equal to Minimum average cost,so Equilibrium is not in long run.