MFE Macroeconomics Option

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MFE Macroeconomics Option

• Week 7, Lecture 1
• Capital Accumulation

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The Solow model (CS 13, Mankiw)
If we have Constant Returns to Scale, we can
write the production function YF(K,L) as yf(k)
where y is output per head, and k is capital per
head. Assume a fixed saving rate, so
IsY Depreciation implies ?KI-dK In steady
state (and ignoring technical progress), capital
must grow at the population growth rate
n ?K/Knsy/k-dsf(k)/k-d So, for given s, d and
n, this defines the steady state level of capital
per head and output per head
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Standard Solow Diagram
Production and saving functions display
diminishing marginal products (per head) This is
crucial in establishing the global stability of
the model Use this diagram to show that a rise in
s, or a fall in n or d, will raise output per
head in the long run, but will only impact on
growth in the short run.
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In the long run we are..much older

• Suppose we parameterise the model as follows
• Date period annual
• Production function y0.5 k 0.5 (Cobb Douglas)
• Savings rate 0.2
• Population growth rate 0.05
• Depreciation d 0
• Normalise steady state output 1, implies steady
  state capital4
• Simulate 10 increase in savings ratio

years
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What does the model tell us?

• Something about output per head, investment and
  savings, but not long run growth
• Long run growth rate determined by population
  growth and (in simple extensions) exogenous rate
  of technical progress
• In steady state, output per head rises if
• Technology improves
• The savings rate increases
• The population growth rate decreases
• The depreciation rate decreases
• Does this imply more savings are always good?

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Is the steady state in the Solow model optimal?
(CS 13.6)
cf(k)-sf(k) In steady state sf(k)(nd)k cf(k)-(
nd)k Maximum consumption achieved when
dc/dkf(k)-(nd)0

• This is called the golden rule, and defines an
  optimum steady state capital stock
• Only by chance will the saving rate deliver a
  steady state where capital is at this optimum
  level
• In this model, therefore, there is a case for
  government intervention to change savings
  behaviour
• But this raises the question why assume s is
  fixed?
• What would happen if the savings rate was the
  result of an optimisation decision?

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A growth model with intertemporal, optimising
consumers (CS 13.7, Romer, Advanced Macro, Ch2)

• In the two period consumption model, consumers
  smooth their consumption between the two periods
• Consumption leans to the first period if
  consumers are impatient, or if interest rates
  fall
• The generalisation to an optimisation problem
  involving an infinite number of periods, when
  u(c)ln(c), implies
• ?c/cr-?-nwhere ? is the rate of time
  preference i.e. impatience, and n is the
  population growth rate. (If there was technical
  progress, this would also appear.)
• In addition, profit maximising by firms implies
  rf(k). As f(k)lt0 (diminishing marginal
  product of k), then reducing k raises r.
• In steady state, income goes to consumption or
  investment, and investment includes providing
  capital to the new born
• yf(k)c(nd)k

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The Infinite Horizon Modele.g.Romer Advanced
Macro, Ch2
Constant consumption line f(k)?n
consumption
Stable saddle path only path which converges to
equilibrium
Lower k, higher interest rates , rising
consumption
Response to a natural disaster that cuts k
interest rates rise, so consumption jumps down to
saddlepath, allowing more rapid recovery
Lower c, more output can be invested, rising k
Constant capital line f(k)c(nd)k As f(k)lt0,
at some point investment will exhaust output
0
capital
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Intertemporal Consumption the power of
expectations

• With log utility, we can write the
  intertemporally optimal consumption function as
  c(t)?y(t)ßy(t1)ß2y(t2). where we ignore
  initial wealth.
• Suppose income os expected to grow at a rate g.
  This then becomesc(t) ?y(t)1
  ß(1g)ß2(1g)2.
• Using the formula for an infinite sum implies

Suppose ß0.9, and g rises from 3 to 4. With
unchanged current income, this implies an
increase in consumption of about 14!
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Implications

• We can show that decentralised economy is
  identical to social planner allocation
• Real rate of interest achieves socially optimal
  level of saving
• No case for trying to alter savings behaviour
• Steady state is not the golden rule, but this is
  a (optimum) consequence of impatience
• Consumption jumps following any disturbance to
  the stable saddlepath line
• Which implies convergence to steady state faster
  than in fixed savings model, but still involves
  one or two decades at least.

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Some Experiments1. Anticipated improvement in
technology
Consumers anticipate income increase, initially
crowding out capital (note closed economy). When
technology improves, marginal product of capital
rises, raising real interest ratse. This
suppresses initial consumption increase.

