ECONOMICS I.
Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,
Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest
Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest
Author: Gergely K®hegyi, Dániel Horn, Klára Major Supervised by Gergely K®hegyi
June 2010
ELTE Faculty of Social Sciences, Department of Economics
ECONOMICS I.
week 11
Economics of time
Gergely K®hegyi Dániel Horn Klára Major
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).
Intertemporal decision
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 1 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;. . .;CT)
short run interest long run interest
P1 P0 =1+r1
1
P1 P0 =1+R1
P2 1
P1 =1+r1
2
P2
P0 = (1+R1
2)2
. . . .
PT
PT−1 = 1+r1
T
PT
P0 =(1+R1
T)T
Denition 2 TheW¯0 endowed wealth is the present value of one's endowmentc¯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
c0+ c1 1 +r1
= ¯W0≡c¯0+ c¯1 1 +r1
U(c0;c1) Optimum:
M RSC = 1 +r Optimal intertemporal decision
In the optimum the intertemporal budget constraint is tangent to the highest possible level of intertem- poral utility.
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 (r01): is the price of changing future money with today's money:
1 +r10 ≡ −∆m1
∆m0
• 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):
P0m≡ −∆m0
∆c0
;P1m≡ −∆m1
∆c1
• Ination rate (a1): The ratio of future price level and today's price level:
1 +a1≡P1m P0m
Note 1 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).
Note 2 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 1 The real interest rate added with the expected ination is a good-enough approximation of the nominal interest rate:
r10 'r1+a1
Proof 1 Discrete version of interest rate calculation Let's look at the following identity:
∆m1
∆m0 ≡ ∆m1
∆c1
∆c1
∆c0
∆c0
∆m0 1 +r01≡P1m
P0m(1 +r1) 1 +r01≡(1 +a1)(1 +r1)
r10 ≡r1+a1+r1a1
Since r1a1 is a very small number, that is r1a1'0, thus r10 'r1+a1
Proof 2 Continuous version of interest rate calculation Ifiis the annual compound interest rate andk is the frequency of payments, then the value of the unit investment in time 0. (H0) is H1 at the end of the rst period:
H1=
1 + i k
k H0
With continuous interest, i.e. ifk→ ∞,limk→∞ 1 +ki
=e, thusH1=ekH0. Therefore
∆m1
∆m0
≡ ∆m1
∆c1
∆c1
∆c0
∆c0
∆m0
er01 =er1ea1 r0 =r +a
Real interest rate and nominal interest rate
Example 1 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.
Income tax versus consumption tax
Consequence 1 Income taxes might not reduce savings as compared to consumption taxes, but they certainly reduce future consumption.
Savings and investment
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 atR∗whereQQis tangent to the highest attainable indierence curve.
Market exchange
The individual here has intertemporal productive opportunities (Production-Possibility curve QQ), as well as exchange opportunities.
Consequence 2 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∗).
Intertemporal equilibrium with productive investment
When productive investment takes place, the equilibrium interest rater∗simultaneously balances (1) the
of lending L with the aggregate demand for borrowing B. The dierence between the two magnitude is nanced out of investor's own savings.
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
Source: Hirshleifer et al., 2009, 614.
Project evaluation
Investment decision and project analysis
Statement 2 The separation theorem A person's production optimum position Q∗ is entirely indepen- dent 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.
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 Denition 3 (Internal) Rate of Return (RoR) (ρ):
0 =z0+ z1
1 +ρ+ z2
(1 +ρ)2 +. . .+ zT
(1 +ρ)T
Statement 3 All projects should be adopted with higher RoR than the market interest rate, i.e. where (ρ > r).
Consequence 3 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 ifV0>0) is equivalent to the rate of return rule (adopt ifρ > r00).
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
Source: Hirshleifer et al., 2009, 629.
Exogenous eects
Exogenous eects
Main factors aecting investments, savings and interest rates
• Time preference
• Time-endowment
• Time-productivity
• Degree of isolation
Eect of time preference
Eect of time-endowment
Eect of time-productivity
