Microeconomics 2

John Hey

Intertemporal Choice

• Chapter 20 – the budget constraint, intertemporal preferences in general and choice in general

• Chapter 21 – intertemporal preferences in particular – the Discounted Utility Model

• Chapter 22 – intertemporal exchange

A question for you

• An observation: to reduce consumption in an economy, the government usually raises the interest rate. Why?

• If interest rates rise …• … an individual is better or worse off?• … saves more or less?• … spends more or less?• The correct answers?....• … it depends…

Framework

• Intertemporal choice.

• Two periods: 1 and 2.

• We consider an individual who receives an income in each of the two periods.

• Might be happy to consume his or her income in the period in which it is received ...

• ... but might prefer to re-distribute it, by saving or borrowing.

• That is what these three chapters of the book are about.

• (We have already talked about allocation within a period to specific goods and services. Here we are talking about allocation between periods.)

• But first some preliminaries about saving and borrowing, rates of interest and rates of return.

When you borrow

Rate of interest

What you borrow in period 1

You must repay in period 2

10% (r=0.1) 100

When you borrow

Rate of interest

What you borrow in period 1

You must repay in period 2

10% (r=0.1) 100 11020% (r=0.2) 100

When you borrow

Rate of interest

What you borrow in period 1

You must repay in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100

When you borrow

Rate of interest

What you borrow in period 1

You must repay in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100 100(1+r)

r m1

When you borrow

Rate of interest

What you borrow in period 1

You must repay in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100 100(1+r)

r m1 m1(1+r)

r m2

When you borrow

Rate of interest

What you borrow in period 1

You must repay in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100 100(1+r)

r m1 m1(1+r)

r m2/(1+r) m2

When you save

Rate of interest

Saving in period 1

What you get back in period 2

10% (r=0.1) 100

When you save

Rate of interest

Saving in period 1

What you get back in period 2

10% (r=0.1) 100 11020% (r=0.2) 100

When you save

Rate of interest

Saving in period 1

What you get back in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100

When you save

Rate of interest

Saving in period 1

What you get back in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100 100(1+r)

r m1

When you save

Rate of interest

Saving in period 1

What you get back in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100 100(1+r)

r m1 m1(1+r)

r m2

When you save

Rate of interest

Saving in period 1

What you get back in period 2

10% (r=0.1) 100 11020% (r=0.2) 100 120

r 100 100(1+r)

r m1 m1(1+r)

r m2/(1+r) m2

Notation and graphical representation

• Intertemporal choice.

• Two periods: 1 and 2.

• m1 and m2: incomes in the two periods.

• c1 and c2: consumption in the two periods.

• r: the rate of interest (10%, r = 0.1; 20%, r = 0.2)

• The rate of return = (1+r)

• We will be drawing graphs with c1 and c2 on the axes, and (m1, m2) as the endowment point.

• First the budget constraint then the preferences.

The Budget Line 1.

• m1 > c1 savings = m1 - c1

• Becomes (m1 - c1)(1+r) in period 2.

• Hence c2 = m2 + (m1 - c1)(1+r).

• Or:

c1(1+r) + c2 = m2 + m1(1+r).

• In the space (c1 ,c2) a line with slope

-(1+r).

The Budget Line 2.

• m1 < c1 borrowings = c1 - m1

• Have to repay (c1 - m1)(1+r) in period 2.

• Hence c2 = m2 - (c1 - m1)(1+r).

• Or:

c1(1+r) + c2 = m2 + m1(1+r).

• In the space (c1 ,c2) a line with slope

-(1+r).

The Budget Line 3.

• maximum consumption in period 2 =

m1(1+r) + m2

• – this is called the future value of the stream of income.

• maximum consumption in period 1 =

m1 + m2/(1+r)

• – this is called the present value of the stream of income.

• Note: we say that the market discounts the income in period 2 at the rate r.

The Budget Line 4.

• The intercept on the horizontal axis =

• m1 + m2/(1+r)

– the present value of the stream of income..

• The intercept on the vertical axis =

• m1(1+r) + m2

– the future value of the stream of income...

• The slope = -(1+r)

Generalisation

• If the individual receives a stream of income:

• m1, m2, m3 … mT

• The present value is

• The future value is

t 1

T mt

( )1 r( )t 1

t 1

T

mt ( )1 r( )T t

An imperfect market (10% and 50%)

Chapter 20

• Let us go briefly to the Maple Chapter 20, but note...

• ... most of Chapter 20 uses general preferences. (So do not spend too much time on studying the rest of this Chapter.)

• But it shows that saving and borrowing depend upon incomes and rate of interest.

• In Chapter 21 we use Discounted Utility Model preferences.

Chapter 20

• Goodbye!