Question

G Housing Supply Assume that housing production is represented by Y = T + L is land and K is capital. Costs are given by C=rL + K

Answer 1

7. Fixing L=1, we have the production function as $\dpi{80} Y = A (0.25 L^{-1} + 0.75 K^{-1})^{-1}$ or $\dpi{80} Y = A (0.25 *1^{-1} + 0.75 K^{-1})^{-1}$ or $\dpi{80} Y = A (0.25 + 0.75 K^{-1})^{-1}$ or $\dpi{80} A (0.25 + 0.75 K^{-1}) = Y^{-1}$ or $\dpi{80} K^{-1} = \frac{1}{0.75 AY} - \frac{1}{3}$ or $\dpi{80} K^* = (\frac{1}{0.75 AY} - \frac{1}{3})^{-1}$ . This is basically the demand for input K.

The cost of production would be $\dpi{80} C = rL + K$ or $\dpi{80} C = r + K$ . The cost function would be $\dpi{80} C = r + K^*$ or $\dpi{80} C = r + (\frac{1}{0.75 AY} - \frac{1}{3})^{-1}$ or $\dpi{80} C = r + (\frac{4}{3 AY} - \frac{1}{3})^{-1}$ .

The marginal cost would be $\dpi{80} MC = \frac{\mathrm{d} C}{\mathrm{d} Y}$ or $\dpi{80} MC = \frac{\mathrm{d} }{\mathrm{d} Y}(r + (\frac{4}{3 AY} - \frac{1}{3})^{-1})$ or $\dpi{80} MC = -(\frac{4}{3 AY} - \frac{1}{3})^{-2}(-\frac{4}{3 AY^2})$ or $\dpi{80} MC = \frac{(\frac{4}{3 AY^2})}{(\frac{4}{3 AY} - \frac{1}{3})^{2}}$ or $\dpi{80} MC = \frac{\frac{4}{3 A}}{(\frac{4Y}{3 AY} - \frac{Y}{3})^{2}}$ or $\dpi{80} MC = \frac{\frac{4}{3 A}}{(\frac{4}{3 A} - \frac{Y}{3})^{2}}$ .