ECON 3070INTERMEDIATE MICROECONOMICTHEORYSpring 2009, Homework 3Due Tuesday Feb 24 (prior to commencementof lecture)Carefully show how you derive you answer and be sure to interpret youranswer where necessary.1. Jason’s preferences over peanuts (y) and almonds (x) are given byU (x, y) = xy + 10 (x + y)Suppose the price of a box of peanuts is $3 and the price of a box ofalmonds is $9. Jason can spend a combined total of $30 a month onthese goods.(a) Write down optimization problem solved by Jason. Formulateoptimality conditions.(b) Find Jason’s optimal consumption bundle.2. Uncle Bob purchases two goods, bow ties, B, and tweed sport coats,C. His preferences are represented by the utility functionU (B, C) = B1/4C3/4where B denotes the number of bow ties purchased and C denotes thenumber of sport coats purchased. Let $25 be the price of bow ties and$60 be the price of sport coats. And finally, let I denote Uncle Bob’sincome dedicated to these two goods.1(a) Write down optimization problem solved by uncle Bob. Formulateoptimality conditions.(b) Solve for the optimal consumption of bow ties and tweed sportcoats.(c) Formulate uncle Bob’s Engel curve for bow ties. Your answershould be an equation that gives B as a function of I.(d) Draw uncle Bob’s Engel curve for bow ties on a graph with B onthe horizontal axis and I on the vertical axis.(e) Are bow ties a normal good?3. Suppose that the production function for lava lamps is given byQ = F (L, K) = KL2− L3where Q is the number of lamps produced per year, K is the machine-hours of capital, and L is the man-hours of labor. Suppose K = 600.(a) Draw a graph of the production function over the range L = 0 toL = 500, putting L on the horizontal axis and Q on the verticalaxis. Over what range of L does the output increase? What is therange of man-hours that you would recommend this firm NEVERto hire if K = 600? Why?(b) Derive the equation for average product of labor and graph theaverage product of labor curve. At what level of labor does theaverage product curve reach its maximum?(c) Derive the equation for marginal product of labor. On the samegraph you drew for part b, sketch the graph of the marginal prod-uct of labor curve. At what level of labor does the marginal prod-uct curve appear to reach its maximum? At what level does themarginal product equal zero?4. Consider again the production function for lava lamps: Q = F (L, K) =KL2− L3.(a) Sketch a graph of the isoquants for this production function.(b) Does this production function have an uneconomic region? Whyor why not?25. For each of the following production functions, graph a typical isoquantand determine whether the marginal rate of technical substitution of la-bor for capital, M RT SL,K, is diminishing, constant or increasing alongthe isoquant.(a) Q = F (L, K) = L2+ K2(b) Q = F (L, K) = L√K(c) Q = F (L, K) = L2/3K1/3(d) Q = F (L, K) = 3L +