Problem Set 4: Producer Theory Core concept questions Question 1: Cobb-Douglas and returns to scale [35 points] Earlier in this class, you learned that the exponents in the Cobb-Douglas utility function had important economic interpretations. In this question, we will learn about the exponents in the Cobb-Douglas production function. Consider the following Cobb-Douglas production function: le, 1) = #113, where a 2 0 and ,8 2 u. 1. Show that the production function is homogeneous of degree n in inputs. What does n equal? [5 points] 2. Using your answer from part [1), if a + ,8 1? [2 points] 3. Now suppose that o: = 1/3 and ,8 = 1/3. What is the rm's marginal product of labor (MP5)? What happens to the M P; when the lm's use of labor increases? Are labor and capital Edgeworth complements or substitutes? [3 points] 4. The rm's prot maximization problem is manKL} pK wL rK, given market prices to, r, and p. Solve for the rm's factor demand functions L\" (w,r, p) and K\" (m, r, p), and its output supply function y' (w, 'r, p). Solve for the lm's prot function 7r(w,r, p). [15 points] 5. Now suppose that a = 1/ 2 and ,8 = 1/ 2. Solve for the rm's factor demand functions L* (w, r, p) and K'(w,r,p), and its output supply function y* (w,r,p). Did you run into any problems? Why or Why not? [10 points] Question 2: Other production functions [graded for completion] Now we will look at the returns to scale for other common production functions. 1. A Leontief production function is given by f(k,l)=min{,:J}. Does it exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale