Archive for December, 2008

Policy implications

Suppose that government wished to encourage individuals to work for longer hours. How might this be done? One method operates by shifting upwards the schedule of additional utility gained from extra hours of paid employment. Suppose that workers currently pay tax on additional labour income. If the tax rate were to be reduced with other things remaining constant, this would have the effect of shifting this function upwards. Note also that if high income levels are taxed at high marginal rates, reductions of these marginal rates will shift the curve upwards and tend to make it fall less steeply from left to right. In either case, the individual will choose to work more hours. The reason for this is that the terms of the trade-off between utility from work-income and utility from leisure have altered; a lower tax rate has increased the attractiveness of work relative to leisure.

This analysis can only be suggestive of the possible effects of changing tax rates. There are very many other effects that one would need to consider before any confidence could be placed in one’s predictions. In addition to inducing existing workers to work longer hours, lower tax rates may also encourage more people to participate in the labour market. Some individuals may choose not to work when tax rates are high, but may choose to do so when tax rate are lower. The effect of tax rates on the incentive to work, and on the choice of hours, has been at the heart of tax reform programmes in the 1980s and 1990s, and has been an important factor contributing to reductions of marginal tax rates in a large number of countries.

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Utility and consumer preferences

Utility refers to the satisfaction or enjoyment which comes from the consumption of goods over some interval of time. For simplicity, suppose that only two goods are available for consumption. It does not really matter what these are: we simply label the two goods x and y. An individual gets utility from consuming particular bundles of the two goods. The amount of utility obtained is described by the individual’s utility function, which can be written as

where U denotes the individual’s total utility, and Qx and Qy denote the quantities of goods x and y consumed per period of time. The qualification “per period of time” is an important one here; we are building a theory of demand, and demand is a flow concept. Different individuals will, of course, obtain utility from various bundles of goods in different ways and to different extents. However, irrespective of what any individual’s preferences might be, we assume that each person has a utility function which incorporates those preferences. We now introduce the idea of an individual’s indifference curve for the goods x and y.

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The individual’s budget constraint

As individuals are utility maximisers, each person seeks to be on an indifference curve as far away from the origin as possible. The extent to which this is possible is limited by the individual’s budget constraint. The budget constraint states that expenditure cannot exceed income. Let m be money income, and Px and Py be the prices of goods x and y. These prices cannot be changed by an individual consumer. The budget constraint can be written as

and is represented diagramatically in Figure 3.5. If the prices of the goods x and y are Px1 and Py1 respectively, and available money income is m1 , the individual’s budget constraint is given by the shaded area. Given this set of prices and money income, it shows all combinations of the two goods that it is feasible for the consumer to purchase. For reasons that will be clear shortly, the individual will choose a combination of goods on the outer diagonal boundary of the budget constraint (the line denoted M1 ). From now on it is this line that we shall call “the budget constraint”.

The slope and position of the budget constraint depend on two factors

  • the relative prices of the two goods: that is the ratio Px : Py . Relative prices determine the slope of the budget constraint

  • the level of money income. This determines, for any given set of prices, how far the budget constraint is located away from the origin.

An individual’s budget constraint will change whenever there is a change in prices or money income. For example, the budget constraint shift from M1 to M2 in Figure 3.6 occurs when money income is increased from m1 to m2 but with prices remaining unchanged. On the other hand, the budget constraint shift from M1 to M3 if the price of good x falls to Px2 but money income and the price of y remain unchanged.



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An indifference curve shows all combinations of the two goods which yield a particular level of utility.

The shape of an indifference curve will depend upon the utility function of the individual concerned. One likely shape for an indifference curve is shown in Figure 3.3. By definition, as the two points labeled 1 and 2 both lie on one indifference curve, they yield an equal utility level, . Indeed, all points along the indifference curve yield that amount of utility. The reason why the term “indifference curve” is used is as follows: since each combination yields the same amount of utility, a rational consumer will be indifferent between alternative bundles along any one such curve.

