3. The Demand Side of Output Markets – Elements of Consumer Behaviour
3.4. The Consumer’s Optimal Choice
Summing up the previous sections, the consumer ranks the various commodity bundles comparing the satisfaction, or utility they offer. The bundles generating the same feeling of satisfaction, or the same utility, belong to the same indifference curve, and the consumer is willing to exchange these bundles. Suppose, that the consumer has two such bundles, the first bundle containing x1 units of commodity x, and y1 units of commodity y – as is shown by point A in Figure 3.7, using the notation A(x1;y1) for this bundle. The second bundle is represented by point B with more of x and less of y: B(x2,y2). Both bundles give the same utility, denoted by U2. Having the same utility, the consumer is willing to substitute one for the other. Let’s have a closer look at the substitution of bundle A for bundle B.
The substitution will be done in two steps. First take bundle A, and decrease the amount of y to the level of y2, keeping the amount of x at x1, so that bundle A is exchanged for bundle C(x1,y2). Obviously, this exchange results in a decrease of total utility, as one commodity has been constant, the other has decreased. Therefore point C lies below our initial indifference curve (in another curve of a lower utility level U1). Then in the second step bundle C is exchanged for bundle B, by keeping the amount y2 constant while increasing the amount of commodity x from x1 to x2, increasing also the total utility level of the bundle, and getting back to the initial indifference curve U2.
In the first step (substituting C for A) a negative change of U1-U2 takes place in total utility, with the amount of x kept constant, so the definition of the marginal utility of commodity y can be applied (MUy = DTUy / Dy ) to measure the resulting decrease in total utility: DTUy=Dy × MUy. In the second step (substituting B for C) a positive change of U2-U1 is obtained in total utility, with a constant amount of commodity y consumed. Thus using
11 In the real world there are consumption situations with increasing marginal utilities: for devoted collectors of something an additional unit of the collection yields increasing satisfaction. The same is true for harmful addictions – as drugs or alcohol. However, increasing marginal utility may be experienced even for normal goods, when a new product is just introduced to the consumer, and the consumption of the first unit creates only moderate satisfaction. Then as the consumer learns to enjoy the commodity, the successive units bring about growing satisfaction, and dinimishing marginal utility described by Gossen will be encountered only after having consumed a substantial amount
the definition of marginal utility of commodity x (MUx = DTUx / Dx) the resulting increase in total utility is: DTUx=Dx × MUx. And finally, the sum of these two successive steps gives the eventual change in total utility – the utility level decreasing from U2 to U1 and then increasing from U1 to U2 – the sum of these changes equals zero.
Therefore the total change in utility is: TUy+DTUx = 0, so Dy×MUy+ Dx×MUx = 0.
Rearranging this equation12 the following formula is obtained: -Dy/Dx = MUx / MUy. However, as it was seen when defining the marginal rate of substitution, and also shown here in Figure 3.7, MRS is equal to the proportion of the changes in y and x (assuming infinitely small change in commodity x – which was also the assumption when defining the marginal utilities of commodities x and y). Therefore MRS = -Dy / Dx, so taking into account the relationships above we get: MRS = MUx / MUy, that is, the marginal rate of substitution is equal to the ratio of the marginal utilities of the commodities x and y at the consumption levels of the actual bundle.
Figure 3.7: The relationship of Indifference Curves and Marginal Utility
Source: Author’s own construction
Now we have all the necessary tools to select the optimal bundle for the consumer.
As it was stated in section 3.1 the consumer’s choice is limited by his/her income and the unit prices of the commodities, and these define the budget constraint (budget line) containing all the commodity bundles available for choice. The question is how to find the best of these bundles, that is, how to choose the point of the budget line that has the highest utility level.
The process of the consumer’s choice is illustrated by Figure 3.8. Plot the budget line and the indifference map in the same diagram. Some of the bundles of the indifference curves lie below the budget line – being cheaper than the consumer’s income -, others lie just in the budget line – costing just the consumer’s income - and others lie above the budget line, being unattainable. There are two bundles in the indifference curve U1 (marked by the two stars in the figure) that are just as expensive as the consumer’s income. However, it is possible to attain the utility level U1 spending less, as the bundles lying in the indifference curve between these two bundles all are falling below the budget line, indicating lower expenditures.
Therefore the consumer can choose better than the bundles denoted by stars. Because the same utility level can be attained cheaper, with the current level of income higher utility is attainable. Such a higher utility level is represented by curve U2, but again, there are bundles
12The equation is rearranged as follows: Dy × MUy+ Dx × MUx = 0, that is, -Dy × MUy= Dx ×MUx . Hence, dividing both sides of the equation by the quantity MUy the following is obtained: -Dy = Dx × MUx / MUy, and finally dividing both sides by Dx the equation becomes: -Dy / Dx = MUx / MUy .
along this curve that lie below the budget constraint, so this curve is not optimal either. The curve U3 however, is unattainable, all the bundles of this curve lie beyond the budget constraint. The consumer should find the indifference curve of the highest utility level among all the curves having at least one attainable bundle, - that is, at least one point of the curve lying in the budget line, and no point of the curve falling below the budget line. This is the indifference curve denoted by U* in Figure 3.8. Therefore the consumer’s optimal choice is the bundle represented by the tangency point of the indifference curve U* and the budget line (assuming well-behaved indifference curves).
