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?
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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?
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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 \(a>\sqrt{3}\)
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 \(\sigma ?\)
John's Lawn Mowing Service is a small business that acts as a price-taker (i.e., \(M R=P\) ). The prevailing market price of lawn mowing is \(\$ 20\) per acre. John's costs are given by total cost \(=0.1 q^{2}+10 q+50\) where \(q=\) the number of acres John chooses to cut a day. a. How many acres should John choose to cut to maximize profit? b. Calculate John's maximum daily profit. c. Graph these results, and label John's supply curve.
Young's theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that \(\partial l(P, v, w) / \partial v=\partial k(P, v, w) / \partial w\). Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that \(\partial q / \partial w=-\partial l / \partial P\). Interpret this result. d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied.
The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for \(\$ 30\) per pound. In bad weather it sells for only \(\$ 20\) per pound. Caviar produced one weck will not keep until the next week. A small caviar producer has a cost function given by $$C=0.5 q^{2}+5 q+100$$ where \(q\) is weekly caviar production. Production decisions must be made before the weather (and the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5 a. How much caviar should this firm produce if it wishes to maximize the expected value of its profits? b. Suppose the owner of this firm has a utility function of the form \\[ \text { utility }=\sqrt{\pi} \\] where \(\pi\) is weekly profits. What is the expected utility associated with the output strategy defined in part (a)? c. Can this firm owner obtain a higher utility of profits by producing some output other than that specified in parts (a) and (b)? Explain. d. Suppose this firm could predict next week's price but could not influence that price. What strategy would maximize expected profits in this case? What would expected profits be?
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