An indifference curve demonstrates that the individual is confronted with trade-offs in consumption. In moving from point 1 to point 2, for example, the consumer can give up some of good y in return for more of good x, without changing the amount of utility he or she attains. If an indifference curve were linear – that is, it could be drawn as a straight line – the terms of this trade-off would not alter as the individual changes the proportions in which the two goods are consumed. However, the indifference curve shown in Figure 3.3 is not linear, and so the terms of this trade-off are not constant. As we have drawn it, an individual has to give up increasingly large amounts of y in return for additional units of x if utility is to remain unchanged.

For any utility function, there will be an indifference curve for each feasible utility level. We show three indifference curves in Figure 3.4. These correspond to the different utility levels, , and , such that > and <. The individual is indifferent between the combinations at points 1 and 2 as they confer the same utility level (). Similarly, the individual is indifferent between the combinations at points 3 and 4 as they confer equal utility (). However, the combinations shown by 3 and 4 are preferred to those at 1 and 2. Any combination on the indifference curve for utility level is preferred to any combination on that for .


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Utility maximisation subject to a budget constraint

For a consumer with a limited income (and that is all of us once our work-leisure choices have been made), the best or efficient allocation of spending is the one which gives the greatest total utility. From all combinations of goods that are feasible to purchase given the budget constraint, the individual seeks that combination which will give the highest level of utility. We illustrate a solution to this problem in Figure 3.7. At the point , utility is maximised at the level U=. To confirm that this is true, consider any point other than which also satisfies the budget constraint. An indifference curve passing through such a point must lie below the highest attainable one, and so would yield lower utility. Note that the individual would prefer to be at a point on a higher indifference curve (such as U= ) but that is not possible as it would require that spending exceed available income.

One characteristic of this solution is of particular interest. Provided that indifference curves are smooth and continuous, and are bowed inwards in the way we have illustrated them (that is, they are convex from below), then the utility maximising combination of x and y purchased will lie at a point of tangency between the budget constraint and an indifference curve.

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The efficient allocation of a consumers budget

We have just seen how a utility maximising consumer allocates a budget between two goods. It is possible to generate a simple rule from this analysis that enables us to get a good intuitive understanding of consumer behaviour. Before we derive this result, another concept is needed – the concept of marginal utility.

Let us return to the notion of an individual’s utility function; we wrote this earlier as

Suppose that, in some period of time, an additional unit of good x is consumed without any change taking place in the amount of y which is consumed. This additional consumption of good x will lead to an increase in the consumer’s total utility. The marginal utility of good x is the change in total utility resulting from the consumption of an additional unit of that good. It can be represented by:

where the symbol [read as “delta”] means “a small change in” some variable of interest, Qx is the quantity of X consumed per period, and MUx is the marginal utility derived from good X. In a similar fashion, MUy = U/Qy, is the change in total utility resulting from the consumption of an additional unit of good y (with consumption of x unchanged).

Now let us return to the tangency condition we noted above. When the consumer is maximising utility, the slopes of the budget line and the indifference curve are equal. The slope of the budget line is given by (the negative of) the price of x divided by the price of y. That is

Slope of budget constraint =

On the other hand, the slope of an indifference curve at any point along it is given by (the negative of) the marginal utility of x divided by the marginal utility of y. That is

Slope of indifference curve =

The derivation of these two results about the slopes is given separately in Box 3.1. If you would prefer to avoid the maths, you will have to take the results on trust.

Finally, as these two slopes are equal at a point of utility maximisation, the following must be true when an individual is optimising his or her expenditure pattern:

=

which after multiplying through by -1 and then rearranging gives

=

What does this result tell us? It shows that maximum utility is achieved by allocating total income between different goods in such a way that the marginal utility derived from each good consumed per unit of money spent is equal over all goods. If we interpret MUx and MUy as the “values” derived at the margin from small, additional amounts of x and y respectively, then we can say the following. A utility maximising individual purchases goods in quantities such that the “value-for-money” derived from each good at the margin is equalised.

