Consider the production function: F(L, K)=L0.2K0.7 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? (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 vertical axis and L in the horizontal axis). Use the first and second derivative to show that this curve is decreasing and convex. Provide a graphical representation of the isoquant indicating at least one combination of labour and capital in this curve. (d) Suppose that instead the production function now is F(L, K)=0.2L +0.7K. The wage rate (price per unit of labour) is w = 3 and the capital rental rate (price per unit of capital) is r = 7. Obtain the cost function C* (y), the marginal cost and the average cost. Derive the equation of the isoquant for y = 2 (with K in the vertical axis and L in the horizontal axis). Provide a graphical representation of this isoquant indicating the intersections with the vertical and horizontal axis.
Consider the production function: F(L, K)=L0.2K0.7 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? (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 vertical axis and L in the horizontal axis). Use the first and second derivative to show that this curve is decreasing and convex. Provide a graphical representation of the isoquant indicating at least one combination of labour and capital in this curve. (d) Suppose that instead the production function now is F(L, K)=0.2L +0.7K. The wage rate (price per unit of labour) is w = 3 and the capital rental rate (price per unit of capital) is r = 7. Obtain the cost function C* (y), the marginal cost and the average cost. Derive the equation of the isoquant for y = 2 (with K in the vertical axis and L in the horizontal axis). Provide a graphical representation of this isoquant indicating the intersections with the vertical and horizontal axis.
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.10P
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Transcribed Image Text: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 vertical axis and L in the
horizontal axis). Use the first and second derivative to show that this curve is decreasing
and convex. Provide a graphical representation of the isoquant indicating at least one
combination of labour and capital in this curve.
(d) Suppose that instead the production function now is
F(L, K) = 0.2L + 0.7K.
%3D
The wage rate (price per unit of labour) is w = 3 and the capital rental rate (price per unit
of capital) is r = 7. Obtain the cost function C*(y), the marginal cost and the average
cost. Derive the equation of the isoquant for y = 2 (with K in the vertical axis and L
in the horizontal axis). Provide a graphical representation of this isoquant indicating the
intersections with the vertical and horizontal axis.
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Step 1: State the given information
VIEWStep 2: Identify returns to scale and find MPL and MPK when F(L, K)=(1, 1)
VIEWStep 3: Explain whether the firm uses when F(L, K)=(1, 1) and find output produced by it
VIEWStep 4: Find L and K when y=2, and analyze costs
VIEWStep 5: Discuss the isoquant when y=2
VIEWStep 6: Check the slope and graph it
VIEWStep 7: Derive cost functions with new data
VIEWStep 8: Derive the isoquant when y=2 with new production function
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