Present versus future

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E CONOMICS I.

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ELTE Faculty of Social Sciences, Department of Economics

Economics I.

week 11

ECONOMICS OF TIME

Authors: Gergely K®hegyi, Dániel Horn, Klára Major Supervised by Gergely K®hegyi

June 2010

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week 11 K®hegyi-Horn-Major

Intertemporal decision Savings and investment Project evaluation Exogenous eects

Prepared by: Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer és David Hirshleifer (2009) Mikroökonómia. Budapest:

Osiris Kiadó, ELTECON-könyvek (henceforth: HGH), and Kertesi Gábor (ed.) (2004) Mikroökonómia el®adásvázlatok.

http://econ.core.hu/ kertesi/kertesimikro/ (henceforth: KG).

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Draft

1 Intertemporal decision

2 Savings and investment

3 Project evaluation

4 Exogenous eects

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Present versus future

E.g.:

Product: C₀ (present corn); C₁ (next year's corn); C₂ (corn two years from now); . . .

Consumed quantities: c₀;c₁;c₂;. . .

Prices (prices paid today for the corn delivered in the given time): P0;P1;P2;. . .

Numeraire: P₀≡1

Denition

r1 annual real interest rate is the additional amount of future corn that have to be paid to receive a unit of present corn:

−∆c1

∆c0 ≡P0

P1 ≡1+r1

Naturally, we can use this line of thinking to compare any two consumptions in dierent points in time (C₀;C₁;. . .;C_T)

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Present versus future (cont.)

short run interest long run interest

P1

P0 =₁₊¹_r

1

P1

P0 = ₁₊¹_R

P2 1

P1 =₁₊¹_r

2

P2

P0 = ₍₁₊¹_R

2)²

. . . .

PT

PT−1 = ₁₊¹_r

T

PT

P0 =₍₁₊_R¹

T)^T

Denition

TheW¯0endowed wealth is the present value of one's endowment c¯₀; ¯c₁) of her present and future claims:

W¯0≡P0c¯0+P1c¯1≡c¯0+ c¯1

1+r₁ Intertemporal budget constraint:

P₀c₀+P₁c₁= ¯W₀≡P₀c¯₀+P₁c¯₁

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Present versus future (cont.)

c₀+ c₁

1+r1 = ¯W₀≡c¯₀+ c¯₁ 1+r1

Intertemporal utility function:

U(c₀;c₁) Optimum:

MRS_C =1+r

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Present versus future (cont.)

Optimal intertemporal decision

In the optimum the intertemporal budget constraint is tangent to the highest possible level of intertemporal utility.

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Present versus future (cont.)

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Real interest rate and nominal interest rate

So far we have only considered real changes behind the "money curtain". That is, the 1000HUF, that we put in the bank with 8%

interest rate, worth 1080HUF in one year. What happens,

however, when living costs increase (exogenously)? Then our 1000 forints might worth lot less...

Real interest rate (r1) is the price of changing a unit of future corn with a unit of today's corn:

1+r₁≡ −∆c₁

∆c₀

Nominal interest rate (r₁⁰): is the price of changing future money with today's money:

1+r₁⁰ ≡ −∆m₁

∆m0

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Real interest rate and nominal interest rate (cont.)

Price level: the amount of money needed to buy a unit of today's goods (some sort of an average of the prices of goods):

P₀^m≡ −∆m₀

∆c₀;P₁^m≡ −∆m₁

∆c₁

Ination rate (a1): The ratio of future price level and today's price level:

1+a₁≡ P₁^m P₀^m

Note

The link relation between the price levels in dierent times are determined by macroeconomics processes (which of course stem from microeconomic processes, but are exogenous for us now).

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Real interest rate and nominal interest rate (cont.)

Note

Since the factual ination rate are usually unknown, because it is determined in the future (ex post), thus we usually talk about expected ination rate.

Statement

The real interest rate added with the expected ination is a good-enough approximation of the nominal interest rate:

r₁⁰ 'r₁+a₁

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Real interest rate and nominal interest rate (cont.)

Proof

Discrete version of interest rate calculation Let's look at the following identity:

∆m1

∆m₀ ≡ ∆m1

∆c₁

∆c1

∆c₀

∆c0

∆m₀ 1+r₁⁰ ≡P₁^m

P₀^m(1+r₁) 1+r₁⁰ ≡(1+a₁)(1+r₁)

r₁⁰ ≡r1+a1+r1a1

Since r1a1is a very small number, that is r1a1'0, thus r₁⁰ 'r1+a1

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Real interest rate and nominal interest rate (cont.)

Proof

Continuous version of interest rate calculation

If i is the annual compound interest rate and k is the frequency of payments, then the value of the unit investment in time 0. (H0) is H1at the end of the rst period:

H₁=

1+ i k

k

H₀

With continuous interest, i.e. if k→ ∞, lim_k→∞ 1+_kⁱ

=e, thus H1=e^kH0. Therefore

∆m1

∆m0 ≡ ∆m1

∆c1

∆c0

∆m0

e^r¹⁰ =e^r¹e^a¹ r₁⁰ =r₁+a₁

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Real interest rate and nominal interest rate

Example

Nominal and real annual yields of USA stocks, 19262002 (percentage)

annual average nominal yield

annual average real yield

variance of the real yield

Treasury bill 3,8 0,8 4,0

long term govt. bonds 5,8 2,9 10,6

long term corp. bonds 6,2 3,2 9,9

large comp. stocks 12,2 9,0 20,6

small comp. stocks 16,9 13,5 32,6

Source: Hirshleifer et al., 2009, 635.

