Show that for the constant returns-to-scale CES production function \\[ q=\left[K f+L_{P} Y^{\wedge}\right. \\] a. \(M P_{K}=\left(^{\wedge} | \sim^{P} \text { and } M P_{L}=\left(j-k^{\prime \prime}\right.\right.\) b. \(\quad R T S=[-) \quad\) Use this to show that \(\mathrm{cr}=1 /(1-\mathrm{p})\) \\[ \left.\right|^{K} \boldsymbol{I} \\] c. Determine the output elasticities for Xand \(L\). Show that their sum equals 1 d. Prove that Hence, show Note: The latter equality is useful in empirical work, because in some cases we may approximate \(d q / d L\) by the competitively determined wage rate. Hence, \(a\) can be estimated from a regression of \(\ln (q / L)\) on in \(w\)

The given constant returns-to-scale CES production function is: \\[q = K^f + L_{P} Y^\wedge\\]

We will find the marginal products by taking partial derivatives with respect to K and L. a. The marginal product of capital (MP_K) is given by the derivative of the production function with respect to K: \\[MP_K = \frac{\partial q}{\partial K} = f K^{f-1}\\] b. The marginal product of labor (MP_L) is given by the derivative of the production function with respect to L: \\[MP_L = \frac{\partial q}{\partial L} = P Y^{\wedge - 1} L_P^{P - 1}\\]

To find the returns to scale, we will multiply the production function by a constant factor, say λ, and find the new total output, and compare it to the previous total output. Let the new total output be \\[q' = (\lambda K)^f + L_{P} (\lambda Y)^\wedge\\] We can rewrite this expression as: \\[q' = \lambda^f K^f + L_{P} (\lambda^{\wedge} Y^{\wedge})\\] Now, divide q' by q: \\[ RTS = \frac{q'}{q} = \frac{\lambda^f K^f + L_{P} (\lambda^{\wedge} Y^{\wedge})}{K^f + L_{P} Y^{\wedge}}\\] Since the production function is constant returns-to-scale, RTS must be equal to λ. We are also given that \\[RTS = \frac{1}{(1-\mathrm{p})}\\] Thus, we have shown the required relationship.

The output elasticity for K is given by: \\[\epsilon_K = \frac{MP_K \cdot K}{q} = \frac{f K^{f-1} \cdot K}{K^f + L_{P} Y^{\wedge}}\\] Similarly, the output elasticity for L is given by: \\[\epsilon_L = \frac{MP_L \cdot L}{q} = \frac{P Y^{\wedge - 1} L_P^{P - 1} \cdot L}{K^f + L_{P} Y^{\wedge}}\\] Now, sum up the output elasticities: \\[\epsilon_K + \epsilon_L = \frac{f K^{f} + P Y^{\wedge} L_P^{P}}{K^f + L_{P} Y^{\wedge}}\\] We notice that the numerator is equal to the total output q. Therefore, \\[\epsilon_K + \epsilon_L = 1\\]

The relationship that needs to be proven is given by: \\[\frac{dq}{dL} = \left(\frac{q}{L}\right)^{p-1}\frac{dq}{dK} \cdot \frac{K}{L}\\] First, we find \\[\frac{dq}{dK} = f K^{f-1} = \frac{f K^f}{K}\\] Next, we find \\[\frac{dq}{dL} = P Y^{\wedge - 1} L_P^{P - 1} = \frac{P Y^{\wedge} L_P^P}{L}\\] Now, divide \(dq/dL\) by \(dq/dK\): \\[\frac{dq}{dL} \cdot \frac{1}{\frac{dq}{dK}} = \frac{P Y^{\wedge} L_P^P \cdot K}{f K^f \cdot L}\\] With some rearranging, we get \\[\frac{dq}{dL} = \left(\frac{q}{L}\right)^{p-1}\frac{dq}{dK} \cdot \frac{K}{L}\\] The mentioned relationship has been proven. By following the steps above, we have successfully shown all the necessary relationships and proved the relevant results for the constant returns-to-scale CES production function.