Derive the expressions of marginal product

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1. Explain following terms:

Production function, Marginal product, Isoquant curve, TRS, Returns to scale, Isoprofit line, Isocost line

2. For each of following production functions comment on the ability to substitute capital for labor. Note that Q, K, and L denote output, capital, and labor respectively.

A: Q = K + L
B: Q = K0.5 L0.5
C: Q = min{K, L}

3. Justify whether the following statement is true or false. "If the production function is y = f(x1, x2) = x12x22 , then it exhibits constant returns to scale."

4. Justify whether the following statement is true or false. "If the value of the marginal product of labor exceeds the wage rate, then a competitive, profit maximizing firm would want to hire less labor."

Problem 1:

A profit-maximizing competitive firm uses just one input, x, to produce output y. Its production function is:

y = 8x1/2

The price of output is 40$ each unit and the factor price is 8$ each unit.

a. Find profit-maximizing output and the amount of the factor that the firm demands.

b. Find also the maximum profits. (Hint: the marginal product of the factor is 4x-1/2.)

Problem 2:

Weeding a vegetable patch only requires labour as an input. The output of such an activity is the number of square meters weeded per day, defined as Y. The relationship between Y and the amount of labour L is:

Y = 20L1/2

a. Draw this relationship on a graph and label it.

b. Is the marginal product of labour always positive? Does this relationship obey the "law" of diminishing marginal returns?

Problem 3:

A farmer uses two inputs (labour:L, and Fertilizer:F) to produce his only output (corn: C). His production function is : C= f(L,F) = 2L1/2F1/2.

a- Derive the expressions of marginal product of labour MPL and marginal product of fertilizer MPF.

b- Use your answers in (a) to check if the decreasing marginal return law is verified.

c- Let WL be the wage per hour and WF the price per one kg of fertilizer. What is the long run cost-minimization condition for this farmer given his technology of production described above? (4 points)

d- What would be the ratio of labour to fertilizer knowing that WL = $8 and WF = $4
e- What are the optimal amounts of labour and fertilizer that this farmer should use to produce 1kg of corn?

f- What would be the cost of producing 1kg of corn?

g- In general, given his production function, if the farmer doubles the amount of the inputs by how much would the output be multiplied? What does this tell you about the returns to scale?

h- Is the answer in (g) in contradiction with the answers found in (b)? Clearly explain.

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