Intertemporal Approach to the Current Account

Part 2

Volatile Investment

Investment is the most volatile part of demand. A 1% change in technology or increase in employment will increase target capital stock.

In any year, investment is one-tenth as large as the capital stock.A 1% change in the capital stock requires a 10%

change in investment.

Investment Curve

Q: Why does the curve slope down? The greater is interest rate, the more profitable capital must be to invest in it.

Q: What shifts the curve? Increases in technology and labor increases profitability of capital.

I

r

( )I r

Investment high but falling in Japan and Korea, Investment high but increasing in China.

Investment as a % ogf GDP

0

5

10

15

20

25

30

35

40

45

China Japan Korea, Rep. United States

1990-1994

1995-1999

2000-2003

Savings high but falling in Japan and Korea, savings high but increasing in China.

Savings as a % of GDP

0

5

10

15

20

25

30

35

40

45

China Japan Korea, Rep. United States

&

1990-1994

1995-1999

2000-2003

Two Consumption Theories

Keynesian: Consumption is dependent on current income.

Permanent Income Theory: Consumption decision is a savings decision so households take into account future income as well as outstanding financial wealth.People prefer smooth consumption and save and

borrow to do so.

Why do People Save?

Life Cycle Motives – Income is Not Smooth Across Time. Households save, in part, to transfer income from high income periods to low income periods.

Precautionary Motives – Households like to achieve a buffer stock of wealth in the case of a possible bad outcome. If households have a buffer stock of saving, bad outcomes in terms of income don’t result in really bad outcomes in terms of consumption.

Life Cycle Motives: Two Period Model

To examine life-cycle theory, we use simplest possible model.

One good consumed by a household that lives two periods, C0 and C1.

Household lives and earns income Y0 and Y21 in each period.

Household pays taxes in each period T0 and T1. Household can buy/sell bonds, B, at real interest rate

r.

Temporal Budget Constrants

First period,B0 = Y0 – T0 – C0 1)

Second period,C1=Y1 –T1+(1+r)B0 2)

Note B can be either > or < 0. If B > 0, household is a saver. If B < 0, household is a borrower.

Intertemporal Budget Constraint

Combine two budget constraints

Present Discounted Value of Lifetime Income equals Present Discounted Value of Lifetime Consumption.

1 1 1

0 0 0

10

1 10 0

1 1

1

1

C Y TB

r rB Y T C

CC W

rY T

W Y Tr

Consider 3 scenarios

Baseline Y1 = Y2 = Y implies

Permanent Consumption Hypothesis

C1=C2

2

1 1

Y rW Y Y

r r

1 1 2[ ]

2 2 1

r r rC W Y Y

r r r

Temporary Rise in Income

The propensity to consume is a fraction of the temporary extra income. The remainder is saved for the future.

1 2

2,

1 1

1 1 2 1 1

2 2 1 2 2

1 1( )

2 2

Y rY Y Y Y W Y Y

r r

r r r r rC W Y Y

r r r r r

rS Y Y

r r

Permanent Rise in Income

The propensity to consume is larger when the increase in consumption is permanent. There is no need to save a permanent rise in income for the future.

1 2

2, ,

1 1

1 1 2

2 2 1

( ) 0

Y rY Y Y Y W Y Y

r r

r r rC W Y Y

r r r

S Y Y

Future Rise in Income

Consumption may rise when future income increases which will also increase W. Savings will fall as people borrow to enjoy future income today.

1 2

2, ,

1 1

1 1 2

2 2 1 1 2

( )2 2

rY Y Y Y W Y Y

r r

r r rC W Y Y

r r r r r

S Y Yr r

Permanent Income Hypothesis

A simplified (and extreme) version of this theory hypothesizes that consumption is equal in each period.

** 21 1 11

[ ]2

C CC W C

r rr

C Wr

Income Stream & Consumption

Consider three hypothetical increases in income of Δ. 1. A Temporary Increase – Y1 increase by Δ, but Y2 is unchanged. This

will increase W by Δ.

2. A Future Increase – Y2 increases by 100, but Y1 is unchanged. W increases by Δ /1+r≈ Δ

3. A Permanent Increase – Y1 & Y2 increase by 100. W increases by Δ(2+r/1+r) ≈2∙ Δ

Cases 1 & 2 increase W by nearly identical amounts. But current consumption depends only on W. Thus, cases 1 & 2 will increase

by similar amounts. Case 3 increases W by nearly double the amount.

* *1 2,C C

Income Stream and Savings

In the first case, future income does not rise but optimal future consumption, C2* does . Current savings must rise.

In the second case, current income does not rise, but optimal current consumption. Current savings must fall.

What happens to savings with a permanent change in income?