• Parameters as earlier, plus
• Rate of time preference initially 0.075 (high)
• 5 improvement in production function , expected
  in five years time

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Some Experiments2. Temporary increase in G

• G initially 20 of output, increases by 5 for
  either 10 or 25 years

Higher G implies higher taxes, but not
permanently, so income effect on consumption less
than positive demand effect of higher G. This
crowds out capital (no Keynesian effects by
assumption), raising interest rates, which also
depresses consumption
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Is the savings rate optimal?

• Previous result depends on a number of
  assumptions
• In particular it assumes infinitely lived
  consumers, or more realistically Barro
  bequests each generation has 2n children,
  cares about their children as much as they care
  about themselves, and leaves bequests to smooth
  utility across generations
• Alternative framework Overlapping Generations
  Models
• If these models involve two periods, then can use
  two period optimal consumption model (assuming no
  bequests) but note each period will involve
  many years.
• If bequests are not allowed, real rate of
  interest only optimal by chance.
• If interest rate too high, economy is
  dynamically inefficient
• If interest rate too low, case for encouraging
  saving but current generation will lose out.
• This analysis is central to debate about how to
  fund state pension schemes
• Nicholas Barr and Peter Diamond The Economics of
  Pensions Oxf Rev Econ Policy 2006 22 15-39

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Discounting and climate change

• Two reasons to discount the future
• Individuals discount future utility
• Impatience, and life chances (but latter need not
  apply in Barro world)
• Future generations will consume more

srtpsocial rate of time preference ?utility
discount rate ?elasticity of marginal
utility ggrowth rate of consumption
Estimates of ? impatience element 0.3 pa. Also
life chances element. Stern Review Ethical
reasons for setting ?0, apart from possibility
of extinction
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What have we learnt?

• Exogenous growth theory tells us something
  about whether national savings might be too high
  or too low.
• It also tells us about how economies in
  disequilibrium may approach their steady state
• However, it tells us rather little about why some
  countries grow fast and others are stuck with low
  (or zero) growth

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Convergence

• If this model applied equally to each economy
  (same technology etc), then all economies would
  tend to converge to the same steady state over
  time. In particular, poor countries would grow
  faster.
• Convergence appears weak across all countries,
  but strong across countries with similar access
  to technology.

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(Conditional) growth convergence among OECD
countries
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Is the Solow framework too pessimistic about the
benefits of investment?

• Recall that in the Solow model, there are
  diminishing returns to investing in capital.
• Alternative the AK model. YAK (CS Ch 14)
• In the AK model, higher savings will permanently
  raise the growth rate of the economy. Possible
  justifications
• Knowledge spillovers
• Individual firms have standard production
  function, but capital investment in one firm has
  benefits for all other firms
• Does produce endogenous growth, but not
  necessarily AK (CS p534)
• Externality that justifies policy intervention
• Human capital
• Research and Development

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Human capital Lucas (1988)
hhuman capital 1-ustudy time youtput kcapital
ktotal capital
Assumption of constant returns in human capital
accumulation required to get AK. Externalities
involved only if there are spillovers
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The problem of unskilled labour (CS 18.3.1)

• Europe Unemployment rose in 1980s and has stayed
  high this unemployment concentrated among
  unskilled
• US Widening of gap between earnings of skilled
  and unskilled
• Krugman (1994) Past and Prospective Causes of
  High Unemployment Kansas Fed Economic Review,
  23-43
• Both due to fall in demand for unskilled workers
  in developed economies
• Is this the result of globalisation/trade, or
  skill-intensive innovation?

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Research and Development

• RD is a non-rival, and (potentially)
  non-excludable good
• It will involve large fixed costs, and low
  marginal costs increasing returns
• Cannot use perfectly competitive set-up
• Romer "Endogenous Technological Change", Journal
  of Political Economy, October 1990.
• Firms use RD to develop new varieties of goods
• Aghion and Howitt "Endogenous Growth", MIT Press,
  1998
• Firms use RD to improve quality of goods
  (quality ladder)
• Creative destruction new goods destroy old
• Plenty of externalities, but is it better to
  create excludability (patents) than to subsidise?
• Imitation or Innovation (CS 14.2.3)

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Growth Regressions

• Empirical analysis too easy danger of data
  mining, lack of robustness. Xavier Sala-i-Martin
  ("I Just Ran Two Million Regressions", American
  Economic Review, Vol. 87, n.2, pp. 178-183,
  1997.)
• Sala-i-Martins summary of evidence
• Its not simple
• Most robust result is conditional convergence
• Its the quality, not the size, of government
  that matters
• Relationship between measures of human capital
  and growth is weak. Correlation with health is
  stronger.
• Institutional factors (free markets, property
  rights, rule of law) are important
• Open economies grow faster