Labor Supply and Intertemporal Choice

• Initially, we want to see how to handle labor supply

ECON 370: Microeconomic Theory Summer 2004 – Rice University

• We will address capital later

Stanley Gilbert

Econ 370 - Labor Supply and Intertemporal Choice

Labor Supply Assumptions

Budget Constraints

• People have H hours of time (per day/week/year as appropriate)

• In this case we have two budget constraints • One for Time

• Time may be spent either on Labor (L) or Leisure (R)

– H=L+R

• Labor is supplied at a wage schedule w(L)

• And one for money or consumption

– We usually assume a constant wage w

– c ≤ m + wL

• People have non-wage income m, which may be zero

• We usually write the latter in terms of Leisure:

• Wage and non-wage income go into aggregate consumption c, which we treat as our numeraire

– c = m + w(H – R) – c + wR ≤ m + wH = M

• People have preferences over consumption and leisure

Ex pen dit ur

• Consumption and leisure are normal goods Econ 370 - Labor Supply and Intertemporal Choice

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En do w e

Econ 370 - Labor Supply and Intertemporal Choice

or

me nt 4

1

Graphically

Including Preferences

c ($)

c ($)

M w 1 hr

c

m

m

Endowment

H

Leisure

Econ 370 - Labor Supply and Intertemporal Choice

R 5

L

H

Econ 370 - Labor Supply and Intertemporal Choice

Labor Supply

Leisure 6

Labor Supply Curve

c ($) w

m

H

Econ 370 - Labor Supply and Intertemporal Choice

L Leisure 7

Econ 370 - Labor Supply and Intertemporal Choice

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Present and Future Values Primer • Take two periods -- 1 and 2 – This month and next, this year and next – Lifetime: earning years and retirement years

• Let r denote the interest rate per period

Primer on Present and Future Value

• If r = 0.1 then $100 saved at the start of period 1 becomes $110 at the start of period 2 • Future Value: Value next period of $1 saved today

Econ 370 - Labor Supply and Intertemporal Choice

Future Value: Formulas

Present Value: Introduction

• Given r, FV of $1 one period from now is

• How much do you save now to get $1 next period?

– FV1 = 1 + r

• v saved now becomes, in next period: v(1 + r)

• Given r, FV one period from now of $m is

• So, want v such that v(1 + r) = 1

– FV1 = m(1 + r)

• Thus, v = 1 / (1 + r) is present-value (PV) of $1 obtained at start of next period

• FV two periods from now is the – FV2 = (1 + r) FV1 = m(1 + r)2

• Then, the present value of $m obtained at start of the next period is:

• FV n periods from now is the – FVn = (1 + r) FVn - 1 = m(1 + r)n

Econ 370 - Labor Supply and Intertemporal Choice

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PV =

11

Econ 370 - Labor Supply and Intertemporal Choice

m 1+ r

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Present Value: Formulas •

• Suppose we have a bond that pays $m each period for n periods

Given r, PV of $m one period from now is PV1 =

•

Present Value: Stream of Income

m 1+ r

• The simplest way to calculate this is to calculate present value of the stream: PV =

Given r, PV of $m n periods from now is PVn =

m

(1 + r )n

m m m + +L+ 2 1 + r (1 + r ) (1 + r )n

• This can be shortened to: PV = m

Econ 370 - Labor Supply and Intertemporal Choice

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Econ 370 - Labor Supply and Intertemporal Choice

Inflation

1+π 1+ r

• That implies that the real interest rate ρ is: 1+ r 1+π

or

ρ=

Given r, PV of $m n periods from now is: PV =

•

Given r, FV n periods from now of $m is FV = m(1 + r )n

•

Given r, PV of $m each period for n periods is

•

Given r, PV of $m each period forever is

r −π 1+π

• This is often approximated as: ρ = r – π Econ 370 - Labor Supply and Intertemporal Choice

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m

•

• Then the real present value of $m in the next period is:

1+ ρ =

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Summary: Formulas

• If the inflation rate is π

PV1 = m

(1 + r )n +1 − 1 r (1 + r )n

Econ 370 - Labor Supply and Intertemporal Choice

PV = m

(1 + r )n

(1 + r )n +1 − 1 r (1 + r )n PV =

m r 16

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Intertemporal Choice – Capital Supply • Households savings serve as the supply of capital to the market • We model the savings decision as a decision between consumption today and consumption in the future

End Primer

Econ 370 - Labor Supply and Intertemporal Choice

Intertemporal Choice Assumptions

Budget Constraint

• People live for two periods: Now and Later

• If the only limitation people have on borrowing is that they must be able to repay it

– Interpretations of these include “working years” and “retirement” – Consumption in the two periods is c1 and c2

• And the interest rate is r

• They have a (usually fixed) income endowment of (m1, m2)

• The the maximum consumption people can have in the present is – m1 + m2 / (1 + r)

• The “price” at which consumption today can be traded for consumption later is the market interest rate

• The budget constraint then becomes: c1 +

• People have preferences over consumption now and later Econ 370 - Labor Supply and Intertemporal Choice

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c2 m ≤ m1 + 2 1+ r 1+ r

Econ 370 - Labor Supply and Intertemporal Choice

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Graphically

Including Preferences

c2

1+r

c2

c2´

1

m2

Endowment

m2

m1

c1´

c1

Econ 370 - Labor Supply and Intertemporal Choice

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m1

Econ 370 - Labor Supply and Intertemporal Choice

Savers

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Borrowers

• Savers move consumption from the present to the future c2

c1

• Borrowers move consumption from the future to the present c2

• That is,

• That is,

– c1 < m1; and – c 2 > m2

– c1 > m1; and – c 2 < m2

c2´ m2

m2 c2´ c1´

m1

Econ 370 - Labor Supply and Intertemporal Choice

c1

m1 c1´ 23

Econ 370 - Labor Supply and Intertemporal Choice

c1 24

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Preferences

Applications

• General preferences take the form U(c1, c2)

• Labor Supply Model – Overtime Pay – AFDC – Religious Participation

• Usually, though, economists use time-separable preferences: • That is U(c1, c2) = u(c1) + ρu(c2)

• Intertemporal Choice

• Monotonicity and Convexity of preferences imply

– Different Interest rates for borrowing and lending – Social Security – Social Security with different interest rates

• u´(c) > 0 and u´´(c) ≤ 0 • Where ρ is the pure rate of time preference

Econ 370 - Labor Supply and Intertemporal Choice

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Econ 370 - Labor Supply and Intertemporal Choice

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