To obtain an intuitive understanding of the condition, it may be helpful to ask what would happen if the consumer’s pattern of purchases did not equalise ‘value for money’ at the margin for each good purchased. If these values for money ratios are not equal, then some goods must offer more value for money than others. A rational individual would switch expenditure between goods, away from those with low value for money and into those giving high value for money. This reallocation would enable the individual to get more total utility from the same income level, demonstrating that the original allocation was not an efficient one.

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DERIVATION OF THE SLOPES OF AN INDIFFERENCE CURVE AND A BUDGET CONSTRAINT


Consider one indifference curve. Let the individual consume a small additional amount of x. The effect of this on total utility is given by the rate of change of utility with respect to a change in x, U/Qx , multiplied by the amount of the change in x, Qx. Next suppose that the individual reduces his or her consumption of y in such a way that the loss of utility caused by having less y exactly compensates for the gain in utility from having more x. The loss in utility from having less y is given by the rate of change of utility with respect to a change in y, U/Qy , multiplied by the amount of the change in y, Qy.

If we constrain the changes in x and y to be along one indifference curve, as illustrated in Panel a in Figure 3.9, then the overall change in utility must be zero. Therefore we have

Rearranging this we obtain

and so

But the final term in this expression is the negative of the ratio of the marginal utility of x to the marginal utility of y. [See panel b in Figure 3.9]. So we conclude that

Slope of indifference curve =

The slope of the budget constraint

The equation of a budget constraint is

Rearrange this to give an equation for Qy in terms of Qx as follows:

Therefore, the slope of the budget line is given by – (Px/Py), as shown in panel (c).



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Price changes and an individual’s demand curve

We now demonstrate that if a good rises in price, the quantity demanded of it is likely to fall. A geometric analysis of maximising utility subject to a budget constraint is used. Begin with a situation in which a consumer is currently in a utility maximising position, at the point indicated by “a” in Figure 3.10. Money income is m, prices are Py1 and Px1, and maximised utility is U1.

Now suppose that the price of good x falls from Px1 to Px2, but money income and the price of good y remain unchanged. This rotates the individual’s budget constraint anti-clockwise; its position changes from the line connecting the points (m/ Py1) and (m/ Px1) to the line (m/ Py1) to (m/ Px2). The set of feasible consumption choices is now enlarged, and the consumer attains a new utility maximum at the point “b” on the indifference curve U2.

What has happened to the quantity of good x that the consumer demands as the price of good x falls? It has increased, from x1 to x2 . The individual’s demand curve for x is, therefore, negatively sloped (at least in this region) as a lower price has resulted in a greater quantity demanded.

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The substitution and income effects of a price change

Although a consumer’s demand curve for any good x is likely to be downward sloping we cannot be certain of this because of the presence of the income effect of a price change. Whenever a price changes, that change will affect the demand for the good in two ways:

  • The price of this good, relative to others, has changed. This induces a substitution effect. The change in relative prices will lead to a re-allocation of spending between goods. Fewer of the good which has become relatively more expensive will be purchased and more of the good which has become relatively less expensive will be bought. The substitution effect will always be negative: a change in the price of a good will lead to a change in the opposite direction in the quantity demanded of it

  • A price change (with a fixed level of money income) will change the consumer’s real income (the purchasing power of the money income). As the consumer’s real income is changed, there will be a change in the amount of this good (and others) purchased. However, the direction of this change is uncertain, for the reasons we explained in Chapter 2. If the good in question is a normal good, higher real income will increase the quantity demanded. Conversely, if the good is an inferior good, higher real income will decrease the quantity demanded.

The demand curve for a good describes the overall relationship between price and quantity demanded, and so incorporates both the substitution effect and the real income effect of a price change.

