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) % (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) % (r=0.2) r100

When you borrow Rate of interest What you borrow in period 1 You must repay in period 2 10% (r=0.1) % (r=0.2) r100100(1+r) rm1m1

When you borrow Rate of interest What you borrow in period 1 You must repay in period 2 10% (r=0.1) % (r=0.2) r100100(1+r) rm1m1 m 1 (1+r) rm2m2

When you borrow Rate of interest What you borrow in period 1 You must repay in period 2 10% (r=0.1) % (r=0.2) r100100(1+r) rm1m1 m 1 (1+r) rm 2 /(1+r)m2m2

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) % (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) % (r=0.2) r100

When you save Rate of interest Saving in period 1 What you get back in period 2 10% (r=0.1) % (r=0.2) r100100(1+r) rm1m1

When you save Rate of interest Saving in period 1 What you get back in period 2 10% (r=0.1) % (r=0.2) r100100(1+r) rm1m1 m 1 (1+r) rm2m2

When you save Rate of interest Saving in period 1 What you get back in period 2 10% (r=0.1) % (r=0.2) r100100(1+r) rm1m1 m 1 (1+r) rm 2 /(1+r)m2m2

Notation and graphical representation Intertemporal choice. Two periods: 1 and 2. m 1 and m 2 : incomes in the two periods. c 1 and c 2 : 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 c 1 and c 2 on the axes, and (m 1, m 2 ) as the endowment point. First the budget constraint then the preferences.

The Budget Line 1. m 1 > c 1 savings = m 1 - c 1 Becomes (m 1 - c 1 )(1+r) in period 2. Hence c 2 = m 2 + (m 1 - c 1 )(1+r). Or: c 1 (1+r) + c 2 = m 2 + m 1 (1+r). In the space (c 1,c 2 ) a line with slope -(1+r).

The Budget Line 2. m 1 < c 1 borrowings = c 1 - m 1 Have to repay (c 1 - m 1 )(1+r) in period 2. Hence c 2 = m 2 - (c 1 - m 1 )(1+r). Or: c 1 (1+r) + c 2 = m 2 + m 1 (1+r). In the space (c 1,c 2 ) a line with slope -(1+r).

The Budget Line 3. maximum consumption in period 2 = m 1 (1+r) + m 2 – this is called the future value of the stream of income. maximum consumption in period 1 = m 1 + m 2 /(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 = m 1 + m 2 /(1+r) – the present value of the stream of income.. The intercept on the vertical axis = m 1 (1+r) + m 2 – the future value of the stream of income... The slope = -(1+r)

Generalisation If the individual receives a stream of income: m 1, m 2, m 3 … m T The present value is The future value is

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!