Question 4 [30 points] a lo: Karthik's preferences are described by the following Cobb-Douglas utility function: U (r, y) = r y , where or 6 (0,1). Let p1 and Pg be the prices for :r and 3; respectively, and m his income. 1. Find Karthik's Marshallian demand functions I; (p3, p,,,m) and y; (pm, p,,,m) by solving his utility maximization problem. Also nd the Lagrange multiplier X\" (pm, pg, m). (Note: you don't have to set up the Lagrangian, but will not receive partial credit if your answer is incorrect.) [10 points] Using your answer from part (1), calculate the numerical values for 22;, (pm, p,,, m), y; (pm, p,,, m), and A\" (pmpy, m) when or = 217,103 = 1, pg = 2, and m = 4. [5 points] If Karthik had a utility function Z(.1;,y) = olog(.r) + (1 or) log(y), where log(u) denotes natural logarithm, would he have the same demand functions? What if the utility function were T(a:, y) = 1%\"y1_\"? Justify all of your answers. (Hint: you don't need to solve the maximization problem again.) [5 points] Using the original utility function U(:r, y) = sawQ, show that the slope of the indilferenoe curve at the bundle (1, 1) is the same as the slope of the indifference curve at the bundle (2, 2). In fact, the slope of the indilferenoe curve is also the same at bundles (3, 3), (4, 4), and so on. To which category of preferences does Cobb-Douglas belong that exhibits this property? [5 points] Using your answer from part (1), write Karthik's indirect utility function, 1:(p_,,, pg, m). [2 points] Suppose Karthik is trying to minimize his expenditure 3932: + pyy, subject to a target utility level [7 = v(pz, py,m) where v(pm, py, m) is the indirect utility function you found in part (5). Using your answer from part (2), nd the numerical values for Karthik's Hicksian demand functions, sf, (103,113., 57) and y; (papaya), and the Lagrange multiplier 11* (mew). (Hint: you don't need to solve the minimization problem.) [3 points]