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Why do the short run and long run average cost curves have different shapes?

Suppose a firm’s technology is described by a production function where the quantity produced is equal to the square root of capital times labor (that is, Q = √(L●K)
a. Using graph paper, draw the isoquant for three different levels of output: 2, 4, and 6 units (hint: you will need to find different combinations of L and K that will make Q equal these output values. For example, both L=1, K=4 and L=2, K=2 make Q=2, so both of these combinations of L and K are on the Q=2 isoquant.)
b. Suppose the price of L and K is $3/hour. On your graph, show isocost lines corresponding to total costs of $12, $24, and $36. Using the isoquants and isocost lines, locate three points on the expansion path and draw the expansion path.
c. Does this production function exhibit increasing, decreasing, or constant returns to scale?
d. Using the three points you found on the expansion path, calculate the firm’s long run total and average costs at each of those points. Summarize your calculations in a table and in a graph.
e. Suppose amount of capital the firm uses in the short run is fixed at 4 units. On your isoquant/isocost diagram illustrate the three points corresponding to output levels of 2, 4, and 6. Calculate the firm’s short run total and average costs and draw them in the same graph with the long run costs.
f. Why do the short run and long run average cost curves have different shapes?

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