Final answer:
The input demand functions L(w,r,P) and K(w,r,P) for the production function q=LK.a can be found by taking partial derivatives of the production function with respect to labor (L) and capital (K). If we solve the cost minimization problem first, we aim to minimize the cost of production while producing a given level of output. If things don't seem to work out, it could be due to various factors, such as unrealistic assumptions or errors in the calculations.
Step-by-step explanation:
The input demand functions L(w,r,P) and K(w,r,P) for the production function q=LK.a can be found by taking partial derivatives of the production function with respect to labor (L) and capital (K). These partial derivatives represent the marginal products of labor (MPL) and capital (MPK) respectively. Thus, the input demand functions can be expressed as:
L(w,r,P) = MPL / w
K(w,r,P) = MPK / r
Where w is the wage rate, r is the rental rate of capital, and P is the price of output.
If we solve the cost minimization problem first, we aim to minimize the cost of production while producing a given level of output. This involves choosing the optimal combination of labor and capital to minimize the cost of producing q units of output. In this case, the input demand functions can be determined by solving the cost minimization problem, considering the wage rate, rental rate of capital, and the price of output.
If things don't seem to work out, it could be due to various factors, such as unrealistic assumptions made in the production function or cost minimization problem, or errors in the calculations. It is important to carefully review the inputs, assumptions, and calculations to identify any potential issues.