# Mises Daily

## Did Rothbard "Borrow" the Income and Substitution Effects?

From the introductory undergraduate to the advanced PhD courses in microeconomics, most students are taught that the concept of an indifference curve is very *useful* in analyzing individual choice. I emphasize the term "useful" for a reason. No professor I have asked about the concept ever said it was literally true. I don't think anyone who understands the concept would ever try to claim that this is how humans actually make choices. Rather, we have been instructed to treat the microeconomic models only as *tools* for prediction of phenomena.

Murray Rothbard and other economists of the Austrian School have rejected the concept of indifference and have developed an approach based on Menger's law of diminishing marginal utility and Mises's axiom of action. This approach, built upon ordinal rankings of human ends, is formulated in Rothbard's treatise *Man, Economy, and State* as the value-scale concept. Professor Rothbard used this concept to derive the laws of diminishing marginal utility and of demand and supply, and he used it to explain price formation and other economic phenomena.

However, Bryan Caplan, a George Mason University professor, claims that this value-scale approach is inadequate for explaining the so-called income and substitution effects of a price change. Caplan claims that, while Rothbard's consumer theory is not capable of explaining the distinction between the income and the substitution effects of a price change, Rothbard still uses these terms in his work. Professor Caplan goes on to argue that Professor Rothbard has "borrowed" these terms from mainstream neoclassical theory.

This claim is based on the observation that the mainstream neoclassical derivations of the income and substitution effects are based on the use of indifference curves and the constrained-optimization framework. The implicit assumption here is that that the derivation of the income and substitution effects *in general* is somehow dependent on the concept of indifference, and thus not possible within Rothbard's theoretical framework.

The thesis of the following article is that Caplan is wrong. It can be shown that a price change does indeed have its income and substitution effects within Rothbard's theoretical framework. A clear indication that Professor Rothbard was aware of this, although he did not go through the steps of explicitly demonstrating the two effects, is the following paragraph of his *Man, Economy, and State*:

Allconsumers' goods are, on the other hand,partialsubstitutes for one another. When a man ranks in his value scale the myriad of goods available and balances the diminishing utilities of each, he is treating them all as partial substitutes for one another. A change in ranking for one good by necessity changes the rankings of all the other goods, since all the rankings are ordinal and relative. A higher price for one good (owing, say, to a decrease in stock produced) will tend to shift the demand of consumers from that to other consumers' goods, and therefore their demand schedules will tend to increase.[1]

Thus, Rothbard notes that individual value scales are formed using one's knowledge of the relevant money prices and that a change in prices will change the ordering of different items on one's value scale. Consequently, a price change will have its substitution effect. Moreover, the idea that money prices are the key element of individual value scales is also at the root of Mises's regression theorem, which Rothbard elaborated at some length.

It also may be obvious that, given a fixed stock of money, an increase (or decrease) in the money price of an item reduces (or increases) the purchasing power of that stock, and thus reduces (or increases) the quantities of different items that can be purchased. Essentially, this is the economic meaning of the income effect of a price change.

However, given the claims that the value-scale approach is somehow inadequate for defining the income and substitution effects, it seems that there are still some benefits to going through the steps of the value-scale approach. I will demonstrate how and why a price change affects the purchasing power of a stock of money, and how and why substitution between different items takes place.

In the following sections, I will first present the mainstream neoclassical framework for deriving the income and substitution effects of a price change, and then develop an example of how Rothbard's value-scale approach could be applied to the problem.

## The Indifference-Curve Approach

The mainstream neoclassical consumer theory rests on the concept of an indifference curve. Within this framework, an indifference curve depicts all combinations (or bundles) of quantities of two goods that are associated with exactly the same level of utility. It is said that, without money prices attached to any of the two goods, an individual would be indifferent among all these bundles — he or she does not prefer any of the bundles.

Next, some assumptions about the properties of the individual utility function and the associated indifference curves are adopted to facilitate mathematical operations, namely, finding a unique solution using calculus. I will not go into the details of these assumptions, as they are elaborated elsewhere at great length. Instead, I will use an example to demonstrate how the effect of a price change is analyzed within this framework.

Suppose that figure 1 represents preferences for apples and oranges, and the budget constraint of a hypothetical consumer, Jim, in a two-dimensional coordinate system.

Figure 1A graphical representation of a two-good constrained-optimization problem when the price of one of the goods changesJim faces the initial price of $1 per apple and $1 per orange, and he has some fixed budget to spend on apples and oranges. Budget line 1 shows all the combinations of apples and oranges that Jim can buy using his entire budget.

The indifference curve U1 depicts all bundles of apples and oranges to which Jim attaches exactly the same level of utility, U1. Any indifference curve closer to the origin of this coordinate system is associated with a lower level of utility. Similarly, any indifference curve further away from the origin is associated with a higher level of utility.

