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Transcript of Consumption 0
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Work Effort, Production and Consumption
We will analyze the economys responses to changed opportuni-ties in terms of wealth effects and substitution effects.
Production Technology
Assume a simple production function whose only input is labor.
Each household plays the role of a producer as well as a con-
sumer. The production function is assumed to be
yt = f(lt) (1)
where f is the production function, y output and t time. A
single commodity will be produced and consumed fully, there are
no inventories and the product is perishable. Assume yl > 0 andyll < 0.
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Taste for Consumption and Leisure
Assume no trade between households. Hence, consumption,ct,will be equal to total production.
ct = yt = f(lt), (2)
A household can increase his consumption by inputting more
labor but will have less leisure time. Assume that both leisure
and consumption brings happiness:
ut = u(ct, lt) (3)
where u denotes utility with uc > 0 and ul < 0.
Somebody who is receiving more consumption would be happyto supply the extra labor.
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Deciding How Much to Work
Each household maximizes his utility function subject to con-sumption is equal to production. If MPL exceeds the slope of
the utility curve, then we know that he will trade his labor for
more consumption which will take him to a higher utility level.
Shifts in the Production Function
A change in production opportunities change induces wealth and
substitution effects. If a change of production opportunities al-
lows one to consume more of both commodities, then wealth is
increasing. A change in the relative costs of commodities allow-
ing one to shift consumption from one to another commodity
leads to substitution effects. Wealth and substitution effectswork in opposite directions for some goods.
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Wealth and Substitution Effect
Let production function shift upward in a parallel manner, mean-ing that consumption increases for a given labor input and no
change in the shape of the function at each level of work, i.e.MPL is not changing. A household would respond by consuming
more of both goods.
In contrast, if MPL changes, i.e. there is a twist in the function,
the household would be willing to trade labor hours for consump-tion. For example, households may want to work more hours formore consumption. In this case, the total effect will be unknown
for labor: depending on the strength of wealth and substitu-tion effect labor can increase or decrease. But consumption will
increase regardless.
New Vocabulary
Superior goods (same as normal goods) and inferior goods.
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Overview
Now we allow consumers to buy and sell commodities. We in-troduce money and credit markets; households can lend and bor-
row. Hence, people will not experience substantial variations in
their consumption patterns even their incomes vary greatly from
period to period (goods cannot be stored). The interest rate
determines the cost of borrowing and return to lending.
The Commodity Market
People specialize in production and consume little of what they
produce and sell the rest to purchase other goods and services.
For simplicity, we will assume only one type of good to be pro-
duced in our economy using only one type of inputlabor.
yt = f(lt) (4)
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Money and the Price Level: The household holds money, mt,during period t. The aggregate quantity of money is Mt, andit does not change over time. The price of the good P is alsoconstant. Each household is assumed to be small so that itcan buy or sell any amount of goods without influencing theestablished price; perfect competition in commodities market.
The Credit Market: The household can borrow or lend through
a credit market. Lender receives a bond; an interest payment,R, in addition to the principal when the bond matures. Forthe issuer of the bond R is the per period cost of borrowing.All bonds are alike. Any household is small to buy or sell any
amount of bonds without effecting the interest rate; perfectcompetition in the credit market. The number of bonds, bt, heldby a household can be positive or negative. The total of bondsby all households, Bt must be zero. There is no government orfinancial institutions, foreigners.
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Mechanics: If bt1 is the dollar amount of bonds held at time
(t 1) by a household. At time
the will receive (1 +
R)
bt1.
Receipts are positive for lenders and negative for borrowers; the
aggregate interest and principal payments for period t is zero.
Someones savings is measured as (bt bt1); since Bt = Bt1 =
0, aggregate savings in bonds Bt Bt1 will be zero in each
period.
Total financial assets of a household is the sum of money and
bonds mt + bt. The stock of financial assets always equal Mtfor Bt = 0. We can define the change in an individuals financial
assets, (mt + bt) (mt1 + bt1), which is the total amount an
individual saves during period t. Given our assumptions, the
aggregate of total savings is zero at all points in time (until weintroduce investment in a later chapter).
