Question

Consider a customer (i) with a utility function u(x, y) = 200x − 25x² + y where the price of good x is $p, and the price of the composite good y is one dollar ($1). Also, assume that each consumer has an income I. (MU_x=200-50x , and MU_y=1)

1. Derive the consumer's demand function for good x.

Now, consider an economy with 100 exact same type of consumers.

2. Calculate an aggregate demand for only good x.

Now, consider a firm which produces good x. This firm's total cost function is

TC(q) = 256 + 16q + q² and MC(q) = 16+2q

3. Calculate the Short-run supply function for this firm.

Answer 1

a)

u(x,y) =200x - 25x² + y

MUx = 200 - 50x

MUy = 1

MRS = MUx/MUy  

= (200 - 50x)/1

= 200 - 50x

Budget constraint

PxX + PyY =I

px + 1y = I

px + y = I  

At optimal choice MRS = Px/Py

200 - 50x = p/1

200 - 50x = p

200 - p = 50x

x = (200 - p)/50

Demand function for x is

  x = (200 - p)/50

b) Number of consumers n = 100

Aggregate demand for X = nx

= 100(200 - p)/50

= 2(200 - p)

= 400 - 2p

c) TC = 256 +16q + q²  

MC = 16 +2q

Short run supply function is given by

p = MC

p = 16 + 2q

p - 16 = 2q

(p - 16)/2 = q  

q =   (p - 16)/2

= p/2 - 8

thus, Short run supply function is

q = p/2 - 8