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# ansf510a - Econ 510a(second half Prof Tony Smith TA...

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Econ 510a (second half) Prof: Tony Smith TA: Theodore Papageorgiou Fall 2004 Yale University Dept. of Economics Solutions for Final Exam 1Q u e s t i o n 1 a) A Competitive Equilibrium with date-0 trading for this economy is a vector of prices { p t } 2 t =0 and a vector of quantities { c it } 2 t =0 for i = A, B such that (1) For i = A, B, { c it } 2 t =0 =a r gm a x 2 P t =0 β t u ( c it ) s.t. 2 P t =0 p t c it = 2 P t =0 p t ω it (2) c At + c Bt = ω At + ω for t =0 , 1 , 2 b) From the f.o.c. of the consumer’s problem, we get: βu 0 ( c i,t + j ) u 0 ( c i,t ) = p t + j p t t, j This together with budget constraint and market clearing condition deter- mines the competitive equilibrium. Now there are 2 ways of solving this prob- lem. The f rst is writing down all the f.o.c. for each of the 2 agents, the market clearing conditions for each of the 3 time periods and the 2 budget constraints and solve out for the prices and quantities (you won’t need to use all of the equations to solve the system). The second and easiest is to use the fact that each agents consumption in every period is going to be a constant share of the aggregate endowment. This follows from the homotheticity of preferences. In other words we have that: c At = γ ( w At + w ) t c =( 1 γ )( w At + w ) t 1

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Now we are going to use the f.o.c. and we get (normalizing p 0 =1 ): βu 0 ( c A 1 ) u 0 ( c A 0 ) = p 1 βc A 0 c A 1 = p 1 = βγw o γw 1 = p 1 1 2 4 16 = p 1 p 1 = 1 8 Similarly: β 2 u 0 ( c A 2 ) u 0 ( c A 0 ) = p 2 β 2 c A 0 c A 2 = p 2 = 1 4 γ 4 γ 4 = p 2 p 2 = 1 4 We can now plug in the prices we found in consumer A’s budget constraint and solve out for his share: γ 4+ 1 8 γ 16 + 1 4 γ 4= μ 4 8 +1 γ (4+2+1) = 11 2 7 γ = 11 2 γ = 11 14 2
and therefore the consumption in each period will be: c 0 A = 11 14 4= 22 7 c 1 A = 11 14 16 = 88 7 c 2 A = 11 14 22 7 c 0 B = 3 14 6 7 c 1 B = 3 14 16 = 24 7 c 2 B = 3 14 6 7 Just to be sure lets verify that the budget constraint for consumer B holds: 1 · 6 7 + 1 8 · 24 7 + 1 4 · 6 7 = 1 8 · 12 21 14 = 12 8 andthusitdoesho ld . c) A Competitive Equilibrium with sequential trading for this economy is a sequence { c it } 2 t =0 , © a i,t +1 ª 2 t =0 , { R t } 2 t =0 (where R t means interest rate from t to t +1 )for i = A, B such that (1) For i = A, B, © c it ,a i,t +1 ª 2 t =0 =a r gm a x 2 P t =0 β t u ( c it ) s.t.

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ansf510a - Econ 510a(second half Prof Tony Smith TA...

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