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Budget Constraints for one Period
The dollar value of income from production is P yt. Interest
income received from the bond market will be Rbt1 positive for
lenders and negative for issuers of bonds and no interest income
from holding money. Purchases of commodity costs P ct. A
household can exchange money for bonds to achieve the desired
composition of assets between bonds and money. Thus for ahousehold at time t we can write
P yt + bt1(1 + R) + mt1 = P ct + bt + mt (5)
We can rearrange this equation to get
(bt + mt) (bt1 + mt1) = P yt + R bt1 P ct (6)
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Budget Constraints for two Periods
Using equation (5), budget constraint for period 1 will be
P y1 + b0(1 + R) + m0 = P c1 + b1 + m1. (7)
For simplicity assume m0 = m1 and get
P y1 + b0(1 + R) = P c1 + b1. (8)
For period two this equation becomes
P y2 + b1(1 + R) = P c2 + b2. (9)
Using equation (9) solve for b1
b1 = P c2/(1 + R) + b2/(1 + R) P y2/(1 + R),
and substitute in (8) to obtain
P y1 + P y2/(1 + R) + b0(1 + R) = P c1 + P c2/(1 + R) + b2/(1 + R)(10)
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The Households Budget Line
Express equation (10) in real termsc1 + c2/(1 + R) = y1 + y2/(1 + R) + b0 (1 + R)/P b2/[P (1 + R)] (11)
Suppose that RHS is fixed to some amount x, then c1 + c2/(1 +
R) = x; for a given quantity x, a household can change todays
consumption, c1, by making adjustments in next periods con-
sumption, c2. The budget line will therefore show all possible
combinations of c1 and c2. The slope of the budget line is
(1 + R). We can interpret interest rate, R, as the premium in
future consumption for saving today rather than consuming.
To complete the discussion on consuming now versus later, we
will next discuss the households preferences for consumptionover time.
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Preferences for Consuming Now versus Later
We want to investigate households choice for c1, c2, l1, l2. Ifthe household has a lot of c1, exchange a unit of c1 in return
for a small increase of c2. Optimal consumption will be achieved
where the budget line is tangent to the indifference curve where
the premium from saving more just balances the willingness to
defer consumption.
Wealth Effects on Consumption appear as shifts in the total
present value of expenditures, x, due to a shift in the production
function
x = c1+c2/(1+R) = y1+y2/(1+R)+b0(1+R)/Pb2/P(1+R)
(12)leads to higher consumption.
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The Interest Rate and Intertemporal Substitution
If interest rates rises to R, the new budget line will be steeper
than the old one. Given we held the overall spending at x, and
draw a new budget line passing through old (c1, c2), the new op-
timal choice, (c
1, c
2), will allow people to raise future consump-tion relative to current consumption by inducing households to
save a larger fraction of current income; a rise in R motivates
households to save more today in return for higher consumption
tomorrow. This is called intertemporal substitution effect.
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Choosing Work Effort at Different Dates and Wealth Effect
If the production function shifts up parallel for periods 1 and 2,
for a given amount of effort l1 and l2, the real sources of fundsis increasing, thus consumption in both periods increases, too.
We know that leisure increases if wealth increases. Hence weexpect that l1 and l2 would decrease.
Choosing Work Effort at Different Dates and Interest Rates
An increase in R reduces current consumption and raise next pe-riods consumption. Todays work effort increases relative to
next periods. Hence, an increase in interest rates motivatespeople to substitute away from this periods leisure raising l1.
Overall, when R raises, todays consumption declines comparedto the next periods. Todays labor, l1, increases relative to next
periods l2. These reflect the positive response to return fromsavings.
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Budget Constraints for an Infinite Horizon
Modify equation (10) to discuss the infinite horizon case as be-low:
P y1 + P y2/(1 + R) + P y3/(1 + R)2 + + b0(1 + R)
= P c1 + P c2/(1 + R) + P c3/(1 + R)2 + (13)
The sums no longer terminates and also the final stock of bonds
does not appear in the equation. Dividing both sides by P, the
real budget constraint becomes
y1 + y2/(1 + R) + y3/(1 + R)2 + + b0(1 + R)/P
= c1 + c2/(1 + R) + c3/(1 + R)2 + (14)
Choices Over Many Periods
Lets assume a given value of total income and investigate the
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consumption behavior.
x = c1 + c2/(1 + R) + c3/(1 + R)2
+ = y1 + y2/(1 + R) + y3/(1 + R)
2 + + b0(1 + R)/P(15)
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The Interest Rate and Intertemporal Substitution: An in-
crease in R motivates people to take less leisure in one periodrelative to that in the next period. Likewise, consumption in one
period will be lower than in the next period. Higher R motivates
people to save.
Wealth Effects: Wealth effects arise from changes in the present
value of real income from the commodity market; the production
function is shifting up parallel to the original one.
Permanent Shifts of the Production Function
If there is a permanent shift of the production function for all
periods, then we can think that x will increase. Thus ct willincrease for all periods. Also labor, lt will decrease.
