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Intertemporal Choice
Frederic Lambert
Business Cycles and Stabilization Policies
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Adding a time dimension
So far we have only studied static choices.
Life is full of intertemporal choices: Should I study
for my test today or tomorrow? Should I save or
should I consume now?
We will consider both the households
consumption/saving decision and the firmsinvestment choice.
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Households consumption/savingchoice
We will present a simple model: the Life-
Cycle/Permanent Income Model of Consumption.
Friedman (Nobel winner 1976).
Will allow us to address several key issues: effectsof government programs including Social Security,
government debts and deficits.
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The model
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More on preferences
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Figure 8.2 A Consumers Indifference Curves
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The Budget Constraint - 1 Abstract from labor/leisure tradeoff. Inelastic labor
supply.
Current incomey >0; future incomey >0; initial
wealth a, say inherited from parents.
in the first period, or it can borrow against future
incomey. Interest rate on both savings and on loans is
equal to r. Let s denote saving.
Choice variables: a= wealth at beginning of future
period; c = current consumption; c = future
consumption
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The consumers current-period budget constraint:
The consumers future-period budget constraint:
The Budget Constraint - 2
atysc +=+
srtyc )1(''' ++=
Summing both budget constraints:
That is called the present value budget constraint.
8
weyar
tt
r
yy
r
cc
PV=+
+
+
+=
+
+
1
'
1
'
1
'
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Figure 8.1 Consumers Lifetime Budget Constraint
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Present values
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Present values - Examples
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Implications of the Euler equation
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A parametric example
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Comparative statics: The effects of
changes in income and wealth - 1
The effect on consumption of a change in income
(current or future) or wealth depends only on howthe change affects the PVLR.
An increase in current income:
Increases PVLR, so shifts budget line out parallel toold budget line
If there is a consumption-smoothing motive, both
current and future consumption will increase Then both consumption and saving rise because of
the rise in current income
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Figure 8.5 The Effects of an Increase in Current
Income for a Lender
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An increase in future income
Same outward shift in budget line as an increase incurrent income
Again, with consumption smoothing, both current
Comparative statics: The effects of
changes in income and wealth - 2
Now saving declines, since current income isunchanged and current consumption increases
An increase in wealth
Same parallel shift in budget line, so both currentand future consumption rise
Again, saving declines, since crises and yisunchanged (wealth effect)
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Figure 8.7 An Increase in Future Income
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Figure 8.9 Stock Prices and Consumption of
Nondurables and Services, 19852006
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Comparative statics: Changes in
interest rate Income effect: if saver s > 0, then higher interest rate
increases income for given amount of saving. Increases
consumption in first and second period. If borrower s < 0,
then income effect negative.
of consumption in period 1 to consumption in period 2.
Current c becomes more expensive relative to c. This
increases c and reduces c.
Hence: for a saver an increase in rincreases c and mayincrease or decrease c. For a borrower an increase in r
reduces c and may increase or decrease c.
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Figure 8.12 An Increase in the Real Interest Rate
for a Lender
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Figure 8.13 An Increase in the Real Interest Rate
for a Borrower
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Life-cycle hypothesis - 1
We can extend to Tperiods: Franco Modiglianis life-
cycle hypothesis of consumption
Individuals want smooth consumption profile over theirlife. Labor income varies substantially over lifetime,
startin out low increasin until the 50th ear of a
persons life and then declining until 65, with no laborincome after 65.
Life-cycle hypothesis: by saving (and borrowing)
individuals turn a very nonsmooth labor income profileinto a very smooth consumption profile.
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Figure 4.A.5 Life-cycle consumption, income, and
saving
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Case of Japan
Japanese saving rate fell from 23% of personal income in1975 to 14% in 1990 down to 5% in 2000.
Over same horizon, US saving rate roughly flat around 6%. Why? One reason: aging of the population in Japan.
Ratio of Japanese over age of 65 to those of working agerose rom n o n . orecas o
increase further to 38% by 2010 and 50% by 2020.
Estimates by HSBC that demographic shift can account forhalf of the decline in the savings rate.
Effects of inflation, slower growth rates, changes ingovernment debt are other factors contributing to savingsdecline.
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US and Japan national saving rates
29Source: Chen, Imrohoroglu and Imrohoroglu, 2006
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Permanent income hypothesis - 1
Future labor income is uncertain.
