Begin with the following assumptions: (i) There are 20,000 identical consumers.

Question

Begin with the following assumptions: (i) There are 20,000 identical consumers. Each consumer has the utility function: U = x2y and I=$450. (ii) There are 10,000 identical perfectly competitive firms producing good X. Each firm has the production function: qX=3L½K½ . Let w=$3, r=$12 and K=1 in the short run for each firm. a) Derive the demand function for X for a typical consumer. b) Derive the market demand function. c) Find the short-run production function for a typical firm. d) Derive the short-run supply function for a typical firm. e) Derive the short-run market supply function. f) Find the short-run competitive market equilibrium price and quantity for good X. g) How much output will the typical firm produce in the short run? How much profit will the typical firm earn?

Explanation / Answer

a) The utility function is given by U = x2y

Marginal rate of substitution is MUx/MUy = 2y/x. Optimum bundle is the one that equates MRS to price ratio Px/Py. Let’s use the relative prices so that Py = 1.

Now each consumers demands a bundle (x*, y*) at which MRS = Px/Py or 2y/x = Px. This gives y = xPx/2.

The budget set of typical consumer is 450 and is composed of:

450 = xPx + yPy

450 = xPx + y

450 = xPx + xPx/2

xPx = 300

Px = 300/x

Consider this as the quantity to be determined:

P = 300/q.

b) With 20000 consumers, the market demand equation is:

P = 300*20000/q

P = 6000000/q

c) The market consists of 10000 firms and 20000 consumers which makes us believe that the market is perfectly competitive. Consider the production system in this market. Given that the value of capital is fixed at K = 1. This makes the production function q = 3L1/2.

d) From this value, find the required amount of labor as L = q2/9.

The cost function should be represented as
C = wL + rK

C = 3L + 12

C = 3*q2/9 + 12

C = q2/3 + 12

From this cost function calculate the average variable cost and marginal cost since the supply curve of a typical competitive firm is its marginal cost function that arises from the minimum of AVC.

AVC = C/q = 1/3(q). MC = 2q/3. Clearly, at q = 1, MC>AVC,

So the supply curve of a typical firm is P = 2q/3.

e) The supply curve of a typical firm is P = 2q/3.

For 10000 firms, the market supply curve is 20000q/3.

f) When the market demand and market supply are intersected, the equilibrium price and quantity is determined

20000q/3 = 6000000/q

q/3 = 300/q

q2 = 900. q = 30 and P = 10.

Hence the equilibrium price is $10 and quantity is 30 units per firm

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