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Answered: - Econ 3102 Section 004 Spring 2016 Problem Set 2 Due April 4,

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Econ 3102 Section 004


Spring 2016


Problem Set 2


Due April 4, 2016



Problem 1: A La?able Claim (50 points)


The hallmark of economic policy for the Reagan presidential administration


(1981-1989) was so-called ?supply-side economics.? The basic idea of this


was that cutting taxes could stimulate economic activity. The hope, in part,


was that the stimulation of economic activity would o?-set lower tax rates


and actually lead to an increase in government tax revenue and thus trim


federal de?cits. This relates to the idea of the La?er curve, which is just a


plot of government tax revenue as a function of the tax rate. This problem


will go through the derivation of the La?er curve in a version of our simple


one period macro model. We will then analyze the claim that lower tax rates


will increase tax revenue.


Consider the following version of our one period macro model. The representative household?s preferences are de?ned over consumption C and labor


N . Their preferences are given by the utility function





N 1+ ?


U (C, N ) = C ?




1+ ?



where ? > 0. The idea of this utility function is that, rather than enjoying


leisure, as the household usually does, it dislikes working. The government


in this model, instead of simply demanding that the household pay a ?xed


amount of taxes T , charges a proportional tax on labor income at a rate ?


where 0 ? ? ? 1. This means that if the household works N hours at a wage


w it must pay taxes equal to ? wN . The household?s budget constaint is thus


C = (1 ? ? )wN + ?





where ? is ?rm pro?ts. The government?s budget constraint is


G = ? wN


Finally, there is a representative ?rm. This ?rm operates a constant return


to scale technology that uses labor N to produce output Y according to


Y =N


There is no capital. Answer the following questions.


i.) For a given tax rate ? de?ne a competitive equilibrium for this model.


ii.) Argue that, in a competitive equilibrium, the ?rm must earn zero pro?ts


(? = 0). Show that in any competitive equlibrium the wage rate w must


equal 1.


iii.) Show that household labor supply in a competitive equilibrium is given




N = (1 ? ? )?


Use this equation and the budget constraint to solve for household consumption C. Show that the household?s utility in equilibrium is decreasing in the tax rate (HINT: Show that for two tax rates ?1 < ?2 the


household can a?ord its equilibrium consumption under ?2 when facing the lower tax rate ?1 while working less than its equilibrium labor


supply under ?2 ).


iv.) We are now ready to de?ne the La?er curve for this model. As stated in


the introduction to this problem, the La?er curve is a plot of government


tax revenue as a function of the tax rate. Government tax revenue in


this model is ? wN . From ii.) we know that w = 1 and from iii.) we


know that N = (1 ? ? )? . This implies that the La?er curve is given by


R(? ) = ? (1 ? ? )?






Notice that R(0) = R(1) = 0. Show that there is a tax rate 0 < ? ? < 1


that maximizes the La?er curve, and show that the La?er curve is


decreasing for tax rates greater than ? ? and increasing for tax rates less


than ? ? . Give a formula for ? ? in terms of ?.


v.) Now pretend that you?re a member of the Reagan administration and


you want to know if lowering tax rates will increase tax revenue. From


iv.), this means that the previous administration would have had to set


a tax rate larger than ? ? . Use your last result from iii.) (that welfare


is decreasing in the tax rate) to make an argument that no government


would choose a tax rate greater than ? ? (HINT: It might be helpful to


try plotting the La?er curve for some values ? in Excel). What does


this tell you about the claims of supply-side economics?



Problem 2: Malthus with Capital (50 Points)


This problem is meant to show that one of the main results of the Malthusian


growth model, that long-run welfare doesn?t depend on the level of technology z, is a result of the fact that growth in output caused by an increase in


the population can?t be used to produce more land T . It does so by replacing


land with capital in the model.


Consider the following variation of the Malthusian growth model. Today


there are N consumers. Consumers each have one unit of time and don?t


value leisure, so all consumers spend their unit of time working. This means


aggregate labor supply today is N . Consumers earn income, which they use


to buy consumption goods and investment goods. They spend fraction s of


their income on investment goods and fraction (1?s) on consumption goods.


This implies that aggregate consumption is C = (1 ? s)Y and aggregate investment is I = sY where Y is GDP. Consumers have children. The number






of children they have is a function of consumption per worker C = N . In






particular, the population tomorrow is given by


N ? = N (1 + gN (?))




where gN is an increasing function. There is a representative ?rm that produces output Y using a CRS technology that uses labor N and capital K,




Y = zF (K, N )


Finally, capital tomorrow is produced using investment goods I and undepreciated capital from today (1 ? ?)K, so that


K ? = I + (1 ? ?)K




Show that in a steady state for this model, capital per worker K =















szf (K)




? gN ([1 ? s]zf (K))













f (K) = F (K, 1)


Note that this implies that steady state capital per worker, and hence steady


state consumption per worker, depends on z.







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