How would you expect an increase in output price, \(P\), to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in \(P\) must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb-Douglas case. c. Use the profit function to show how the presence of inferior inputs would lead to ambiguity in the effect of \(P\) on input demand.

Let's assume that a firm produces output using two inputs: capital (K) and labor (L). The production function can be represented by the isoquant curve. Isoquants represent combinations of inputs that produce the same level of output. When the output price, \(P\), increases, the firm seeks to maximize profits by producing more output. To explain graphically, we can draw an isoquant map showing isoquants for different levels of output. However, since higher isoquants represent higher outputs, as the output price increases and firms seek to produce more, they will move to a higher isoquant. If neither input is inferior, which means that as output increases, the firm will need to use more of both K and L, the firm will move towards a higher isoquant that requires more capital and labor. Consequently, a rise in \(P\) must not reduce the demand for either factor.

Suppose the firm produces output according to the Cobb-Douglas production function: \[F(K,L) = A K^\alpha L^\beta\] where A, \(\alpha\) and \(\beta\) are constant parameters. In the Cobb-Douglas case, the input demand functions for capital and labor are given by: \[K^* = \alpha \frac{w}{r}L^*\] \[L^* = \beta \frac{r}{w}K^*\] where \(K^*\) and \(L^*\) are the optimal demand for capital and labor, and w and r represent the cost of labor and capital, respectively. Notice that both \(K^*\) and \(L^*\) are increasing functions with respect to P. Thus, the graphical presumption from part (a) is demonstrated by the input demand functions in the Cobb-Douglas case.

Let's assume the profit function is given by: \[\pi = PF(K,L) - wL - rK\] where \(\pi\) represents profit, P is the output price, F(K,L) is the production function, w and r are the costs of labor and capital as before. Taking partial derivatives of the profit function with respect to K and L, we get the following conditions for profit maximization: \[\frac{\partial \pi}{\partial K} = P\frac{\partial F(K,L)}{\partial K} - r = 0\] \[\frac{\partial \pi}{\partial L} = P\frac{\partial F(K,L)}{\partial L} - w = 0\] When neither input is inferior, an increase in P leads to an increase in output, which in turn increases the demand for both K and L. However, if one of the inputs is inferior, an increase in output may reduce the demand for that particular input. In that case, we may observe that \(\frac{\partial F(K,L)}{\partial K}\) or \(\frac{\partial F(K,L)}{\partial L}\) is negative. The presence of inferior inputs creates ambiguity in the effect of \(P\) on input demand because the relationship between output and input demand is no longer straightforward. An increase in \(P\) might increase or decrease the demand for a particular input, depending on whether it is inferior or not. Thus, in the presence of inferior inputs, conclusions drawn from graphical analysis in part (a) and Cobb-Douglas input-demand functions in part (b) may not hold.