Consider a production function of the form \\[ q=\beta_{0}+\beta_{1} \sqrt{K L}+\beta_{2} K+\beta_{3} L \\] where \\[ 0 \leq \beta_{i} \leq 1 \quad i=0 \ldots .3 \\] a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters \(\beta_{0} \ldots . \beta_{3}\) b. Show that in the constant returns-to-scale case this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree zero. c. Calculate \(\sigma\) in this case. Is \(\sigma\) constant?

To exhibit constant returns to scale, we need to show that doubling the inputs (capital K and labor L) results in a doubling of output q. Let's denote this scaled version of the inputs as \(K' = 2K\) and \(L' = 2L\). Then, we can rewrite the production function with these scaled inputs: \(q' = \beta_0 + \beta_1 \sqrt{K'L'} + \beta_2 K' + \beta_3 L'\) We want this output, q', to be twice the original output q: \(2q = \beta_0 + \beta_1 \sqrt{2K \times 2L} + \beta_2 (2K) + \beta_3 (2L)\) Substituting the original production function, we have: \(2(\beta_0 + \beta_1 \sqrt{KL} + \beta_2 K + \beta_3 L) = \beta_0 + \beta_1 \sqrt{4KL} + 2\beta_2 K + 2\beta_3 L\) Simplifying: \(\beta_0 + 2 \beta_1 \sqrt{KL} + \beta_2 K + \beta_3 L = \beta_0 + \beta_1 \sqrt{4KL} + 2 \beta_2 K + 2 \beta_3 L\) For this equation to hold, we must have: \(\beta_1 = \frac{1}{2}\) So the constraint for constant returns to scale is \(\beta_1 = \frac{1}{2}\).

To show that the marginal productivity functions exhibit diminishing returns and are homogeneous of degree zero, we first need to calculate the marginal product of capital and labor: \(\frac{\partial q}{\partial K} = \frac{1}{2} \beta_1 \frac{L}{\sqrt{KL}} + \beta_2 = \frac{\beta_1 L}{2\sqrt{KL}} + \beta_2\) \(\frac{\partial q}{\partial L} = \frac{1}{2} \beta_1 \frac{K}{\sqrt{KL}} + \beta_3 = \frac{\beta_1 K}{2\sqrt{KL}} + \beta_3\) For diminishing returns, we need the second derivatives to be negative: \(\frac{\partial^2 q}{\partial K^2} = -\frac{\beta_1 L}{4K\sqrt{KL}} < 0\) \(\frac{\partial^2 q}{\partial L^2} = -\frac{\beta_1 K}{4L\sqrt{KL}} < 0\) Since these second derivatives are negative, we have diminishing marginal productivities. Now, to show that the marginal productivity functions are homogeneous of degree zero, we need to show that the functions are invariant under scaling: \(\frac{\partial q'}{\partial K'} = \frac{\beta_1 L'}{2\sqrt{K'L'}} + \beta_2 = \frac{\beta_1 (2L)}{2\sqrt{2K \times 2L}} + \beta_2 = \frac{\beta_1 L}{2\sqrt{KL}} + \beta_2 = \frac{\partial q}{\partial K}\) \(\frac{\partial q'}{\partial L'} = \frac{\beta_1 K'}{2\sqrt{K'L'}} + \beta_3 = \frac{\beta_1 (2K)}{2\sqrt{2K \times 2L}} + \beta_3 = \frac{\beta_1 K}{2\sqrt{KL}} + \beta_3 = \frac{\partial q}{\partial L}\) Thus, the marginal productivity functions are homogeneous of degree zero.

Elasticity of substitution \(\sigma\) is given by: \(\sigma = \frac{d(\ln(\frac{K}{L}))}{d(\ln(\frac{\frac{\partial q}{\partial K}}{\frac{\partial q}{\partial L}}))}\) We first find the ratio \(\frac{\frac{\partial q}{\partial K}}{\frac{\partial q}{\partial L}}\): \(\frac{\frac{\partial q}{\partial K}}{\frac{\partial q}{\partial L}} = \frac{\frac{\beta_1 L}{2\sqrt{KL}} + \beta_2}{\frac{\beta_1 K}{2\sqrt{KL}} + \beta_3}\) Now, we take the natural logarithm of the ratio and differentiate it with respect to \(\ln(\frac{K}{L})\): \(\frac{d(\ln(\frac{\frac{\partial q}{\partial K}}{\frac{\partial q}{\partial L}}))}{d(\ln(\frac{K}{L}))}\) Due to the complexity of the expression, this derivative cannot be simplified, and hence it is not constant. Thus, \(\sigma\) is not constant in this case.