4. Joe has the following utility function:U(x, y) = x2y'. Let I denote Joe's income, Px the price of x, and py the price of y. (a) Set up the Lagrangian function for Joe's utility maximization problem and solve it to derive Joe's optimal demand for x and y. (b) -.... Compute the own price elasticity of Joe's demand for x. If pr increases by 5% what will be the approximate change in Joe's demand for x? Use the elasticity of x to answer this part. (c) ... Compute the income elasticity of Joe's demand for x and y. Based on your answer, what do you deduce about Joe's preferences regarding these two goods._' Suppose Julia currently consumes positive amounts of cheesecake and coffee and her marginal rate of substitution is four slices of cheesecake for one cup of coffee. If the price of a slice of cake is $3 and the price of coffee is $1.5, Julia is maximizing her welfare. The short run cost functiOn for a rm is C(v,w,q,) = tug; + 1112'. The underlying k: production function exhibits constant returns to scale