Suppose a firms long-run production function is given by. Let the price of labor be w and the price of capital be r.

a) [2 point] Write the condition that must hold at an optimal input combination. That is, find the MRTS and set it equal to the input price ratio. b) [2 point] Suppose a = b. Graph several isoquants for the production function given in part (a). Put K on the vertical axis and L on the horizontal axis. c) [2 point] Suppose w = 2 and r = 1, and that a = b as in the previous part of this problem. Rewrite the MRTS = (w/r) condition from part (a) using this new price information. What must be true about the ratio of capital to labor if the firm is minimizing the cost of producing a given quantity or maximizing production subject to a cost constraint? d) [2 points] Recall that the firms expansion path describes how the mix of capital and labor evolves as production expands (or total cost increases). Use your answer to part (c) to graph the firms expansion path. Show two different optimal input combinations along the expansion path by drawing in their associated isoquant curves and isocost lines (you do not need to find the values of K and L, just locate the optimal input combinations on your graph).