microeconomics ii. b - regi.tankonyvtar.hu€¦ · microeconomics ii. b week 4 economics of time...

MICROECONOMICS II.

B

ELTE Faculty of Social Sciences, Department of Economics

Microeconomics II.�B�

week 4ECONOMICS OF TIME

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

February 2011

week 4

Gergely K®hegyi

Intertemporaldecision

Savings andinvestment

Project evaluation

Exogenous e�ects

Prepared by: Gergely K®hegyi, Dániel Horn and Klára Major, usingJack 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).

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Draft

1 Intertemporal decision

2 Savings and investment

3 Project evaluation

4 Exogenous e�ects

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Present versus future

E.g.:

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

Consumed quantities: c0; c1; c2; . . .

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

numeraire: P0 ≡ 1

De�nition

r1 annual real interest rate is the additional amount of future cornthat 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 twoconsumptions in di�erent points in time (C0;C1; . . . ;CT )

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Present versus future (cont.)

short run interest long run interest

P1P0

= 11+r1

P1P0

= 11+R1

P2P1

= 11+r2

P2P0

= 1(1+R2)2

. . . . . .PT

PT−1= 1

1+rTPT

P0= 1

(1+RT )T

De�nition

The W̄0 endowed wealth is the present value of one's endowmentc̄0; c̄1) of her present and future claims:

W̄0 ≡ P0c̄0 + P1c̄1 ≡ c̄0 +c̄1

1 + r1

Intertemporal budget constraint:

P0c0 + P1c1 = W̄0 ≡ P0c̄0 + P1c̄1

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


c0 +c1

1 + r1= W̄0 ≡ c̄0 +

c̄11 + r1

Intertemporal utility function:

U(c0; c1)

Optimum:

MRSC = 1 + r

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Optimal intertemporaldecision

In the optimum theintertemporal budgetconstraint is tangent tothe highest possible levelof intertemporal utility.

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Real interest rate and nominal interest rate

So far we have only considered real changes behind the "moneycurtain". 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 1000forints might worth lot less...

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

1 + r1 ≡ −∆c1∆c0

Nominal interest rate (r ′1): is the price of changing futuremoney with today's money:

1 + r ′1 ≡ −∆m1

∆m0

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Real interest rate and nominal interest rate(cont.)

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

Pm

0 ≡ −∆m0

∆c0;Pm

1 ≡ −∆m1

∆c1

In�ation rate (a1): The ratio of future price level and today'sprice level:

1 + a1 ≡Pm1

Pm0

Note

The link relation between the price levels in di�erent times aredetermined by macroeconomics processes (which of course stemfrom microeconomic processes, but are exogenous for us now).

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Note

Since the factual in�ation rate are usually unknown, because it isdetermined in the future (ex post), thus we usually talk aboutexpected in�ation rate.

Statement

The real interest rate added with the expected in�ation is agood-enough approximation of the nominal interest rate:

r ′1 ' r1 + a1

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Proof

Discrete version of interest rate calculation

Let's look at the following identity:

∆m1

∆m0

≡ ∆m1

∆c1

∆c1∆c0

∆c0∆m0

1 + r ′1 ≡Pm1

Pm0

(1 + r1)

1 + r ′1 ≡ (1 + a1)(1 + r1)

r ′1 ≡ r1 + a1 + r1a1

Since r1a1 is a very small number, that is r1a1 ' 0, thus

r ′1 ' r1 + a1

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

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 ofpayments, then the value of the unit investment in time 0. (H0) isH1 at the end of the �rst period:

H1 =

(1 +

i

k

)k

H0

With continuous interest, i.e. if k →∞, limk→∞(1 + i

k

)= e,

thus H1 = ekH0. Therefore

∆m1

∆m0

≡ ∆m1

∆c1

∆c1∆c0

∆c0∆m0

er′1 = er1ea1

r ′1 = r1 + a1

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Real interest rate and nominal interest rate

Example

Nominal and real annual yields of USA stocks, 1926-2002(percentage)

annual

average

nominal

yield

annual av-

erage 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.

