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- Suppose the production function for widgets is given by KL – 0.5K2 – 0.1 L2 , where q represents the annual quantity of widgets produced, K represents annual capital input, and L represents annual labor input. (a). Suppose K=5; what is the average productivity of labor (Average product of Labor, MPL) (b). Suppose K=10; at what level of labor input does the total output reach the maximum?100VKL, where q is 4. The production function for puffed rice is given by q = the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour. (a) Calculate the q = 1,000 isoquant for this production function and show it on a graph. (b) If K = 10, how many workers are required to produce q =1,000? What is the average productivity of puffed-rice workers? (c) Suppose technical progress shifts the production function to q = 200 VKL. Answer parts (a) and (b) for this new situation.Suppose that the production function is given as follows:TPL = 10L + 5L2 + L3Find the total product, Marginal product and average product when L = 5.
- Given the following aggregate production function: Y = K0.25 (AL) 0.75, where technology A grows at a fixed rate: = g> 0 (a) Obtain the marginal product of capital algebraically, also discussing the second derivative. (b) Transform the production function into efficiency-worker terms, showing how =ỹ depends AL on =k. K ALK 0.25 (AL) 0.75, where technology A grows at Given the following aggregate production function: Y = a fixed rate: = g> 0 (a) Obtain the marginal product of capital algebraically, also discussing the second derivative. (b) Transform the production function into efficiency-worker terms, showing how =ỹ depends AL K on =k. ALConsider the following production function: q = (KL)“, where a > 0. Answer the following questions: (a) Under what conditions (i.e. values of a) will the production function exhibit decreasing returns to scale? Under what conditions will it exhibit constant returns to scale? Under what circumstances will it exhibit increasing returns to scale? (b) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale. (c) Derive an expression for the cost function of a firm using the production function to produce output of a good. (d) Find the first and second partial derivatives of the cost function with respect to q. Interpret the second partial derivative and relate the sign of the derivative to the returns to scale.
- 2-1 Consider the production function Y = X+4x2 – 0.2x3 a. Derive the exact marginal physical product and average physical product equationsSuppose that a rock star gives concerts. The production function of the concert is the following: Q(C, A,G) = (C – 10)/ªAª/4G+/4 where C represents hourly contribution of guitarist Chris; A represents hourly contribution of guitarist Ataman and G represents hourly contribution of guitarist George.Consider the following production function:q = (KL)^α, where α > 0.Answer the following questions:(a) Under what conditions (i.e. values of α) will the production function exhibit decreasing returns to scale? Under what conditions will it exhibit constant returns to scale? Under what circumstances will it exhibit increasing returns to scale? (b) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale. (c) Derive an expression for the cost function of a firm using the productionfunction to produce output of a good. (d) Find the first and second partial derivatives of the cost function with respect to q. Interpret the second partial derivative and relate the sign of the derivative to the returns to scale.
- Suppose that a rock star gives concerts. The production function of the concert is the following: Q(C,A,G) = (C – 10)'/4A/4G/4 where C represents hourly contribution of guitarist Chris; A represents hourly contribution of guitarist Ataman and G represents hourly contribution of guitarist George. a) If Chris has 26 units of hours, is there increasing returns to scale in production of concerts? b) Assume George has fixed amount of hours, G=16, and hourly wage of G is 5. Denote the wage of Chris= r and the hourly wage of Ataman =w. Derive Rock star's demand for inputs-C and A, respectively-as a function of output (Q). c) Assume that George has fixed amount of hours G=16 and hourly wage of G is 5. Additionally, you are given that the wage of C = r=100 and the unit wage of Ataman=w=100. Show that Rock star's short run cost function is given by TC(Q) = 1000 +50Q²-The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C)=²1C²2 where co. ₁, and care constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co5, c, 0.25, and c₂-0.75. Assume sach unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, devalop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. L, CZO € 70,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0S LS 3,000 and 0 s Cs 1,000. Round your answers to the nearest integer when necessary.) units at (L. C)=((1) q=35L + 40K (2) q=L.5K.5 a) For the production functions of (1) and (2) create the isoquants for q = 100, and for q = 144 with K on the vertical axis and L on the horizontal axis. (Use excel's scatter plot to draw these pictures. So, two pictures with two isoquants each.) b) Suppose that K is fixed at K = 9. Using excel draw the total product curves for equation (1). In a separate excel chart draw the marginal product and average product for K = 9 in equation (1). Give an explanation for the shapes of the total product, marginal product, and average product curve.