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Chapter 11: Problem 6

Would a lump-sum profits tax affect the profit-maximizing quantity of output? How about a proportional tax on profits? How about a tax assessed on each unit of output? How about a tax on labor input?

Short Answer

Expert verified
Answer: The profit-maximizing quantity of output will not change for lump-sum profits tax or proportional tax on profits, as these taxes do not affect the firm's marginal cost or marginal revenue. For taxes assessed on each unit of output or taxes on labor input, the profit-maximizing quantity of output will change due to shifts in the marginal cost curve.

Step by step solution

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1. Lump-sum profits tax

A lump-sum profits tax is a fixed amount of tax that is independent of the firm's level of output or profits. Since it doesn't change based on output, it will not affect the firm's marginal cost or marginal revenue. Therefore, it will not change the profit-maximizing quantity of output.
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2. Proportional tax on profits

A proportional tax on profits is a tax that is a percentage of the firm's profits. In this case, the tax will affect the firm's overall profit, but it will not change the firm's marginal cost or marginal revenue. Hence, the profit-maximizing output level remains unchanged.
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3. Tax assessed on each unit of output

A tax assessed on each unit of output is a per-unit tax. This tax will increase the marginal cost of production, shifting the marginal cost curve upward. As a result, the profit-maximizing output level will change. To determine the new profit-maximizing quantity of output, we need to find the new intersection point between the marginal cost curve (including the tax) and marginal revenue curve.
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4. Tax on labor input

A tax on labor input will affect the firm's total cost of production, as it raises the cost of labor. This leads to a higher variable cost that will affect the firm's marginal cost. Consequently, the marginal cost curve will shift upward. Like with the tax assessed on each unit of output, we need to find the new intersection point between the marginal cost curve (including the tax) and marginal revenue curve to determine the new profit-maximizing quantity of output.

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Most popular questions from this chapter

This problem has you work through some of the calculations associated with the numerical example in the Extensions. Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors (GM), who we imagine are deciding between remaining as separate firms or having GM acquire Fisher Body and thus become one (larger) firm. Let the total surplus that the units generate together be \(S\left(x_{F}, x_{G}\right)=x_{F}^{1 / 2}+a x_{G}^{1 / 2},\) where \(x_{F}\) and \(x_{G}\) are the investments undertaken by the managers of the two units before negotiating, and where a unit of investment costs \(\$ 1 .\) The parameter \(a\) measures the importance of GM's manager's investment. Show that, according to the property rights model worked out in the Extensions, it is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, in particular, if \(a>\sqrt{3}\)

Universal Widget produces high-quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production ( \(q\) ) is given by total cost \(=0.25 q^{2}\) Widgets are demanded only in Australia (where the demand curve is given by \(q_{A}=100-2 P_{A}\) ) and Lapland (where the demand curve is given by \(q_{L}=100-4 P_{L}\) ); thus, total demand equals \(q=q_{A}+q_{L}\). If Universal Widget can control the quantities supplied to each market, how many should it sell in each location to maximize total profits? What price will be charged in each location?

The production function for a firm in the business of calculator assembly is given by \\[ q=2 \sqrt{l} \\] where \(q\) denotes finished calculator output and \(l\) denotes hours of labor input. The firm is a price-taker both for calculators (which sell for \(P\) ) and for workers (which can be hired at a wage rate of \(w\) per hour). a. What is the total cost function for this firm? b. What is the profit function for this firm? c. What is the supply function for assembled calculators \([q(P, w)] ?\) d. What is this firm's demand for labor function \([l(P, w)] ?\) e. Describe intuitively why these functions have the form they do.

With a CES production function of the form \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) a whole lot of algebra is needed to compute the profit function as \(\Pi(P, v, w)=K P^{1 /(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)},\) where \(\sigma=1 /(1-\rho)\) and \(K\) is a constant a. If you are a glutton for punishment (or if your instructor is), prove that the profit function takes this form. Perhaps the easiest way to do so is to start from the CES cost function in Example 10.2 b. Explain why this profit function provides a reasonable representation of a firm's behavior only for \(0<\gamma<1\) c. Explain the role of the elasticity of substitution ( \(\sigma\) ) in this profit function. What is the supply function in this case? How does \(\sigma\) determine the extent to which that function shifts when input prices change? e. Derive the input demand functions in this case. How are these functions affected by the size of \(\sigma ?\)

How would you expect an increase in output price, \(P\), to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in \(P\) must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb-Douglas case. c. Use the profit function to show how the presence of inferior inputs would lead to ambiguity in the effect of \(P\) on input demand.
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