Macroeconomics
William Scarth
Chapter 11 Questions
1. Consider the two-sector model of economic growth (involving a
manufactured goods sector involving both physical and human capital
and a Cobb–Douglas production function, and an education sector
involving no physical capital, the remainder of the human capital, and a
linear production function). Assume no government in your model, and
assume that capital’s share of income in the manufactured goods sector
is one-third. Assume that there is an exogenous increase in the marginal
productivity of human capital when it is employed in the education
sector. (Perhaps this is due to the invention of overhead projectors for
lectures.)
Explain the equations that make up this model, and derive the effect of
this invention on both the level of the full-equilibrium consumption-tophysical-capital ratio and the full-equilibrium growth rate of
consumption. Show that the former effect must be exactly double the
latter effect.
2. This question gives you the opportunity to consider the effects of one
dimension of an ageing population: that there will be a higher
proportion of the population retired, denoted as p below. Use the
following endogenous growth model (in which K and H denote physical
and human capital, and (for simplicity) there is no depreciation).
Y  K  [(1  p) H ]1
Y / K  r
(1   )Y /((1  p ) H )
C / C  r  
r  (1  p ) w
Y  C  K  H
production function
profit max: K’s marginal product equals interest rate
profit max: H’s marginal product equals wage
utility max: optimal consumption–savings choice
utility max: equal yields on two forms of capital
resource constraint
Use this model to determine how an increase in p shifts the graph
showing how ln(C) grows over time. To do so, assume a balanced
growth equilibrium (C / C  K / K  H / H  n) , define x = C/K, and
determine (dn/dp) and dx/dp).
3. (a) Consider the following two options facing the government. Policy I
raises the level of living standards in an immediate, once-for-all fashion
(by 15 per cent), but it has no effect on the ongoing growth rate of
consumption. Policy II has no immediate consumption-level effect, but it
raises the ongoing growth rate of consumption immediately and
permanently by one-half of 1 percentage point. Assume that, just before
either policy is implemented, the value of total consumption is unity.
Assume also that the rate of time preference of the agents who populate
this economy is 6 per cent, and that the pre-existing growth rate is 2 per
cent. Which policy would you support? Explain why.
(b) When assessing the growth-rate implications of more investment in
education in section 11.5 of the text, we assumed that the annual private
rate of return on education is 7.5 per cent. How much would this
analysis have been affected if we had assumed 12 per cent instead?
Explain your reasoning.