6/ The Impact of Interest Rates
(a) The Big Picture:
As we have already established, a rise in the rate of interest causes the budget constraint to swivel round the endowments point in a clockwise direction. When examining the effect of a rise in interest rates on consumption patterns, we require to analyse the case of severs and borrowers separately. Figures A and B set out possible impacts for both cases. For the saver the initial utility maximising equilibrium in denoted by pt A. The underlying indifference cure has been omitted to facilitate clarity. The borrower has an initial equilibrium at B. 1
A” Figure A: Possible new equilibria for a saver following an interest rate rise
1
Figure B: Possible new equilibria for a borrowerer following an interest rate rise
A’ A
Y1
Y1
B” B B’
Y0
0
Y0
0
SAVER – New possible equilibria arise at A’ and A”. At A’ the increase in interest prompts the saver to raise both C0 and C1. However, an equally plausible alternative is A” where the saver reduces C0 but substantially increases C1. For all net savers in the economy the effect of a rise in r on C0 is uncertain. BORROWER – New possible equilibria are denoted by B’ and B”. At B’ the borrower is so impatient for C0 that he/she reduces C1. At B”, the borrower reduces C0 and increases C1. However, for a borrower a rise in interest rates will reduce C0. The critical lesson for us is to derive what will happen to aggregate current consumption, C0 as interest rates rise. For the borrower
current consumption will increase but for the lender the impact in indeterminate. Thus, at the macro level, the simple 2 period model cannot predict in overall effect of a rise in r on current consumption.
(b) Income and Substitution Effects.
We can further examine the underlying behaviour of the agent by decomposing the move from the initial equilibrium to the new equilibrium following a change in r into 2 effects. substitution effect:- if r rises, this means that an agent would have to give up less C0 to achieve a one unit rise in C1. In other words, C1 is relatively cheaper in terms of C0 following a hike in interest rates. This change in relative prices will provoke a rational consumer to substitute the ‘cheaper C1 for the relatively expensive C0. Thus, in the case of a rise in r, the substitution effect serves to reduce C0 and increase C1. If interest rates fall, this logic would be reversed, C0 would become relatively cheaper in terms of C1 and agents would tend to substitute C0 for C1. Thus, if r rises the substitution effect will result in lower current consumption and vice versa. Income effect:- if r rises the income effect will impact differently on savers and borrowers ♦ saver – higher r results in greater interest income which will make the saver feel ‘richer’ in the sense that the saver will have extra unearned income. A ‘rise’ in income will induce the saver to raise both C0 and C1. ♦ borrower – higher r leads to the borrower facing higher interest charges on debt. Thus the borrower will be ‘poorer’ and have lower income. This will serve to reduce both C0 and C1. If r falls the income effect reverses. Savers are poorer and reduce both C0 and C1 whilst borrowers are richer and will raise both current and future consumption. The income and substitution effects of a change in interest rates for both a saver and a borrower are summarized in Table 1
TABLE 1 Substitution effect Income Effect Overall Effect
RISE IN R saver borrower ↓C0 ↑C0 ?C0 ↓C0 ↓C0 ↓C0
FALL IN R saver borrower ↑C0 ↓C0 ?C0 ↑C0 ↑C0 ↑C0
Considering the case of a rise in interest rates represented in Figures A and B. ♦ saver:- If the substitution effect is small relative to the income effect then the saver will choose consumption bundle A’ thus increasing C0. If the substitution effect is large relative to the income effect the agent will choose consumption bundle A” with reduced C0. ♦ borrower:- Both the income and substitution effect reinforce each other and the borrower will reduce C0.
(c) Graphical Representation
We can represent the income and substitution effects graphically by drawing a line parallel to the new, post interest rate change budget constraint through the initial equilibrium consumption bundle. This decomposition follows the procedure due to Eugene Slutsky. Figure C sets out the case for the saver. The substitution effectis isolated by line XY drawn parallel to the new, post interest rate rise budget constraint. XY sets out the trade off in the capital market now available at the higher rate of interest. Agents for whom the substitution effect is relatively strong will substitute a large amount of future consumption for C0. The income effect will serve to increase both C0 and C1 resulting in a final equilibrium at A”. In this case a rise in r will reduce C0.
1 x
A”
A’ A
Y1
Y
Figure C: The substitution effect is represented along XY. If the effect is weak it will result in a small movement along XY. The income effect will cause a move to A’. If the substitution effect is large, it will engender a large move along XY and the income effect will result in a move to A”
Y0
0
If the substitution effect is relatively weak, it will prompt a small substitution of C0 for C1 (i.e. a small move along XY). The income effect will raise both C0 and C1 resulting in the equilibrium consumption bundle changing from A to A’. At A’, C0 has risen compared with A.
7/ Conclusions
Our study of the 2 period model is motivated by a desire to establish a consumption function for a forward looking consumer. Capital markets allow the agent to shift consumption across time by lending or borrower. Our key results are ↑Y0 → ↑C0 and ↑C1 ↑Y1 → ↑C0 and ↑C1 ↑r → ?C0 and ↑C1 The consumption is the relationship between C0 and the relevant influences. Thus, we can write C0 = f(Y0,E(Y1),r) ( + , +, ?) This states that current consumption, C0, depends on current income, Y0, expected future income, E(Y1) and the rate of interest. The theory informs us that a rise (fall) in current or expected
income will raise (reduce) C0 but that a change in r cannot be determined, a priori. This consumption function is different to that set out by Keynes in the General Theory. A second key objective was to establish what restrictions required to be placed on the model to derive Keynes’ AIH. We established that liquidity constraints were necessary but not sufficient to establish the AIH. Not only is it necessary to eliminate borrowing via an imperfect capital market but this liquidity constraint has to bind on the agent both before and after a rise in current income. A binding liquidity constraint implies an agent for whom wants and needs are high relative to current level. This ensures that the agent will consume all current income in the period it is received and consume any increase in current income in the current. To establish a theory of aggregate consumption we can imagine that such an agent is representative of all agents in the economy. However, it is not plausible to argue that in industrialised capital markets all agents face no borrowing and are so poor that they spend all of current income in the period it is received. This may be a better approximation of the UK in the 1930s or of a poor LDC. It is not a realistic view of the UK, US or any other industrialised economy. Jim Stevens February 2004