You are given the following table for a production process which has two variable inputs:
Capital Labor (L)
(K) 1 2 3 4 5
1 35 60 70 85 95
2 60 70 85 95 105
3 70 85 95 105 115
4 85 95 105 115 125
5 95 105 115 125 135
1. Draw the isoquants corresponding to the following output levels: 60, 70, 85, 95, 105, 115. What returns to scale does the production function exhibit? (Hint: Start from an input combination of L=1 and K=1). What can be said of the MRTSLK?
2. Analyze the marginal productivity of labor when capital is fixed at 3 units. Does it exhibit diminishing, constant, or increasing productivity? Analyze the marginal productivity of capital when labor is fixed at 4 units. Does it exhibit diminishing, constant, or increasing productivity?
Given the production function
Q = 10 KL.5
1. Draw the isoquant for Q = 40 using four points corresponding to the following units of labor: 1, 2, 3, 4.
2. Find the equations for MPL and MPK. Does this production function exhibit diminishing returns to labor? Explain.
3. Find the equations for APL and APK.
1.Find the equation for MRTSLK. What can be said of the MRTSLK we move down the isoquant? Support your answer.
2.Determine whether this production function exhibits constant, increasing, or decreasing returns to scale. Show your computations.
Joe produces a specialized spare part used in the manufacture of tractors. His operation relies more on highly skilled workers rather than on the use of capital. His labor cost is $10,000/unit of labor while capital costs $4,000/unit of capital. A consultant has recently estimated Joe's production function to be:
Q = 100 KL
Joe has just received an order for 1000 units of the spare part.
1. Determine what combination of inputs (L and K) will minimize Joe's cost of production. (Hint: Minimize cost subject to a required output).
a) Solve the problem graphically.
b) Solve the problem using the method of Lagrange. (Show your computations).
2. Joe has a budget of only $40,000 for this order. Determine the maximum output Joe can produce given this budget using the method of Lagrange. Show your computations. (Hint: Maximize output subject to a budget constraint).
1. What will happen to the demand for variable inputs X1 and X2 used to produce output Y if the price of X2 increases and output is maintained at the original level? Illustrate your answer with a graph. Plot X1 in the x-axis and X2 in the y-axis.
2. What will happen to the demand for inputs x1 and x2 if the desired production level decreases but the prices of inputs and output remain the same? Illustrate your answer with a graph. Plot X1 in the x-axis and X2 in the y-axis.
3. A farmer produces wheat (Q1) and sheep (Q2). If the price of sheep decreases, what will happen to the optimal combination of outputs. Illustrate your answer with a graph. Plot Q1 in the x-axis and Q2 in the y-axis.