Jim will not want to buy any old combination of apples and oranges but the one that he values the most. This bundle is depicted by the point where his budget line is tangent to the indifference curve U1 (point A). Any other point on Jim's budget line would be on an indifference curve that is closer to the origin than U1, and thus associated with a lower level of utility. Consequently, point A depicts the combination of apples and oranges that Jim would purchase had he decided to spend his entire budget on apples and oranges.

Next, let us examine what happens when the price of oranges increases from $1 to $2 per orange. Now, Jim's budget line is represented by budget line 2. Oranges are now more expensive. Thus, for any given quantity of apples, Jim can now buy fewer oranges, using up his entire budget. For example, if Jim's budget was $5, he was able to buy, say, 1 apple and 4 oranges before the price increase. Now he can buy only 2 oranges if he wants to buy 1 apple.

In order to determine his optimal bundle, Jim will now find the point at which his new budget line is tangent to another indifference curve, U2, (point C). Point C depicts the combination of apples and oranges that Jim would buy had he decided to spend his entire budget on apples and oranges, given the new price of oranges.

It is evident that Jim is buying more apples and fewer oranges now. Let us focus on his consumption of oranges to further examine the effects of the price increase. First, we can see that his consumption of oranges has *decreased* by Δ oranges. This decrease could next be decomposed into the income effect and the substitution effect.

Figure 2 illustrates this decomposition process. Point B divides the total price effect (Δ oranges) into the income effect (IE) and substitution effect (SE). Point B belongs to a hypothetical budget line (BL 3) tangent to the indifference curve U2. The slope of BL 3 is determined by the initial prices of apples and oranges ($1 per apple and $1 per orange).

Figure 2A graphical representation of the income and substitution effects within the constrained-optimization frameworkThis line shows by how much the initial income would have to be reduced in order to bring Jim's utility down to U2. In a sense, this is intended to show that a price increase reduces individual utility in a similar way as does a reduction in income.

Another important characteristic of the point B is that this is a point of tangency between the hypothetical budget line BL 3 and the indifference curve U2. This means that if only the income-reducing effect was taken into account, with the relative prices staying unchanged, Jim's new optimal bundle would be B. Thus, the reduction in the quantity consumed between the points A and B can be attributed to the income-reducing effect of the price increase. This is the *income effect* of the price change.

However, in addition to the income-reducing effect, the increase in the price of oranges also makes apples more desirable relative to oranges. This is why Jim substitutes oranges with apples and "moves" from point B to point C as his budget line becomes steeper under the new price of oranges. This is the *substitution effect* (SE) of the price change.

Many neoclassical economists would argue that this is the end of the story and that nothing more or nothing different could be said. However, Murray Rothbard had important criticisms of the above-presented approach. In essence, these criticisms boil down to the fact that this is not how we, humans, actually make decisions.[2]

Taken literally, this approach would imply that humans solve complicated mathematical problems, involving differentiation of functions and many other operations, without even being aware of doing so. The typical defence of this mechanistic approach goes along the lines that the approach is useful as long as we don't interpret it literally but interpret it *as if* the consumers were making their choices in this manner.[3]

However, Rothbard was not satisfied with this justification and thus developed a different approach based on Menger's law of diminishing marginal utility and Mises's axiom of action. The next section presents an application of this approach to individual purchasing decisions where the individual is put in a similar context as before.

## The Value-Scale Approach

The consumer, Jim, from the previous section, would, according to Rothbard's framework, rank his ends in the order of diminishing marginal importance or utility. Next, he would assess the means available to satisfy these ends. In this case, Jim's specific ends are unknown to us, but we know that whatever they may be, the only means to satisfy them are the money that he owns and the apples and oranges that can be bought using that money. In the same fashion, he would then rank all his means in the order of diminishing marginal importance. In fact, he orders his actions in *time*, with the most important being performed first.

In the example presented above, Jim has some stock of money, say $5, that he wants to exchange for apples and oranges. The price of both apples and oranges is $1. Jim's particular ordering of apples, oranges, and money on his value scale is an empirical question, known only to him, but suppose that it looks like the distribution in figure 3.

Figure 3A hypothetical value scaleWe can see that he would exchange the first dollar for the first apple, since the first apple is more important to him than the first dollar. Similarly, he would exchange the second dollar for the second apple, and the third dollar for the third apple.

At this point, Jim would stop exchanging his money for apples. This is because acquiring the fourth apple is less important to him than owning the fourth dollar — the fourth apple is lower on his value scale compared to the fourth dollar.

Moreover, the first orange is more important than the fourth dollar. This is why Jim would benefit from exchanging the fourth dollar for the first orange. Finally, he would exchange his fifth dollar for the second orange.

Jim has now exchanged the entire stock of money that he allocated for purchasing apples and oranges. He bought 3 apples and 2 oranges. In mainstream neoclassical language, this would be his "optimal" bundle of apples and oranges.

The next question that needs to be answered is what happens when the price of oranges increases from $1 to $2.

