1 p n q 1 q n w it is easy to check that pqw k b pqw

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1 , . . . , p n ; q 1 , . . . , q n , w ). It is easy to check that [ ( p,q,w ) K B p,q,w B p ,q , w . Note that B p ,q , w is the union of two bounded sets (i.e. B p , w and B q , w ), hence bounded. Thus S ( p,q,w ) K B p,q,w is also bounded. We conclude that B p,q,w is upper hemicontinuous. Now to prove lower hemicontinuity we use the sequential definition. Take a domain sequence ( p m , q m , w m ) ( p, q, w ). Pick x B p,q,w . We want to construct a sequence ( x m ) for which x m B p m ,q m ,w m for each m and such that x m x . If w = 0, then clearly x = 0, so we set x m = 0 n and we are done. Assume that w > 0. By definition of B p m ,q m ,w m , for each m we have either p m · x m w m or q m · x m w m . Then at the limit we have either p · x w or q · x w . In the first case, let x m w m ( p · x ) w ( p m · x ) x. Then, for each m , p m · x m = w m ( p · x ) w ( p m · x ) ( p m · x ) = w m p · x w w m , since p · x w by assumption. Hence x m B p m ,q m ,w m . Furthermore, x m w ( p · x ) w ( p · x ) x = x, by the continuity of the dot product and of scalar multiplication. For the case where q · x w , just take the sequence x m w m ( q · x ) w ( q m · x ) x. Again, we obtain the same conclusions as before: x m B p m ,q m ,w m and x m x . Thus B p,q,w is LHC.
(c) Prove or disprove the following: If the consumer’s utility function is strictly qua- siconcave, then | x * ( p, q, w ) | ≤ 1. Page 2 of 7
Econ 201A Fall 2010 Problem Set 3 Suggested Solutions
2. Suppose a government has to decide on a budget. Its tax revenue this year in 2010 is Y and its tax revenue next year in 2011 will be kY , where k > 1 is the growth rate of the economy. The government has to decide how much to spend this year, x 1 , and how much to spend next year, x 2 . Because of the poor economy this year, the government will certainly run a deficit of x 1 - Y > 0. For each dollar deficit, the government will have to pay bond holders 1+ r dollars next year, where r > 0 is the interest rate. Next year, the government can spend its 2011 tax revenue kY , minus what it will owe bond holders to finance the 2010 deficit. (The government cannot continue to run a deficit in 2011). The government’s objective function is U ( x 1 , x 2 ) = v ( x 1 )+ v ( x 2 ), where v refers to the annual social benefit generated by government spending. Assume an interior solution. (a) Write the government’s budget constraint over 2010 and 2011 spending.
(b) Suppose v is continuously differentiable. Using the Implicit Function Theorem, write a matrix equation which defines the comparative statics of 2010 and 2011 spending with respect to the interest rate and the growth rate: ∂x * 1 /∂r , ∂x * 1 /∂k , ∂x * 2 /∂r , and ∂x * 2 /∂k . You do not have to explicitly solve or simplify this matrix equation; for example, inverses do not have to be computed.