Figure 3.8: The Consumer’s Optimal Choice
Source: Author’s own construction
The budget line is tangent to the indifference curve U* in the optimal point.
The slope of the tangent line to the indifference curve is defined by the marginal rate of substitution: dy/dx = -MRS. On the other hand, the slope of the budget line is –by definition - determined by the price ratio of the two commodities, that is - px / py . In the point of the optimal bundle the two slopes must be equal (as the tangent line to the indifference curve is the budget line itself), so the equation MRS = px / py holds. From the previous sections we know that MRS = MUx / MUy , the marginal rate of substitution is equal to the ratio of the marginal utilities of the two commodities. Then, for the optimal bundle the following relationship must hold: MUx / MUy = px / py, meaning that the ratio of the marginal utilities – representing the consumer’s opinion – must be the same of the ratio of the market prices of the commodities, representing the opinion of the market.
Rearranging the relationship:
MU
x/ p
x= MU
y/ p
y. This formula can be generalised to more than two commodities:MUx / px = MUy / py= MUv / pv= MUw / pw ...etc
.
Thus Gossen’s Second Law can be stated as (Farkasné Fekete – Molnár, 2007;
Kopányi, 1993; Samuelson-Nordhaus, 1987): A consumer will spend his/her income in an optimal way achieving maximum satisfaction when the marginal utility of the last unit of income spent on each good is exactly the same (this is called the equimarginal principle).
The above relationship offers a simple way of deciding whether a particular bundle of commodities is optimal or not. An example is shown in Table 3.1.
As the table shows, the consumer buys two products, chocolates and oranges. For both products the table shows the changes in utility generated by increasing consumption,
that is, the marginal utilities. Thus, for instance, eating 2 units of chocolates instead of 1, the consumer’s utility increases by 50 units, so the marginal utility is 50 (assuming no other change in the circumstances). As we know, the consumer spends his/her income in the best possible way when the ratio of the marginal utility of the last consumed unit and the unit price of chocolate is equal to the ratio of the marginal utility of the last consumed unit and the unit price of oranges. As the table shows, this is the situation with consuming 3 units of chocolates and 5 units of oranges. The decision process is the following: At first the consumer decides whether to choose an orange or a chocolate as the first unit to consume. As the value of MU/p is higher for the first unit of oranges (2.00), than for the first unit of chocolates (1.33), the first consumption choice is to buy 1 orange, spending 20 HUF. The second purchase is another orange, as the MU/p value of the second orange is 1.50, is still higher than the same value for the first chocolate (1.33). The consumer spends again 20 HUF. The third purchase, however, is a unit of chocolate, because the MU/p value for the first chocolate is 1.33 while it is only 1.00 for the third orange. Similarly, the fourth purchase is an orange again, as the ratio of marginal utility to price is 1.00 for the third orange, and it is only 0.83 for the second chocolate. The fifth purchase is a chocolate (with 0.83 for the second chocolate while it is only 0.6 for the fourth orange), while the order of the sixth and seventh purchases cannot be decided, because the MU/p values for third chocolate and the fifth orange are the same 0.5, so the order of these purchases does not matter for the consumer.
Table 3.1: Example for Maximising the Consumer’s Utility
Chocolates Oranges
Chocolate
s MUchoc’s Pchoc’s MUchoc’s/Pchoc’s Orange
s
MUoranges Poranges MUoranges/Poranges
1 db 80 60 1,33 1 db 40 20 2,00
Source: Adapted by Farkasné Fekete – Molnár (2007), page 82.
Naturally, to attain the optimal choice the consumer must have sufficient income, that is 280 HUF in the example (with three chocolates costing 180 HUF and five oranges costing 100 HUF). With income less than this amount the decision process described above will stop earlier, while with more income the process continues, as shown in the table, where the purchase of 6 chocolates and 7 oranges also satisfy the equimarginal principle.
When the consumer wishes to decide whether a given commodity bundle is an optimal bundle or not, a similar method is followed. Suppose that the consumer’s income is 500 HUF, and the bundle he/she intends to buy is 5 chocolates (costing 300 HUF) and 10 oranges (costing 200 HUF). With this bundle, however, MUchoc’s/Pchoc’s = 0,17 >
MUoranges/Poranges= 0,025. Therefore the decision is not optimal. How to rearrange the consumer’s bundle? The MU/P value should be decreased for the chocolates and increased for the oranges. As the consumer cannot change the unit prices of the commodities, the marginal utilities should be changed, namely decreased for chocolates and increased for oranges. This can be attained, as is shown in the table, by increasing the amount of
chocolates and decreasing the amount of oranges. Thus the consumer should buy more chocolates, and the additional money for that will be spared by buying less oranges.
3.5. The Impact of Changes in Incomes and Prices on the Consumer’s Optimal