08 Jun Question Consider a two-commodity economy with
Question
Consider a two-commodity economy with 400 identical consumers. Each has the utilityfunction U = x1x2 and each has an income of 20. The x1-commodity is produced by 100price-taking profit-maximizing firms. Forty of them have the total cost function C(q) = q2and sixty of them have the total cost function C(q) = 2q2.
(a) Find a formula for the demand curve for x1−commodity. Your formula will tell us thequantity demanded at any price P.
(b) Find the supply curve for the x1−commodity.
(c) Find the price at which demand for the x1−commodity equals supply.
(d) Graph the supply and demand curves.
Hints:• Note that if an individual’s utility function is given by U(x1, x2) = xα1 xβ2 with budgetconstraint of p1x1 + p2x2 = I, then the individual’s demand for each commodity isgiven by:x∗1 =αα + βIp1x∗2 =βα + βIp2(See the Overview Notes posted on Angel, Section 11, for a proof of this shortcut)• Market demand is the sum of individual demand curves• To find the quantity supplied by a profit maximizing firm in a competitive market, setprice equal to marginal cost• Market supply is the sum of firm supply curves
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