Consumption 0

8/3/2019 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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