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Income tax versus consumption tax

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Income tax versus consumption tax (cont.)

Consequence

Income taxes might not reduce savings as compared to

consumption taxes, but they certainly reduce future consumption.

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Savings and investment

Autarchy

Robinson Crusoe has intertemporal exchange opportunities, but can engage in productive transformation between consumption this year and consumption next year.

QQ is the

Production-Possibility curve through his

endowment E. The Crusoe optimum is at R∗ where QQ is tangent to the highest attainable indierence curve.

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Savings and investment (cont.)

Market exchange

The individual here has intertemporal productive opportunities

(Production-Possibility curve QQ), as well as exchange opportunities.

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Savings and investment (cont.)

Consequence

In a regime of pure exchange, a person can achieve a preferred intertemporal patter of consumption only by borrowing or lending.

At the equilibrium interest rate the overall market supply of lending equals the overall market demand for borrowing

(L^∗=B^∗). But when intertemporal production (investing) is also possible, each individual chooses his or her optimal scale of investment and lending or borrowing. The equilibrium interest rate balance the optimum supply of saving with the aggregate demand for investment (S^∗=I^∗), and also equates the aggregate supply of lending with the aggregate demand for borrowing (L^∗=B^∗).

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Savings and investment (cont.)

Intertemporal equilibrium with productive investment

When productive investment takes place, the equilibrium interest rate r^∗ simultaneously balances (1) the aggregate supply of saving S with the aggregate demand for investment I, and (2) the aggregate supply of lending L with the aggregate demand for borrowing B. The

dierence between the two magnitude is nanced out of investor's own savings.

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Savings and investment (cont.)

Growth, investment and saving (1973-1984, percent) growth rate investment

rate savings rate

The ve highest growth rate

Egypt 8,5 25 12

Yemen 8,1 21 -22

Cameroon 7,1 26 33

Syria 7,0 24 12

Indonesia 6,8 21 20

The ve lowest growth rates

Zambia 0,4 14 15

Salvador -0,3 12 4

Ghana -0,9 6 5

Zaire -1,0 n.a. n.a.

Uganda -1,3 8 6

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Investment decision and project analysis

Statement

The separation theorem A person's production optimum position Q∗ is entirely independent of his or her personal preferences.

Present value for two periods:

V0≡z0+ z1

1+r₁

1 Present value rule (Independent projects). Adopt any project with positive present value, and reject any project with negative present value.

2 Present value rule (Mutually exclusive projects). Adopt the project with the largest present value V0, provided it is positive.

3 Present value rule. Tabulate all the possible combinations of projects available, including doing nothing. Then choose the set of projects that maximizes overall present value.

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Investment decision and project analysis (cont.)

Present value for more periods:

V0≡z0+ z1

1+r₁+ z2

(1+r₂)(1+r₁)+. . .+ zT

(1+r_T). . .(1+r₂)(1+r₁) with identical interest rates:

V₀≡z₀+ z₁

1+r + z₂

(1+r)²+. . .+ z_T (1+r)^T with long term interest rates:

V0≡z0+ z1

1+R₁+ z2

(1+R₂)²+. . .+ zT

(1+R_T)^T

Denition

(Internal) Rate of Return (RoR) (ρ):

0=z0+ z₁

1+ρ+ z₂

(1+ρ)²+. . .+ z_T

(1+ρ)^T

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Investment decision and project analysis (cont.)

Statement

All projects should be adopted with higher RoR than the market interest rate, i.e. where (ρ >r).

Consequence

For independent projects, if the payment stream has only a single reversal of signs (an investment followed by a payo phase), then the present value rule (adopt if V0>0) is equivalent to the rate of return rule (adopt ifρ >r⁰⁰).

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Investment decision and project analysis (cont.)

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Investment decision and project analysis (cont.)

Social rates of return to education

Region Primary Secondary Higher

Asia (non-OECD) 16,2 11,1 11,0

Latin-America 17,4 12,9 12,3

OECD 8,5 9,4 8,5

Sub-Saharan Africa 25,4 18,4 11,3

World 18,9 13,1 10,8

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Exogenous eects

Main factors aecting investments, savings and interest rates Time preference

Time-endowment Time-productivity Degree of isolation Risk

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Eect of time preference

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Eect of time-endowment

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Present versus future

E CONOMICS I.

Economics I.

Draft

Present versus future

Denition

Present versus future (cont.)

Denition

Present versus future (cont.)

Present versus future (cont.)

Optimal intertemporal decision

Present versus future (cont.)

Real interest rate and nominal interest rate

Real interest rate and nominal interest rate (cont.)

Note

Real interest rate and nominal interest rate (cont.)

Note

Statement

Real interest rate and nominal interest rate (cont.)

Proof

Real interest rate and nominal interest rate (cont.)

Proof

Real interest rate and nominal interest rate

Example

Income tax versus consumption tax

Income tax versus consumption tax (cont.)

Consequence

Savings and investment

Autarchy

Savings and investment (cont.)

Market exchange

Savings and investment (cont.)

Consequence

Savings and investment (cont.)

Intertemporal equilibrium with productive investment

Savings and investment (cont.)

Investment decision and project analysis

Statement

Investment decision and project analysis (cont.)

Denition

Investment decision and project analysis (cont.)

Statement

Consequence

Investment decision and project analysis (cont.)

Investment decision and project analysis (cont.)

Exogenous eects

Eect of time preference

Eect of time-endowment

Eect of time-productivity