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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!