# X 2 was optimal thus we have two conditions that

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, x * 2 ) was optimal. Thus we have two conditions that characterize the set of optimal bundles ( x * 1 , x * 2 ): p 1 x * 1 + p 2 x * 2 = w, x * 1 = x * 2 . Solving this system of linear equations yields x * ( p, w ) = w p 1 + p 2 , w p 1 + p 2 . In particular, x * ( p, w ) is a singleton in R 2 + for each p R 2 ++ and w R + .
Prove that x * ( p, w ) (viewed as a function) is continuous. Page 2 of 9
Econ 201A Fall 2010 Problem Set 2 Suggested Solutions
4. Suppose the consumer lives for two periods, 1 and 2, and can consume a weakly positive amount of a single good in both periods. The price of the good is \$1 in both periods. Before period 1, she is endowed with wealth w > 0. Any wealth she does not spend in period 1 is saved and accrues interest at rate r . Her utility for lifetime consumption is the discounted sum of her per-period utilities, i.e. U ( x 1 , x 2 ) = u ( x 1 ) + δu ( x 2 ), where 0 < δ < 1. The per-period utility function u meets the Inada conditions: u (0) = 0; u is continuously differentiable everywhere; u 0 ( x ) > 0 for all x ; u 00 ( x ) < 0 for all x ; and lim x →∞ u 0 ( x ) = . (a) Formally describe this consumer’s Walrasian budget set.
(b) Prove that U is strictly monotone and strictly quasiconcave in R 2 + .
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Econ 201A Fall 2010 Problem Set 2 Suggested Solutions U ( x ) = u ( x 1 ) + δu ( x 2 ) > u ( y 1 ) + δu ( y 2 ) = U ( y ). The other case is similar. So U is strictly monotone. To prove strict quasiconcavity we prove something stronger. Here we prove that the function is strictly concave, which implies that it is strictly quasiconcave. A twice differentiable function U on R 2 + is strictly concave if and only if the Hessian matrix is negative definite. 1 H = " 2 U ∂x 2 1 2 U ∂x 1 ∂x 2 2 U ∂x 2 ∂x 1 2 U ∂x 2 2 # = u 00 0 0 δu 00 This is clearly negative definite since u 00 ( x ) < 0 for any x . Note: A more direct (but also more tedious) proof that relies only on the defini- tion of quasiconcavity given in class (and the fact that a function with a strictly negative second derivative is strictly concave) is possible as well: Suppose U ( x ) U ( y ) and x 6 = y . Let α (0 , 1). By definition, we have u ( x 1 ) + δu ( x 2 ) u ( y 1 ) + δu ( y 2 ) . Given that u ( · ) is strictly concave, we have u ( αx 1 + (1 - α ) y 1 ) αu ( x 1 ) + (1 - α ) u ( y 1 ) (= iff x 1 = y 1 ) , (3) δu ( αx 2 + (1 - α ) y