Application: Life Cycle of Saving

Permanent Income Hypothesis suggests that households like to keep a constant profile of consumption over time.

Age profile of income however is not constant. Income is low in childhood, rises during maturity and reaches a peak in mid-1950’s and drops during retirement.

This generates a time profile for savings defined as the difference between income and consumption.

Time Path of Savings

time

C,Y

S>0

S<0C

Y

S<0

East Asian Demographics

During last 25 years, East Asian Nations had a sharp decrease in their ‘dependency ratio’.

Dependency ratio is the % of people in their non-working years (children & seniors.

Dependents are dis-savers and non-dependents are savers.

East Asian Demographics

Due to plummeting birth rates, East Asia had a plummeting ratio of youths as a share of population

This put a large share of population in high savings years.

Share of prime age adults has hit its peak in most Asian countries and will fall over the next half century.

Change in Age Shares%Below 15 % Prime Age 20-591950-1990 2005-2025

China -13.56 0.41Hong Kong -20.64 NAIndonesia -7.26 5.52Japan -16.72 -4.03South Korea -18 -4.12Malaysia -7.7 7.5Singapore -20.22 8.35Taiwan -18.82 NAThailand -14.74 0.25

Interest Rates: Incentives and Effects

A rise in interest rates increases the payoff to savings and increases the incentive to save. Substitution Effect (Plus Factor for All)

A rise in the interest rate reduces the amount of savings you need to do to meet target level of future consumption. Income Effect (Minus Factor for Net Savers).

A rise in the interest rate reduces the amount of borrowing you can do and still meet some target lever of future consumption. Income Effect (Plus Factor for Net Borrowers)

Aggregate Savings & Interest Rates

Interest rates have a positive impact on savings by borrowers, i.e. borrowers reduce their borrowing.

Interest rates have an ambiguous effect on savings by savers.

Since there is positive net savings, interest rates have ambiguous effect on aggregate savings.

Empirically, impact of interest rates on savings are hard to detect.

Saving Curve

Q: Why does the curve slope up? Empirical work suggests substitution effect is slightly more powerful than income effect.

Q: What shifts the curve? Changes in current income relative to future income.

( )I r

I

r( )S r

Household born in period 0 and lives until period T. (T+1 period lives)

Household begins with real financial wealth F

Present value of consumption equals present value of financial & human wealth

31 2

0 2 3 ...1 1 1 1

TT

CC C CC W F

r r r r

31 2

0 2 3 ...1 1 1 1

TT

YY Y YW Y

r r r r

If Y grows at constant rate

Yt = (1+g)tY0

2 30 0 0 0

0 2 3

2 10 0

11011

1

(1 ) (1 ) (1 ) (1 )...

1 1 1 1

1(1 ... ) 1

11

11

T

T

T T

Tgrg

r

g Y g Y g Y g YW Y

r r r r

x x x Y x Yx

Y

Annuity Value

Just as any stream of future payments has a present value, so does it have an annuity value.

An annuity is an asset that makes a constant payment every period, for a number of years, T. Such an annuity has a present value.

The annuity value of any amount is the size of the payment of an annuity whose present value is equal that amount.

Present Value of an Annuity Payment: Annuity Value of Present Wealth

The real present value of an annuity with payment Y.

Off-the-shelf formula for geometric sum

Solve for present value of an annuity Y

5)

2 3

2 3

...1 1 1 1

1 1 1 11 ...

1 1 1 1

Tt T

T

Y Y Y YPV Y

r r r r

Yr r r r

1

2

11

11 1 11 ...

11 1 1 11

T

T

r

r r rr

1

11

11

11

T

Tt

rPV Y

r

Annuity Value of a Present Value

If you have some current lump sum, PV, payment and you want to buy a annuity for T periods.

Q: How big an annuity payment Y can you get.

A: Invert Equation 5)

1

1111

11

T

T

rY PV

r

Permanent Income

The permanent income theory says that households keep consumption smooth consuming the annuity value of their financial wealth, F, plus the present value of lifetime income, W.

1

1111

11

[ ]T

r

r

C W F

Example

The fraction is referred to as the propensity to consume out of wealth.

A household lives for = 40 periods and the real interest rate is .02. In every period they would consume a fraction of their wealth equal to

11.02

4111.02

1

1.0353

11

111

1

1

rT

r

Applications: Wealth Effect

Changes in asset prices will change the current value of financial wealth.

The effect of an increase in financial wealth on consumption is called the wealth effect.

According to the PIH, a one dollar increase in the value of a stock portfolio should lead to an increase in consumption equal to the propensity to consume out of wealth.

Econometricians estimate that the wealth effect to be less than $.05 consistent with our theory.