Suppose the production function for widgets is given by \\[ q=K L-.8 K^{2}-.2 L^{2} \\] where \(q\) represents the annual quantity of widgets produced, \(K\) represents annual capital input, and \(L\) represents annual labor input. a. Suppose \(K=10 ;\) graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? b. Again assuming that \(K=10\), graph the \(M P_{L}\) curve. At what level of labor input does \\[ M P_{L}=0 ? \\] c. Suppose capital inputs were increased to \(K=20 .\) How would your answers to parts (a) and (b) change? d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?

Calculate the first derivative of the average productivity with respect to L: \[ \frac{d(AP)}{dL} = \frac{-0.2(2L) - 10}{L^2}\] Now, set the derivative equal to 0 and solve for L: \[-0.4L - 10 = 0\]

Solve for the value of L that maximizes average productivity: \[L = \frac{10}{0.4} = 25\] Now we need to find the number of widgets produced at this point: \[ q = 10(25) - 0.8(10)^2 - 0.2(25)^2 = 250 - 80 - 125 = 45\] At the level of labor input of 25, the average productivity reaches a maximum and 45 widgets are produced. b. To graph the Marginal Product of Labor (MP_L) curve, we'll find the derivative of the Total Productivity (TP) with respect to L: \[MP_L = \frac{d(TP)}{dL} = 10 - 0.4L\] Now we need to find the level of labor input where MP_L = 0: \[ 10 - 0.4L = 0\]

Solve for L: \[L = \frac{10}{0.4} = 25\] When the labor input is 25, MP_L = 0. c. If capital inputs are increased to K = 20, we need to see how it changes the answers in part (a) and (b). When K =20, Total Productivity (TP) = q = 20L - 0.8(20)^2 - 0.2L^2 Following the same steps as we did for K = 10, we would find the maximum average productivity, and the level of L. Similarly, we would also find the level of labor input where MP_L = 0. d. To determine if the production function exhibits constant, increasing, or decreasing returns to scale, we'll analyze the degree of homogeneity of the production function. The widget production function can be written as: \[q = \alpha K^{\beta_1} L^{\beta_2} - \gamma K^{\delta_1} - \theta L^{\delta_2}\] To analyze returns to scale, we will calculate the sum of the exponents for the inputs K and L: Returns To Scale = β1 + β2 In this case, β1=1 and β2=1, therefore, Returns To Scale = 1 + 1 = 2 Since the sum of the exponents is greater than 1, the widget production function exhibits increasing returns to scale.