Looking at Jim's value scale in figure 3, it can be seen that he would not be able to buy the second orange. This is because, after spending the third dollar on the third apple, Jim would need to pay 2 dollars for the first orange. This would exhaust his entire stock of money, and thus, Jim would now have 3 apples and only 1 orange.

But this analysis ignores the fact that Jim's value scale is price-dependent. Any change in the price would involve a reordering of different items, or, to reiterate Rothbard's words:

A higher price for one good (owing, say, to a decrease in stock produced) will tend to shift the demand of consumers from that to other consumers' goods, and therefore their demand schedules will tend to increase.[4]

In fact, what I have done here is to demonstrate that, *ceteris paribus*, due to the increase in the price of oranges, Jim would have less money to spend on other items that are ranked lower on his original value scale. This is the *income effect* of a price increase.

The next thing that needs to be done is to take into account the fact that Jim prefers paying less for the same thing. This is because paying less for the same thing leaves more money available for satisfying other ends. Since it is always better to satisfy more ends compared to less, it is also better to pay less for the same thing. Consequently, Jim would choose a $1 orange over a $2 orange, given the chance to do so.[5] This implies that the first $2 orange must be lower on his value scale compared to the first $1 orange.

We could extend this reasoning and conclude that the first $3 orange would be ranked lower than the first $2 orange; the first $4 orange would be ranked lower than the first $3 orange, etc. The same applies for the second, third, and so on. This reordering could be continued indefinitely. Consequently, there is no lower bound on how low an item can move down the value scale as its price increases.

Thus, it may well be that the first $2 orange ends up not only below the first $1 orange but also below the fifth apple. Again, where, exactly, different items will end up is an empirical question that depends entirely on Jim's subjective valuations of his means and ends. At this point, I will hypothesize that Jim's new value scale looks like figure 4.

Figure 4A hypothetical value scale after a price increaseAccording to this value scale, Jim will still exchange the first 3 dollars for the first 3 apples. However, the $1 oranges are not available anymore, and the $2 oranges are placed below the fifth dollar. This means that the fourth dollar will be exchanged for the fourth apple, and the fifth dollar will be exchanged for the fifth apple, since each of these two consecutive dollars is valued less than each of the two apples.

Finally, Jim ends up exchanging his $5 for five apples. Consequently, the quantity of oranges bought dropped from the initial 2 to 0. This is the total effect of the increase in the price of oranges. This total effect can be decomposed into the *income effect*, where, as shown earlier, the quantity of oranges bought fell from 2 to 1, and the *substitution effect* where the quantity of oranges bought fell from 1 to 0. However, these two effects happen simultaneously. The purpose of the chronological presentation is just to indicate the elements of the two effects with more transparency.

A similar exercise could be done for a price decrease. Then, the amount of money available for all other goods, placed lower on the original value scale, would increase. In addition, the relative desirability of the good whose price decreased would then increase. Thus, the income and substitution effects work both ways within the value-scale framework.

In this particular example, the effect of a price change on the quantity of oranges purchased was observable. But it is also possible that we do not observe any change in the individual quantity purchased of some good whose price increased or decreased. In these cases, we would likely observe only the income effect on the purchased quantity of other goods.

Suppose Jim was a passionate smoker. Then, for him, an increase in the price of cigarettes would likely have an impact only on the quantity of other goods that he would buy, but not on the quantity of cigarettes. This is because cigarettes are very high on his value scale. The price change may not be sufficient to "push" some quantity of cigarettes down his value scale to become less important than the last dollar in his stock of money. Rather, the reduction in the purchasing power of his stock of money would bring about a change in, say, the number of haircuts Jim has, and/or the number of evenings out, etc.[6]

Clearly, more-complicated examples involving not only quantity but also quality of goods could be thought of. But this would be outside of the scope of this article, where the purpose was to demonstrate the derivation of the income and substitution effects using the value-scale framework.

## Conclusion

The value-scale concept, when applied with all its elements, is capable of explaining the income and the substitution effects of a price change. Caplan's claim that the value-scale approach is inadequate for explaining these effects is simply an incorrect interpretation of Professor Rothbard's theoretical framework.

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##### Notes

[1] *Man, Economy, and State*, p. 282.

[2] For more details, see Rothbard's *Man, Economy, and State*.

[3] Nicholson. 2005. *Microeconomic Theory: Basic Principles and Extensions*. 9th Edition. Thomson: Willard, OH.

[4] *Man, Economy, and State*, p. 282.

[5] Except in the case that Jim has some peculiar need for paying a higher price for oranges. This would, in turn, imply that the act of paying a higher price for oranges is one of Jim's ends. This would imply that the act of exchange is an end to itself, which removes the conventional economic meaning of exchange. Thus, it is assumed that paying a higher price for oranges is not one of Jim's ends.

[6] Neoclassical economists using the indifference-curve framework might approach this problem using a Leontief utility function. The interested reader is encouraged to see a depiction of this function here.

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