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Unformatted text preview: Fall, 2003 ARE202A FINAL EXAM DECEMBER 8 2003 Tackle first the question you think is the easier one. It’s always a good strategy to make attempts at all parts of the question, because then you always have a chance at partial credit. If you omit a part, then you lose that chance! Don’t attempt either bonus part till you’ve done what you can on the nonbonus parts of both questions. Problem 1 . (60 points) Consider a consumer with CobbDouglas preferences: U = A . Q α L β with A > 0, α = 1 / 3 and such that β = 1 / 2, and where Q is a composite consumer good and L is leisure (not labor). The consumer maximizes his utility subject to nonnegativity constraints ( Q ≥ 0, L ≥ 0), a time constraint L ≤ T (where T is the total time available) and subject to his budget constraint: pQ ≤ ( 1 τ ) w ( T L ) + Y where p is the price of the composite commodity, τ is the tax rate on wage income ( τ ∈ [ , 1 ) ), w is the wage rate per unit of labor time, and Y is other nontaxed income. We assume that the consumer’s other income represents a small share of his maximum wage budget, i.e., that Y < w ( 1 τ ) T 2 . (a) Write the utility maximization problem in the usual form. Max Q , L U ( Q , L ) = AQ 1 / 3 L 1 / 2 subject to pQ + ( 1 τ ) wL ≤ ( 1 τ ) wT + Y Q ≤ L ≤ L ≤ T (b) Draw the feasible set. The graph is shown on Figure 1. (c) On a graph, identify geometrically a segment representing the set of points that could potentially satisfy the Mantra (i.e., be candidates for solution), given what you know about the utility function. (Hint: look at the picture you’ve drawn for (b)) See Figure 2. The gradient of U is defined as: 5 ( U ) = ( U / 3 Q U / 2 L ) 2 FS g 1 g 2 g 3 g 4 Q T L w ( 1 τ )+ Y p T + Y w ( 1 τ ) FIGURE 1. Feasible set FS A B 5 g 1 5 g 2 5 g 3 5 g 4 U ( K , L ) T 5 U Q L w ( 1 τ )+ Y p T + Y w ( 1 τ ) FIGURE 2. Potential solutions according to the Mantra: the segment [AB] (for Q > and L > ), so both components of the gradient of U are strictly positive. Thus the gradient vector of U will point towards the NorthEast. The gradients of the constraints are the following: 5 g 1 = ( p w ( 1 τ )) 5 g 2 = ( 1 ) 5 g 3 = ( 1 ) 5 g 4 = ( 1 ) 3 So we see that the gradient of U will never be in the positive cone of constraints (2), (3) or (4) by themselves. In fact the gradient of U may only be in the positive cone of constraint (1) or in the positive cone spanned by the gradients of constraints (1) and (3) or (1) and (4). This means that without further computations, the set of points candidate for solution according to the Mantra is the segment of points noted [AB] in Figure 2. (d) In your answer to (c), there should be exactly two points at which two constraints are satisfied with equality. Use the Mantra to check arithmetically (by finding nonnegative coefficients that enable you to write the gradient of U as a nonnegative linear combination of the gradients of the two con straints) whether or not the KKT conditions can be satisfied at either of these points.straints) whether or not the KKT conditions can be satisfied at either of these points....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Simon

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