Income of an individual household,y consists of a
permanent part,yp and a transitory partyty = yp + yt
Permanent art : ex ected avera e future income
(usual salary) Transitory partyt: random fluctuations around this
average income (bonus)
Permanent meansy andy change. Transitory: onlyy
changes.
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Figure 8.8 Temporary Versus Permanent Increases
in Income
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Application: Ricardian equivalence
What are the effects of government deficits in the
economy?
A first answer: none (Ricardo (1817) and Barro
(1974)). How can this be?
All that matters is present value of governmentexpenditures and taxation. Timing does not
matter. Deficits today imply higher taxes in future.
The answer outside our small model is tricky.
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Ricardian equivalence
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Households problem
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Figure 8.15 Ricardian Equivalence with a Cut in
Current Taxes for a Borrower
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Empirical evidence
Many instances of temporary tax cuts.
President George H. W. Bush (1992): withholding cut.
Pure timing issue. Little effect on consumption. President George W. Bush (2001): tax rebate. Timing
mixed with reduction in tax rates. Modest increases in
President George W. Bush (2007-08): Stimulus rebate.Seems to have mostly led to increased saving, modestconsumption increase.
In its exact form, Ricardian equivalence fails. Evidence thatconsumption does respond to temporary tax cuts, buteffects not substantial.
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The firms investment choice
Why is investment important?
Investment fluctuates sharply over the business
cycle, so we need to understand investment to
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un ers an e us ness cyc e
Investment plays a crucial role in economic
growth
Production function and law of
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Production function and law of
motion for capital Firms current-period production function:
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Firms future-period production function :
Evolution of the firms capital stock:
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Profits
Firms current-period profits:
Firms future-period profits:
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Present value of profits:
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Firms dynamic problem
r
KdNwYIwNYV
INN +
++=
with1
)')1('''()(max
,',
43
IKdK
NKFzYNKzFY
+=
=
=
)1('
)','(''),(
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The Firms Labor Demand
As before, the firmslabor demand schedule
is the marginal product
of labor for the firm,
44
w c s ownwar
sloping.
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The Representative Firms Investment
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The Representative Firm s Investment
Decision
The firm invests to the point where the
marginal benefit from investment equals themarginal cost.
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Marginal cost of investment
The marginal cost of investment is 1, as the
firm gives up one unit of current profits foreach unit it invests, so:
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Marginal benefit of investment
The marginal benefit of investment is the
marginal product of future capital plus thequantity of capital that will be left in the
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uture a ter eprec at on, a scounte acto the present:
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Optimal investment rule
The firms optimal investment rule, obtained
by equating the marginal benefit andmarginal cost of investment:
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O
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Simplified optimal investment rule:
Optimal investment rule
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Optimal Investment Schedule for the
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Optimal Investment Schedule for the
Representative Firm
50Copyright 2008 Pearson Addison-Wesley. All rights reserved.
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A li ti
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Applications
Housing booms and busts
Investment and the stock market
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Time to build and aggregate fluctuations
H i b d b t
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Housing boom and busts
''
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rIB +=
1)(
Housing boom and busts
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Housing boom and busts
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The Relative Price of Housing
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The Relative Price of Housing
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Residential Construction as a Percentage of
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g
GDP
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Investment and the stock market
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Firms change investment in the samedirection as the stock market: Tobins qtheory of investment
Investment and the stock market
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firm should invest more Tobins q= capitals market value divided by
its replacement cost If q< 1, dont invest
If q> 1, invest more
Investment and the stock market
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Stock price times number of shares equalsfirms market value, which equals value offirms capital Formula: q= V/(pKK), where Vis stock market
Investment and the stock market
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va ue o rm, s rm s cap ta , pK s pr ce onew capital
So pKKis the replacement cost of firms capitalstock
Stock market boom raises V, causing qto rise,increasing investment
Investment and the stock market
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Data show general tendency of investmentto rise when stock market rises; but
relationship isnt strong because many otherthings change at the same time
Investment and the stock market
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This theory is similar what we said: Higher MPKf increases future earnings of firm, so Vrises
A falling real interest rate also raises Vas people buystocks instead of bonds
A decrease in the cost of capital, pK, raises q
Investment and the stock market
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Investment and the stock market
10000
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14000
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Dow Jones Industrial (left scale)
Private investment/GDP (right scale)
Lags and investment
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Some capital can be constructed easily, butother capital may take years to put in place.
Acknowledging that it may take time to get
g
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the business cycle.See Kydland and Prescott (1982), Time tobuild and aggregate fluctuations,Econometrica