Marginal Products and Returns to Scale
- IV.A.3: Determine whether a production function exhibits increasing, decreasing or constant returns to scale and whether it is homogeneous of any degree.
- IV.A.4: Determine for each of a production function’s inputs whether there is an increasing, constant or decreasing marginal product.
- IV.A.5: Explain how marginal product and returns to scale differ conceptually.
Suppose there is a perfectly competitive firm with the production function \[f(L,K) = L^\alpha K^\beta\]
Suppose $\alpha = 0.5$ and $\beta = 0.5$ . Remember that the marginal return of an input is the quantity of additional output a firm is able to produce if they increase the quantity of an input by one unit.
The graph above shows a solid blue isoquant curve (bundles of input that yield that same quantity of output) for $\hat{K}=\hat{L}=20$ and a dashed blue isoquant for the bundles that yield twice as much output. The orange dot represents the inputs $\hat{K}=\hat{L}=40$.
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Use the 3D graph on the above left to determine whether the marginal product of labor is increasing, constant, or decreasing.
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Use algebra to determine whether the marginal product of capital is increasing, constant, or decreasing.
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Using 2D graph on the above right, predict if this firm has increasing, decreasing, or constant returns to scale.
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Confirm your answer from (c) algebraically. Is the production function homogenous of any degree? If yes, of what degree?
Now suppose $\alpha = 0.8$ and $\beta = 0.4$, and use the graphs to answer e and f below.
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Is the marginal product of labor increasing, constant, or decreasing?
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Does this firm have increasing, decreasing, or constant returns to scale?
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Confirm your answer to (f) algebraically. Is the production function homogenous of any degree? If yes, of what degree?
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How do the returns to scale and the marginal product of labor differ from what they were when $\alpha=0.5$ and $\beta=0.5$? Why?