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Chapter 12: Problem 1

In a famous article [J. Viner, "Cost Curves and Supply Curves," Zeilschríl fur Nationalokonomie \(3 \text { (September } 1931 \text { ): } 23-46]\), Viner criticized his draftsman who could not draw a family of \(S A T C\) curves whose points of tangency with the U-shaped \(A C\) curve were also the minimum points on each \(S A T C\) curve. The draftsman protested that such a drawing was impossible to construct. Whom would you support in this debate? \\[

Short Answer

Expert verified
Answer: No, it is not possible to draw a family of SATC curves with points of tangency to the U-shaped AC curve at their respective minimum points. The tangency points on the AC curve represent the lowest average total cost but not necessarily at the minimum points of the respective SATC curves.

Step by step solution

01

- Understand the AC and SATC curves

The Average Cost (AC) curve represents the costs of a firm per unit of output. Its shape is usually U-shaped, indicating that at first, average costs decrease as the output expands, reaching a minimum point, and then increase the output. The SATC curves refer to the short-run average total costs at different scales of production. As the firm expands or contracts, the SATC curves shift.
02

- Relationship between AC and SATC curves

To clarify the debate, we need to understand the relationship between AC and SATC curves. The AC curve is the envelope of the family of SATC curves. In other words, for any given level of output, the AC curve represents the lowest possible average total cost.
03

- Visualize the scenario

Imagine a U-shaped AC curve, which represents the firm's lowest average costs at different output levels. The family of SATC curves should show varying shapes following the envelope the AC curve creates. The student can draw a U-shaped AC curve on a graph paper and then try to draw some SATC curves tangent to the AC curve at different points. They'll notice that at the minimum points of each SATC curve, they will not necessarily be tangent to the AC curve.
04

- Reach a conclusion

Based on the understanding of the relationship between the AC and SATC curves, the draftsman is correct. It is not possible to draw a family of SATC curves whose points of tangency with the U-shaped AC curve are also the minimum points on each SATC curve. The tangency points on the AC curve represent the lowest average total cost but not necessarily at the minimum points of the respective SATC curves.

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Most popular questions from this chapter

Suppose the total cost function for a firm is given by \\[ T C=q w^{2 / 3} v^{1 / 3} \\] a. Use Shephard's lemma (footnote 8 ) to compute the constant output demand functions for inputs \(L\) and \(K\) b. Use your results from part (a) to calculate the underlying production function for \(q\)

Suppose that a firm's fixed proportion production function is given by \\[ q=\min (5 K, 10 L) \\] and that the rental rates for capital and labor are given by \(v=1, w=3\) a. Calculate the firm's long-run total, average, and marginal cost curves. b. Suppose that \(K\) is fixed at 10 in the short run. Calculate the firm's short-run total, average, and marginal cost curves. What is the marginal cost of the 10 th unit? The 50 th unit? The 100 th unit?

Suppose that a firm produces two different outputs, the quantities of which are represented by \(q_{1}\) and \(q_{2}\). In general, the firm's total costs can be represented by \(T C\left(q_{1}, q_{2}\right) .\) This function exhibits economies of scope if \(T C\left(q_{1}, 0\right)+T C\left(0, q_{2}\right)>T C\left(q_{1}, q_{2}\right)\) for all output levels of either good. a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two single-product firms producing each good separately. b. If the two outputs are actually the same good, we can define total output as \(q=q_{1}+q_{2}\) Suppose that in this case average cost \((=T C / q)\) falls as \(q\) increases. Show that this firm also enjoys economies of scope under the definition provided here.
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