Diminishing Marginal Rate of Technical Substitution Assignment Help
1.A cost-minimizing firm’s production function is given by Q = LK, where MPL = K and MPK = L. The price of labor services is w and the price of capital services is r. Suppose you know that when w $4 and r $2, the firm’s total cost is $160. You are also told that when input prices change such that the wage rate is 8 times the rental rate, the firm adjusts its input combination but leaves total output unchanged. What would the cost-minimizing input combination be after the price changes?
2.A firm has the production function Q = LK. For this production function, MPL = K and MPK = L. The firm initially faces input prices w $1 and r $1 and is required to produce Q = 100 units. Later the price of labor w goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the firm’s price elasticity of demand for labor over this range of prices.
3.Suppose the production of airframes is characterized by a Cobb–Douglas production function: Q LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.
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4.A plant’s production function is Q = 2KL+ K. For this production function, MPK = 2L +1 and MPL = 2K. The price of labor services w is $4 and of capital services r is $5 per unit.
a) In the short run, the plant’s capital is fixed at K = 9. Find the amount of labor it must employ to produce Q = 45 units of output.
b) How much money is the firm sacrificing by not having the ability to choose its level of capital optimally?
5.Contrast the production functions given below:
a. Suppose that the production function faced by a 30-weight ball bearing producer is given by Q = 4 K0.5L0.5, where MPK = 2K– 0.5L0.5 and MPL = 2K0.5L– 0.5. Do both labor and capital display diminishing marginal products? Does the production function display a diminishing marginal rate of technical substitution?
b. Suppose that the production function faced by a 40-weight ball bearing producer is given by Q = 4KL, where MPK = 4L and MPL = 4K. Do both labor and capital display diminishing marginal products? Does the isoquant you drew in (a) display a diminishing marginal rate of technical substitution?
c. Compare your results. Must labor and capital display diminishing marginal products in order for the MRTS to diminish?
6.a)Isoquants can be convex, linear, or L-shaped. What does each of these shapes tell you about the nature of the production function? What does each of these shapes tell you about the MRTS?
b)Can an isoquant ever slope upward? Explain.
c)Explain the term “marginal rate of technical substitution.” What does a MRTS = 4 mean?
7.a)What is the marginal rate of technical substitution? What does it imply about an isoquant’s shape?
b)What does the curvature of an isoquant imply about the two inputs, capital and labor?
c)What is an isocost line? What does its slope tell us about the relative cost of labor and capital?
d)How will a firm react to an increase in the price of one input relative to another?
8.Suppose a firm’s production function initially took the form Q = 500(L + 3K ). However, as a result of a manufacturing innovation, its production function is now Q = 1,000(0.5L + 10K ).
a) Show that the innovation has resulted in technological progress in the sense defined in the text.
b) Is the technological progress neutral, labor saving, or capital saving?
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