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Econ 3102 Section 004
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 ?
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
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
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 =
? 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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