We can obtain a graphical representation of the decomposition of a price change into substitution and income effects in the following way. First, the substitution effect of the price change is identified. The income effect is then obtained as the difference between the total effect and the substitution effect of the price change.

The substitution effect can be identified by asking how much demand for the good would change if

  1. its price changes, and

  2. the consumer is compensated for a price increase (or financially penalised for a price fall) by just the amount required to prevent the consumer from gaining more utility than he or she had prior to the price change.

Refer to Figure 3.11 which illustrates the reasoning.. Suppose the consumer initially allocates his or her money income of m by purchasing y1 units of good y (at the price Py1 ) and x1 units of good x (at the price Px1). This is a utility maximising expenditure pattern obtains utility level U1 . We next suppose that the price of good x falls, with money income and the price of good y remaining constant. This causes the individual’s budget constraint to rotate anti-clockwise. The consumer now switches expenditure to the allocation shown by point b, at which utility is maximised at the higher level, U2.

The consumption of x increases from x1 to x3. This is the total effect of the price change. Let us now conduct the following hypothetical experiment. Starting from the new (post price change) budget constraint, we take away from the consumer the maximum amount of income that is just compatible with him or her being able to get the original utility level U1 at the new set of prices. This results in the consumer’s budget constraint moving to the line de. With this budget constraint (and at the new set of prices, Py1 and Px2 ), the highest level of utility level attainable is U1, its original level. This is achieved by purchasing x3 units of x and y3 units of y.

This has removed the income effect of the price change. Any change in the demand for good x can, therefore, be attributed to the substitution effect alone. For good x, the substitution effect (SE) of the price fall consists of the change from x1 to x3. The income effect (IE) of the price change consists of the change in quantity from x3 to x2. To see why this is so, note that if we were to begin at the point c and, without changing relative prices, return to the consumer the money income previously taken away, the individual would move from c to b, thereby raising consumption of x from x3 to x2.

The following can be said by way of conclusion.

Total effect of a price change = substitution effect + income effect

(x1 to x2) (x1 to x3) (x3 to x2)

The case we investigated in Figure 3.11 was one in which the two component parts of the overall effect work in the same direction, so reinforcing one another. It can be deduced from this that x is a normal good, as an increase in real income has resulted in an increase in the quantity demanded of the good.

But as some goods are inferior, there are other possibilities. These are illustrated in Figures 3.12 and 3.13. In these two cases, x is an inferior good, and so the income effect of a price change works in the reverse direction to the substitution effect, thereby reducing the size of the overall effect. Figure 3.12 shows the case where the income effect only partially offsets the substitution effect. Overall, a price fall of good x results in a greater quantity being demanded, even though the size of the overall change has been reduced by the income effect.

Figure 3.13 illustrates a theoretically plausible, but very unlikely, situation. The extent of the income effect is so great that it not only reduces the overall effect, it actually reverses it. Here, a fall in the price of good x results in a fall in the quantity demanded of that good. The demand curve has a positive slope!

Empirical evidence shows that there are few, if any, Giffen goods in practice. Theory also suggests the probability of Giffen goods is extremely small. Most goods are normal goods. But even where a good is inferior, the income effect is likely to be small in magnitude in comparison with the substitution effect. Unless a very large proportion of income is spent on one good, the income effect will be small for moderate sized price changes. Only in very exceptional circumstances would an income effect dominate the substitution effect in the way required for Giffen goods.

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From individual demands to the market demand for a good

To this point , our focus has rested entirely on the individual consumer. For economic and business analysis, though, it is market demand that matters. How does market demand relate to the demands of individual consumers? The answer is very simple. The market demand for a good is simply the sum of all individual consumer demands for that good. To obtain this sum, take one price and find the quantity demanded by each individual at that price. Then sum these quantities to find the market demand at that price. Repeating this analysis for all possible prices, we obtain the market demand schedule.

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