In Example 6.4, wheat is produced according to the production function: q = 100 (K0.5L0.5). Beginning with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing. (Round your responses to two decimal places.) The MPK at 5 units of capital is 156.52. The MPK at 6 units of capital is 142.89 . The MP, at 50 units of labor is 14.14. The MP, at 51 units of labor is 14. Does this production function exhibit increasing, decreasing, or constant returns to scale?

The following equations represent the long-run production functions for four different technologies, where capital and labor are both variable inputs. For each function, indicate whether it exhibits increasing, decreasing, or constant returns to scale. Clearly show your derivations. (Note that in parts a and c, the exponents are decimals.) f(L,K)= 8L²K4 b. f(L,K)= L+7K c. f(L,K)= L³K d. f(L,K)= 4L+K? а. с.

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Consider the production function: F(L, K) = L0:2K0.7. %3D The wage rate (price per unit of labour) is w = 2 and the capital rental rate (price per unit of capital) is r = 7. (a) Does this production function exhibit increasing, decreasing or constant returns to scale? Explain. What is the marginal productivity of labour and the marginal productivity of capital for (L, K) = (1,1)? Would a firm (which minimises costs) use this combination of labour and capital? Explain. If your answer is yes, then what would be the quantity of production for which the company would use this combination? r (b) Compute the quantity of labour and capital that this firm would use to produce y = 2 at the minimum cost. How much would this cost be? What is the average cost and the marginal cost for that production level? Hint: for this part, you can use directly (without providing the derivation) any results derived in the lecture or tutorials. (c) Derive the equation of the isoquant for y = 2 (with K in the…

The figure shows the production function of a farmer who produces grain by working certain hours each day.     Which of the following statements is correct about this production function?         a. The slope of the production function represents the marginal rate of substitution. b. The ceteris paribus assumption means that the farmer becomes more productive as he works more hours. c. A rightward movement from B along the production function implies that the farmer has better technology. d. This production function exhibits economies of scale. e. A technological improvement would shift up the production function

A farmer has 1 acre of land. If he uses X pounds of fertilizer he can harvest √x tons of wheat. What is his production function if he has A acres of land and can just replicate the same production process on each piece of land? XXX A * A VXA √XA

Consider an economy that produces goods and services according to the production function F (L, K) = 90L Ki,where L represents the number of workers in an economy and K represents the combined units of physical and human capital. The marginal products of labour and capital are: (登) MPL = 60 30(£) MPK = Question 1 Initially: there are 27 workers and 64 units of capital. Part (a): What is the economy's real GDP and GDP per capita? 1st scenario Suppose L increases to 125 while capital remains constant at 64: Part (b): Calculate the economy's real GDP and GDP per capita. Part (c): Relative to the initial scenario (L = 27 and K = 64), how much did MPL change by? 2nd scenario Now suppose L remained constant at 27 workers and capital increased to 216 units: Part (d): Calculate the economy's real GDP and GDP per capita. Part (e): Relative to the initial scenario (L = 27 and K = 64), how much did MPK change by? 3rd scenario Finally, suppose L = 54 and K = 128. Part (f): Calculate the economy's…

2. Consider a Cobb-Douglas production function with three inputs. K is capital (the number of machines), L is labor (the number of workers), and H is human capital (the number of college degrees among the workers). The production function Y = K2/6 L3/6 H1/6 a) Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor? (Hint: The marginal product of labor MPL is found by differentiating the production function (Y) with respect to labor (L)) b) Derive an expression for the marginal product of capital. How does an increase in the amount of human capital affect the marginal product of capital? (Hint: The marginal product of capital MPK is found by differentiating the production function (Y) with respect to capital (K)).

Output is produced according to Q = 4LK, where Lis the quantity of labor input and K is the quantity of capital input. If the price of K is $10 and the price of L is $5, then the cost-minimizing combination of K and L capable of producing 32 units of output is: L = 2 and K = 4. L = 4 and K = 2. L = 1 and K = 8. L = 8 and K = 1. L = 2 and K = 2.