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Marginal Propensity to Save and Consume: If yt increasesby one unit each period and if ct increases by less than one unit,
then the mpc will be positive but less than one and the mps willbe positive. Households use this extra saving in future during
which the increase in consumption will be more than one unit.Generally, if the change is permanent, we expect that mpc will
be close to one and mps will be near zero.
Temporary Shifts of the Production Function: Change inreal income occurs once only. Households spread this over their
life time, and consume very little of the extra income and savethe rest. Thus mpc will be small and mps will be high.
Friedmans (1957) permanent income model says that consump-
tion depends on a long-term averages of income. If change is
temporary, then permanent income and consumption changesvery little. Hence, mpc will be small if change is temporary.
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Wealth Effects from an increase in the Interest Rate
Given the budget constraint, y1 +y2/(1+R)+ +b0(1+R)/P =
c1 + c2/(1 + R) + , if spending falls more, the household can
save; i.e wealth increases and vice versa. Lenders observe an
increase in wealth and borrowers have a decline in wealth. In the
aggregate, the wealth effect is nil: neglect wealth effects but
focus on intertemporal substitution effects.
An Increase in the Schedule for Labors Marginal Product
Work and real income will increase by equal amount in each
period. In this case mpc would be close to one and saving will
not change.
If the increase in productivity is temporal, people work today
more to expand leisure or consumption tomorrow. Thus, tem-porary improvement of productivity stimulates saving.
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Consumption under certainty: PIH Consider a consumer who
lives for T periods whose lifetime utility is
U =T
t=1
u(Ct), u() > 0, u() < 0 (16)
where u() is the utility function and Ct is consumption in period
t. The individual has a given initial wealth of A0 and laborincomes of Y1, Y2, . . . , Y T in the T periods of his life. Individualcan borrow or save at an exogenous interest rate, subject to
the constraint that any outstanding debt be repaid at the end
of his life. For simplicity interest rate is set to 0. The budget
constraint will be
Tt=1
Ct A0 +T
t=1Yt (17)
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Behavior
Since marginal utility of consumption is always positive, budget
constraint will be satisfied with equality. the Lagrangian for themaximization problem is therefore
L =T
t=1
u(Ct) +
A0 +
Tt=1
Yt T
t=1
Ct
(18)
The first order condition for Ct isu(Ct) = (19)
Since equation (19) holds every period, the marginal utility ofconsumption is constant and so the consumption; hence, usingthe fact that C1 = C2 =
Ct = 1TA0 +
Tt=1
Yt (20)
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Implications
Consumption is not determined by current income but by the
permanent income (Friedman). The difference between currentand permanent income is called transitory income. If there is
a windfall gain of Z, permanent income increases by Z/T and
consumption increases little. Saving will be more effected:
St = Yt Ct =
Yt
1
T
T
t=1
Yt
1
T
A0, (21)
Saving is high when income is high relative to its average. Saving
and borrowing is used to smooth the path of consumption. This
is the lifecycle/permanent income hypothesis of Modigliani and
Brumberg (1954) and Friedman (1957). In here saving is ba-
sically future consumption, which is driven by preferences over
current and future consumption and information about futureconsumption prospects.
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Consumption under Uncertainty: The random walk hy-
pothesis
Suppose there is uncertainty about labor income. Assume a
quadratic utility function. The individual maximizes
E[U] = E
T
t=1
Ct
a
2C2t
, a > 0. (22)
Individuals pay off any debt at the end of life yielding the samebudget constraint,
Tt=1 Ct A0 +
Tt=1 Yt. Suppose that individ-
ual has chosen firstperiod consumption optimally given the in-
formation available and suppose that consumption in each future
period optimally given the information then available. Hence, an
optimizing individual would set
1 aC1 = E1[1 aCt], f or t = 2, 3, . . . , T , (23)
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which implies
C1 = E1[Ct], f or t = 2, 3, . . . , T . (24)
Since budget constraint holds, the above equation takes the form
Tt=1
E1[Ct] = A0 +T
t=1
Yt = T C1 (25)
hence, we obtain
C1 =1
T
A0 +
Tt=1
E1[Yt]
(26)
Individual consumes 1/T of his expected lifetime resources.
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Implications
Generally,
Ct = Et1[Ct] + t, (27)
where Et1t = 0. Thus, since Et1[Ct] = Ct1 by equation 9,
we can write
Ct = Ct1 + t. (28)
This result due to Hall (1978) implies that consumption fol-
lows a random walk, implying that changes in consumption is
unpredictable. The individual adjusts his current consumption
to the point where consumption is not expected to change, i.e.
smoothing consumption.
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