Consider a single-market analysis of welfare (with production). We use the notation qi to denote consumer i's demand for good 1, and qj be firm J's supply of good 1.

There are 500 consumers. Let mi be the income of a particular consumer i. The income of the 500 consumers are distributed evenly in (0,1).

The utility of all consumers are the same. For consumer i, his utility is:

qi +x2i

The price of good 1 is denoted by p and that the price of good 2 is normalized to 1. The demand of Good 2 by consumer i is denoted by x2i. Hint:

MRSi=1/2(qi)^1/2

There are 1000 firms producing Good 1 and they are identical. Firm j has a cost function:

Cj(qj)=q2j

Part a:

Find qi (the demand of good 1 by consumer i) in terms of p.

Part b: Find the market demand function of good 1, denoted by D(p)

Part c:

Find the supply function for Good 1 for Firm j.

Part d:

Find the market supply function S(p).

Part e:

Find the equilibrium price p.