Question

A) Suppose q = min[2K,3L]. Find the short-run cost function.

B) Can total cost ever be lower in the short run than the long run? Explain.

Answer 1

A) To produce 1 unit of output, you need atleast 1/2 units of K and 1/3 units of L. To produce 2 units of output, you need atleast 1 unit of K and 2/3 units of L. You can accordingly reason that to produce q units of output, you need atleast q/2 units of K and q/3 units of L.

Let r show the price of K and w show the price of L. Then 1 unit of output, in requiring 1/2 units of K and 1/3 units of L, costs atleast r/2 + w/3. Similarly, 2 units of output costs atleast r + 3w/2. Finally, q units of output costs atleast q(r/2 + w/3). So, the cost function is as follows.

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The cost function shows the minimum cost required to produce q units of output.

B) Short run costs can never be less than long run costs. Why?

Say there are only two factors of production- K and L. Let K be fixed in the short run and L be variable in the short run. Now, say some combination of K and L, call it (K₁,L₁) allowed to the firm to produce 4 units of output at the cost of $100.

In the long run, the firm can choose any and every combination of K and L to produce the 4 units of output. As you know, it will always choose that combination of K and L to produce 4 units which minimizes its cost. If any combination of K and L can produce 4 units of output for less than $100, the firm will choose that. But in the long run, why will the firm every choose to produce 4 units of output for with a combination of K and L that costs more than $100? It always has the option of using (K₁,L₁), which will allow it to produce 4 units for exactly $100. If every other combination is costlier, it will stick with its short run combination.

So, because the firm can always use its short run (K,L) combination even in the long run to produce q units of output, it will never use a (K,L) combination in the long run that is more expensive than its short run choice.