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econ11_08_ps8_sol

econ11_08_ps8_sol - Eco11 Fall 2008 Simon Board Economics...

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Eco11, Fall 2008 Simon Board Economics 11: Solutions to Practice Problems 8 (Weeks 9–10) December 3, 2008 1. First Welfare Theorem There are two goods ( x and y ) and two agents (A and B). The agents’ utility functions are u A = v A ( x ) + y u B = v B ( x ) + y where v A = 2 ln( x ) and v B ( x ) = 4 x 1 / 2 , The agents have incomes m A = 10 and m B = 10. The price of good y is p y = 1; the price of good x is to be determined. A single firm produces good x . It has cost function c ( q ) = q 2 / 2. (a) Show that p = 2 is an equilibrium price. Find the equilibrium allocations ( x A , x B , q ). (b) Suppose a social planner chooses ( x A , x B , q ) to maximise the total surplus, v A ( x A ) + v B ( x B ) - c ( q ) subject to q = x A + x B . Verify your allocations from part (a) satisfy the FOCs from this optimisation problem. Solution (a) Substituting her budget into her utility, A maximises 2 ln( x ) + ( m - px ) The FOC implies that demand is given by x = 2 /p . Similarly, B maximises 4 x 1 / 2 + ( m - px ) The FOC implies that demand is given by x = 4 /p 2 . 1
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Eco11, Fall 2008 Simon Board The firm’s marginal cost is MC = q . Hence the supply curve is q = p . The equilibrium price thus solves 4 p 2 + 2 p = p Substituting in, p = 2 solves this equation. At this price the allocations are ( x A , x B , q ) = (1 , 1 , 2) (b) Using the fact that q = x A + x B , the social planner wishes to maximise S = 2 ln( x A ) + 4 x 1 / 2 B - 1 2 ( x A + x B ) 2 (where the S stands for ‘surplus’). The FOCs are dS dx A = 2 x - 1 A - ( x A + x B ) = 0 dS dx B = 2 x - 1 / 2 B - ( x A + x B ) = 0 Substituting in x A = 1 and x B = 1 we see that both are satisfied.
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