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Income tax versus consumption tax

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Income tax versus consumption tax (cont.)

Consequence

Income taxes might not reduce savings as compared toconsumption taxes, but they certainly reduce future consumption.

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Savings and investment

Autarchy

Robinson Crusoe hasintertemporal exchangeopportunities, but canengage in productivetransformation betweenconsumption this year andconsumption next year.QQ is theProduction-Possibilitycurve through hisendowment E. The Crusoeoptimum is at R∗ whereQQ is tangent to thehighest attainableindi�erence curve.

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Savings and investment (cont.)

Market exchange

The individual here hasintertemporal productiveopportunities(Production-Possibilitycurve QQ), as well asexchange opportunities.

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Consequence

In a regime of pure exchange, a person can achieve a preferredintertemporal patter of consumption only by borrowing or lending.At the equilibrium interest rate the overall market supply oflending equals the overall market demand for borrowing(L∗ = B∗). But when intertemporal production (investing) is alsopossible, each individual chooses his or her optimal scale ofinvestment and lending or borrowing. The equilibrium interest ratebalance the optimum supply of saving with the aggregate demandfor investment (S∗ = I ∗), and also equates the aggregate supplyof lending with the aggregate demand for borrowing (L∗ = B∗).

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Intertemporal balancewith productiveinvestment

When productiveinvestment takes place,the equilibrium interestrate r∗ simultaneouslybalances (1) the aggregatesupply of saving S withthe aggregate demand forinvestment I, and (2) theaggregate supply oflending L with theaggregate demand forborrowing B. Thedi�erence between the twomagnitude is �nanced outof investor's own savings.

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Growth, investment and saving (1973�1984, percent)

growth rateinvestmentrate savings rate

The �ve highest growth rateEgypt 8,5 25 12Yemen 8,1 21 -22Cameroon 7,1 26 33Syria 7,0 24 12Indonesia 6,8 21 20

The �ve lowest growth ratesZambia 0,4 14 15Salvador �0,3 12 4Ghana �0,9 6 5Zaire �1,0 n.a. n.a.Uganda �1,3 8 6


week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Investment decision and project analysis

Statement

The separation theorem A person's production optimum positionQ∗ 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 projectwith positive present value, and reject any project withnegative present value.

2 Present value rule (Mutually exclusive projects). Adopt theproject with the largest present value V0, provided it ispositive.

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

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Investment decision and project analysis (cont.)Present value for more periods:

V0 ≡ z0+z1

1 + r1+

z2(1 + r2)(1 + r1)

+. . .+zT

(1 + rT ) . . . (1 + r2)(1 + r1)

with identical interest rates:

V0 ≡ z0 +z1

1 + r+

z2(1 + r)2

+ . . .+zT

(1 + r)T

with long term interest rates:

V0 ≡ z0 +z1

1 + R1

+z2

(1 + R2)2+ . . .+

zT(1 + RT )T

De�nition

(Internal) Rate of Return (RoR) (ρ):

0 = z0 +z1

1 + ρ+

z2(1 + ρ)2

+ . . .+zT

(1 + ρ)T

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Investment decision and project analysis (cont.)

Statement

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

Consequence

For independent projects, if the payment stream has only a singlereversal of signs (an investment followed by a payo� phase), thenthe present value rule (adopt if V0 > 0) is equivalent to the rate ofreturn rule (adopt if ρ > r ′′).

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects


Social rates of return to education

Region Primary Secondary Higher

Asia (non-OECD) 16,2 11,1 11,0Latin-America 17,4 12,9 12,3OECD 8,5 9,4 8,5Sub-Saharan Africa 25,4 18,4 11,3World 18,9 13,1 10,8


week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

Exogenous e�ects

Main factors a�ecting investments, savings and interest rates

Time preference

Time-endowment

Time-productivity

Degree of isolation

Risk

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

E�ect of time preference

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

E�ect of time-endowment

week 4

Gergely K®hegyi



Project evaluation

Exogenous e�ects

E�ect of time-productivity

microeconomics ii. b - regi.tankonyvtar.hu€¦ · microeconomics ii. b week 4 economics of time...

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