Consider a competitive, closed economy with a Cobb-Douglas production function with parameter α = 0.25. The parameter A is equal to 60. Assume also that capital is 100, labor is 100. Does the production function exhibit constant returns to scale? Demonstrate with examples. Determine if the production function exhibits diminishing marginal returns to capital. Demonstrate with calculus

Consider a competitive, closed economy with a Cobb-Douglas production function with parameter α = 0.25. The parameter A is equal to 60. Assume also that capital is 100, labor is 100. Does the production function exhibit constant returns to scale? Demonstrate with examples. Determine if the production function exhibits diminishing marginal returns to capital. Demonstrate with calculus

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.8P

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Consider the following production function: f (A, B) = gamma multiply A^alpha multiply B^Beta. where A and B are the inputs and alpha, Beta, gamma are in the set (0,1). Let wA and wB the price of the two inputs. Assume wA, wB > 0. Is the production function separable?Does the production function exhibit constant returns of scale?Compute the cost function and the conditional input demand function.How do these three functions react to a change in wA? Suppose the price of both inputs double, what happens to the conditional input demand function? And to the cost function? Suppose the desired level of output double, what happens to the conditional input demand function? And to the cost function?…
Consider 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.
Production is described by the function f(K, L) = AL0.3K 0.3, A > 0.a. Interpret the exponents of the function f( K, L) and the parameter A. b. Explore the effects of scale of this production function. Does the answer depend on A?c. What is the degree of homogeneity of this function? Does the answer depend on A? d. Consider a production function given by F(K, L) = f 2(K, L ). How do the answers to thequestions in b. and c. change?e. Consider a production function given by F(K ,L) = f(K, L) + 2. How do the answers to thequestions in b. and c. change
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Suppose the production function for widgets is given by: Q = f (K, L) = 2 ∗ KL − K2/2 − L2/2 (a) Suppose L=5 (is fixed), derive an expression for and graph the total product of capital curve (the production function for a fixed level of labor) and the average productivity of capital curve. (b) At what level of capital input does the average productivity reach a maximum? How many widgets are produced at this point? (c) Again, assuming L=5, derive an expression for and graph the MPK curve. At what level of capital input does MPK =0? (d) Does this production function exhibit constant, increasing or decreasing returns to scale?
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) = COLc1Cc2 where c0, c1, and c2 are 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 c0 = 5, c1 = 0.25, and c2= 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. = 0 (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 0 < = L < = 3,000 and 0 <= C <= 1,000. Round your answers to the nearest integer when necessary.)________ units at (L, C) = Please use Excel to get answers. Show steps on Excel Please!
Consider a Production Function given by the form z=F(K,L)=(K^(0.3) + L^(0.7) )^2 where K is the amount of capital and L is the amount of labor. Further, we assume that K>0 and L>0. Take the first order partial derivative of the production function with respect to capital and labor. What does the first order partial derivative imply? (Hint: Remember the definitions of the variables)