Macroeconomics
William Scarth
Chapter 10 Questions
1. One aspect of an ageing population is a drop in the population growth
rate (since elderly people do not have babies), and this question allows
you to examine the effect of this development on average living
standards, as predicted by a ‘new’ (micro-based) exogenousproductivity-growth-rate model. The model is defined by the following
equations:
utility max: Ramsey consumption with
population growth rate θ
economy’s resource constraint
k  f (k )  c  (   )k
f (k )  r  
profit maximization
c  (r   )c  k
The model involves three endogenous variables: c and r (jump
variables) and k (a sticky-at-point-in-time variable). You are to derive a
phase diagram in c-k space, and to use that analysis to determine the
effect on living standards of a drop in θ. You should focus on solutions of
the model that involve positive values for all endogenous variables, and
you should assume that (both before and after the drop in θ) the
interest rate exceeds the population growth rate (r > θ). Finally, assume
that f   0; f   0. Fully explain both the short-run and the long-run effect
of the lower population growth rate on material living standards.
2. In 2010, the Mirrlees task force on tax reform in the UK issued its report.
One recommendation concerning the personal income tax system was that
the cost of generating income should not be taxed. Since forgoing
consumption to save can be thought of as the cost of acquiring future
income, the report argued that the risk-free return on saving should not be
taxed. Use analysis that you have learned in this chapter to assess this
recommendation.
3. This question concerns the environment (part (a)) and nonrenewable resources (part (b)).
(a) If we are to be more active in pursuing a cleaner environment, we
will have to withhold more of each year’s output from both
consumption and investment (in traditionally defined capital). More
current output will be needed to install such things as ‘scrubbers’ on
smoke stacks (to ensure that contaminants do not pollute the
atmosphere). Many individuals are concerned that this commitment
may lead to lower levels of consumption (material living standards as
traditionally measured). Use the following micro-based, exogenousgrowth, small-open-economy model to examine this issue:
C  ( r *   )C  p( p   )( K  A)
  r * A  (F(K)  C  G)
A
To simplify notation, since labour supply is constant, no labour term is
included in the production function. There is perfect international
capital mobility, so this economy’s interest rate equals the constant
foreign yield, r*. Profit maximization ( F ( K )  r*) means that perfect
capital mobility fixes the level of capital (and, therefore, overall output
as well). Thus, K is an exogenous constant. The first equation is the
standard consumption function (p and  stand for the death probability
and the rate of time preference). A denotes the amount of domestically
employed capital that is owned by foreigners. (This is why domestic
wealth in the first equation is (K − A)). The second equation is the
accumulation identity for A: foreigners increase their ownership of
domestically employed capital whenever the interest payments that this
economy must pay to foreigners (as rent for using the pre-existing level
of foreign-owned capital) exceed that period’s export earnings. (Exports
equal output minus the sum of domestic consumption and government
spending on environmental protection.)
Assume zero underlying economic growth. This two-equation model
determines the time paths for C and A, given exogenously determined
values for the other variables and parameters. In your analysis, assume
that foreigners do not own all the capital, so that K > A. Use the two
equations to draw a phase diagram in C-A space (C is the jump variable,
while A adjusts sluggishly through time.) Use your phase diagram to
determine how the time path for consumption is affected by a once-for-
all increase in the level of government spending on environmental
protection. Explain your reasoning.
(b) Explain Hotelling’s analysis of the determination of the time path for
the price of a non-renewable resource. Explain why this analysis has led
some economists to be complacent about the dwindling supply of nonrenewable resources. Explain what this analysis says pro-savings
initiatives (such as a reduction in the tax rate applied to interest
earnings) will do to the rate at which we use up our non-renewable
resources. Finally, explain why, despite Hotelling’s analysis, a tax on the
use of non-renewable resources can be defended (on efficiency
grounds). Explain why such a tax can be viewed as preferred to a payroll
tax (on efficiency grounds).