Q = L1/2K1/2
so that the marginal product of labor and capital are
MPL = (1/2)(K/L)1/2
MPK = (1/2)(L/K)1/2
If w = $8 and r = $4, determine the necessary conditions for the input choices, K and E to be cost-minimizing. In other words, what is the cost-minimizing ratio of K to E for this firm? Your answer will be in the form of 2L: 5K. You do not need to solve for specific values of L&K.
Answer
In order to minimize cost a firm hires that amount of Inputs(L and K) such that MRTS = Slope of Isocost curve.
MRTS = Marginal rate of Technical Substitution
Total cost(TC) = wL + rK = 8L + 4K.
Let Labor be on horizontal axis and Capital be on Vertical axis.
Then MRTS = Slope pf Isoquant = -MPL/MPK = -(1/2)(K/L)1/2 / [(1/2)(L/K)1/2] = -K/L
Also TC = 8L + 4K and at isocost curve TC is constant. and as we assume Labor be on horizontal axis and Capital be on Vertical axis. Lets Calculate Slope of Isocost curve :
TC = 8L + 4K => 0 = 8 + 4dK/dL
=> dK/dL = -2 --------------Slope of isocost curve
Thus MRTS = Marginal rate of Substitution
=> -K/L = -2/1
=> K/L = 2/1
=> K : L = 2 : 1
Hence, the cost-minimizing ratio of K to L for this firm is 2 : 1
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