Consumption

1
Consumption
2
Utility Functions
Economic agents are assumed to have utility
functions which represent their preferences over
consumption C. They are assumed to be ordinal
and provide rankings among alternative
consumption choices. Any function U which has a
positive first derivative and negative second
derivative can be a utility function. A positive
first derivative says that more is preferred to
less or that marginal utility is positive. A
negative second derivative is the property of
diminishing marginal utility and makes the
function concave. The amount by which utility
goes up with an additional unit of consumption
decreases as you have more consumption.
Intuitively, the first loaf of bread tastes much
better to a starving man than to one whose belly
is full.
Two examples of utility functions are ln(C) and
the quadratic, aC-ßC²,with a2ßC.
3
Smooth Consumption is Best
Suppose the agent has 8 units of consumption to
divide between two periods. We can show that with
diminishing marginal utility, you will prefer to
divide the consumption equally between the two
periods. We graph the utility function Uln(C).
We can see that since U(5)-U(4)desire to allocate away from periods with high
consumption to ones that are low.
4
Diminishing Marginal Utility
The loss in utility going from 4 units of
consumption to 3 units, 1.396-1.0990.297, is
greater than the gain from going from 4 to 5,
1.609-1.3960.213. The desire for smooth
consumption will provide us reasons to hold
insurance (life, auto, etc.) and save for
retirement.
5
Life Cycle Theory
Consider a person who expects to live for NL
years and earn income for WL years, and be in
retirement for (NL-WL) years. A person will earn
labor income of YL annually will earn over his
working life, WLYL in wage income. This income
has to last him into his retirement years as
well. In his youth, he lives off his parents. It
can be shown rigorously that an individual would
desire to have a smooth consumption profile. The
basic idea is one of diminishing marginal utility.
6
Optimal Consumption
Planned consumption per year is then proportional
to labor income
In the previous section's example, NL2, WL1,
and YL8, so C4.
Now consider a more detailed example. Let a
person begin working at 0, retire at 50, live to
62.5, and earn 25,000 a year. His total resources
are
(WL50)(YL25,000)1,250,000
What is his consumption in any given year,
WL/NL50/62.525,00020,000.
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Graphical Life Cycle
8
Assets
The maximum level of assets is achieved precisely
at the end of the working life. From our example
(C-YL)WL(25,000-20,000)5020,00012.5250,000,
which finances 12.5 years of retirement
consumption at 20,000 a year. Now let's
introduce assets into the model. Let WR represent
the agent's stock of wealth. His lifetime
consumption possibilities are at some point T
C(Age at death-T)WR(Age at retirement-T)YL.
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Adjustment of Consumption
From our numerical example, with T40, he has
saved 40(25,000-20,000)200,000, plus we will
give him a bequest of 50,000 for WR250,000.
Total consumption over the remaining 22.5 years
is 500,000, or 22,122 a year. The fractions
coming from wealth and income are
CaWRcYL
where a1/(age of death -T) and c(age of
retirement-T)/(age of death-T), so from our
example a1/22.5, and c10/22.5.
From our numerical example, with T40, he has
saved 40(25,000-20,000)200,000, plus we will
give him a bequest of 50,000 for WR250,000.
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Simple Extensions
This model is of course very simple. It rules out
most importantly any uncertainty over date of
death or retirement. You spend your last penny to
seal your coffin shut. Also missing is bequest
behavior. You don't consume your last penny when
you have heirs. How might fluctuations in the
stock and bond markets affect consumption in this
model? Are expectations important?
11
Application Stock Crashes
On October 19, 1987 the Dow Jones Industrial
Average (DJIA) fell 508.32 points to close at
1,738.40. This was a drop of 22.6 in the index.
The magnitude of the 1987 stock market crash was
much more severe than the October 28, 1929 crash
(12.8), the 554.26 point drop of October 27,
1997 (7.2) or the record 617.78 drop of April
14, 2000 (5.7). The loss to investors in 1987
amounted to US500 billion. While the 1929 crash
is commonly believed to have led to the Great
Depression, the 1987, 1997 and 2000 crashes
seemed to have no lasting effect on the real
economy.
12
Stock Crashes and Consumption
The life cycle model highlights is whether these
changes were permanent or transitory. In 1929,
stocks continued to fall while in 1987, 1997 and
2000, they rebounded. On October 29, 1929, the
Dow Jones fell another 11.7. The DJIA fell 89
from its September 29, 1929 peak of 379.61 to its
July 8, 1932 trough of 41.22. The Dow did not
return to its 1929 peak until November 23, 1954!
By contrast, stocks returned to their pre-1987
crash levels by January 24, 1989. The index rose
337.17 points on October 28, 1997, the day after
the crash, a gain of 4.7, and it crossed its
pre-crash high on February 6, 1998. This past
year, the Dow rose 276.74 points on April 17,
2000, the first trading day after the crash. The
pre-crash close was reached a mere 6 trading days
later. Investors continue to "buy the dips"
13
Liquidity Constraints
Most people's income patterns are not so smooth.
The life cycle theory then predicts that people
will save more in years that their income is
high. Assume our agent earns 10,000 for 15 years,
20,000 for 15 years, and 40,000 for 20 years, for
the same total income of 1,250,000. He still
desires to consume 20,000 every year but now he
can't, at least not until he has worked for 15
years! The agent's consumption now has the
"humped" profile. The problem is, when you most
need to borrow, you can't.
Discussion Why can't you borrow against future
income?
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Liquidity Constraints Example
Numerical example WL3, YL(YL1,YL2,YL3), NL4.
Let C be the optimal (smooth) consumption
enjoyed by an unconstrained consumer. We have a
new consumption rule. C C if CYL, CYL if
CYL.
To calculate C, we compute lifetime resources
So in period 1, the consumer is constrained and
chooses C110. In the second period, he
reoptimizes
so he is no longer constrained.
15
Can Fiscal Policy Help?
These liquidity constraints can influence the
effects of government policy. Consider the tax
and transfer program TR(5,0,0,0), TA(0,0,5,0),
a transfer payment of 5 in period 1 and a tax of
5 in period 3. Introduce an unconstrained
consumer Ms. B with YL(20,20,20,0). If she
receives a transfer in period 1, she merely saves
and repays it as tax in period 3. Mr. A however
loves the new policy. His after tax/transfer
income is (15,20,25,0) and he now enjoys a
unconstrained consumption throughout his
life.can't.
Ironically, the U.S. Social Security system works
precisely in the opposite direction of yuppie
desires.
16
National Savings Rate
Savings must come from one of four sources (1)
Households (2) Business (3) Federal government
(4) State and local government.
Unfortunately, both households and the government
now have negative savings.
17
Personal Savings Rate
Turns negative in 2005!
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Consequences?
What, if any, are the consequences of the decline
in national saving. In the short run, maybe not
very much at all. We can borrow from abroad when
are savings is inadequate. The long run
consequence is growing indebtedness to foreign
investors. We may be mortgaging your future!
Lets look at other countries
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Draw Your Own Conclusion
You can do your own analysis using the
standardized estimates of GDP compiled by Robert
Summers and Alan Heston of the University of
Pennsylvania that account for price differences
between countries.