A firm's production function follows ...

Question

A firm's production function follows Q = 2L + 3K

The price of capital is r = 6

If the price of labor is exactly 4, it should always hire as much labor as capital (true / false)

Answer 1

FALSE.

In the scenario where the firm's production function is Q = 2L + 3K, and the prices of labor and capital are 4 and 6, respectively, it's not necessarily true that the firm should always hire as much labor as capital.

To decide the most cost-effective combination of labor (L) and capital (K), we consider the marginal products of each input relative to their costs:

•       Marginal Product of Labor (MPL) = 2 (the coefficient of L in the production function).

•       Marginal Product of Capital (MPK) = 3 (the coefficient of K in the production function).

We compare the ratio of the marginal product to the price for each input:

•       For labor: MPL/Price of Labor = 2/4 = 0.5

•       For capital: MPK/Price of Capital = 3/6 = 0.5

Since the marginal product per dollar spent is the same for both labor and capital, there's no inherent cost advantage in favoring one over the other. The firm should consider other factors like total input budget, availability of resources, and specific production conditions to decide the mix of labor and capital.

Therefore, the statement "If the price of labor is exactly 4, it should always hire as much labor as capital" is false.

The optimal mix of labor and capital depends on factors beyond their prices and marginal products

Answer 2

False, although the question is worded strangely. Two approaches to the answer

1. Intuition. By the production function, labor is always ⅔ as productive as capital no matter how much labor or capital is used. Note that 3 labor and 0 capital produces as much as 0 labor and 2 capital. Also note that labor costs exactly ⅔ as much as capital. That means no matter how much you want to produce, the cost will be the same no matter how much labor or capital you hire. Example: Suppose you need to produce 24 units of output. The following combinations all produce 24 output: 12 L, 0K. 6L, 4K. 0L, 8K. All 3 of the options cost 48 = 12*4 + 0*6 = 6*4 + 4*6 = 0z + 8*6. Thus any combination of (L,K) that produces the required output is cost minimizing since they all cost the same. So it is not REQUIRED to have labor = captital. So FALSE
2. Mathematic. To be cost minimizing, the ratio of the marginal product of the inputs should equal the ratio of the price of the inputs. MPL = 2. MPK = 3 so ratio of MPL to MPK is always ⅔ regardless of the quantity of the inputs. Since the ratio of the input prices is also 4/6 = ⅔, then the ratio fo the MP and prices will be equal regardless of how many inputs are used. Thus in this case, cost minimization does not REQUIRE equal quantites of the inputs. Thus the answer is FALSE