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1. Let (p', y') for t = 1,.... N be a set of observed choices that satisfy WAPM, let YI and YO be the inner and outer bounds to the true production set Y". Let a (p), a(p), and a (p) be profit functions associated with YO, Y, and Y I correspondingly. (a) Show that for all p. a (p) > *(p) 2 #-(p). (b) If for all p, a(p) = "(p) = * (p), what you can say about YO, Y, and Y/? Provide formal arguments. (c) For (p'. y' ) = ([1, 1]. [-3, 4]), and (p3, y') = ([2, 1], [-1, 2]) construct YI and YO (graphically). What can you say about returns to scale in the technology these observations are coming from? Hint: think y = (-c,y). 2. Given the production function f(21, 12, 13) = 17 min (x2, 23)", (a) Calculate profit maximizing supply and demand functions, and the profit function. What re- striction you have to impose on a? (b) Fix y. Calculate conditional demands and the cost function c(w1, w2. (). (c) Solve the problem py - c(un, w2, y) - maxy, do you obtain the same solution as in 20? Explain your findings. 3. Given the production function f(21, 12) = 21 + 29, where b > 0, calculate the cost function c(1, 1, y). How would costs respond to the changes in wi, wa, and y? How would factor demands respond? 4. Consider a firm with conditional factor demand functions of the form (output has been set equal to 1 for convenience): 12 = 1+ dufw5. What are the values of the parameters a, b, c, and d and why? 5. The cost function is c(w1, w2, y) = wjugg. (a) What do we know about a and b? (b) What are the conditional factor demands? What is the production function? (c) What can you tell about returns to scale? 6. Let e(w1, u2. 7) = 2(g) be the isocost and y = f(21, 12) = 7 be the isoquant corresponding to a fixed output level 7 = 1. (a) What are the slopes of these lines? (b) Draw the isocost and isoquant for Cobb-Douglas technology ? = 27 2; corresponding to J = 1. (c) Suppose c(w1, w2, 1) = aw + buy. Draw the isocost and the corresponding isoquant (use the slopes to obtain the shape of the isoquant). (d) Repeat for c(w1, w2, 1) = minfawn, buy}. (e) Draw conditional demand z, as a function of " for 7c and 7d. Hint: If you have trouble in 7c - 7e, think what technology these costs came from.Consider a variation to the baseline Solow growth model without population or technological progress. The per-capita production function is given by yt = f(kt) = kta, where 0