(Solved): Exercise 2: For the following production function: \[ y=K^{\alpha} L^{\alpha} \] a. Show whether th ...

Exercise 2: For the following production function: \[ y=K^{\alpha} L^{\alpha} \] a. Show whether the function exhibits constant, increasing or decreasing returns to scale for different parameter values (of \( \alpha \) ). b. Consider the following three parameter values: \( \alpha=1, \alpha=1 / 2 \) and \( \alpha=1 / 4 \). For each of these parameter values, draw two isoquants in a graph, one for \( y=1 \) and one for \( y=2 \). Use the graph to show the returns to scale of the production function. c. Define the long-run cost minimization problem for the firm. d. Obtain the long-run input demands, total cost and average coet all as a function of output \( y \), input prices \( w_{1}, w_{2} \) and technology parameter \( \alpha \). Also, obtain the short-run average costs for \( \bar{K}=1 \). e. Graph the (long-run) average cost functions for all three cases in part (b). They should have a very essential difference. Why? (your answer should point owards returns to scale). For this exercise replace ' \( A \) ' with the last digit of your ASU ID# and 'B' with the second-to-last digit of your ASU ID#. Consider the following production function: \[ y=\left[(A+1) x_{1}^{\rho}+(B+1) x_{2}^{\rho}\right]^{\gamma / \rho}, \quad \text { where } \gamma>0,1>\rho>0 \] a. Determine the relationship between \( \gamma \) and returns to scale. b. Define the cost-minimization problem, obtain the long-run input demand unctions and the total cost function when \( w_{1}=(A+1) \) and \( w_{2}=(B+1) \). Determine the relationship between \( \gamma \) and both the average and marginal ast function. Explain briefly why this is consistent with your answer to (a).