Income and Substitution Effects: A Rejoinder to Professor Joseph Salerno

Abstract: Professor Joseph Salerno (2019) has commented on my recent reconstruction of the income effect from a causal-realist perspective (Israel, 2018b). In this rejoinder, I clarify my position and show that the main points of criticism in Salerno’s response are unfounded. In particular, I show that my argument does not involve a claim of greater “realism of assumptions” and it by no means contradicts the law of demand. Moreover, I work out in more detail the similarities and differences of my approach to the standard neoclassical decomposition of income and substitution effects. I show that my approach is closer to the Slutsky decomposition as opposed to the Hicks decomposition.

JEL Classification: B53, D11

---

Karl-Friedrich Israel ([email protected]) is senior researcher at the Institute for Economic Policy at Leipzig University, Germany. The author would like to thank Kristoffer Hansen, Dr. Tate Fegley and Professor Guido Hülsmann for fruitful discussions and some very helpful comments on an earlier version of this rejoinder.

---

1. INTRODUCTION

Starting from Professor Salerno’s (2018) refutation of the income effect, I have recently argued that the income effect should not be discarded. Rather, the neoclassical theory of the income effect can be reconstructed along causal-realist lines (Israel, 2018b). Salerno (2019) has honored my paper with a critique, which provides a welcome opportunity to clarify my position.

This rejoinder is structured as follows. Section 2 contains a brief exposition of the standard microeconomic analysis of income and substitution effects. This will highlight the backdrop against which our debate arose. Section 3 contains a discussion of the causal-realist point of view on demand analysis. Here I discuss the assumptions underlying the imaginary construct of the demand curve and show to what extent the causal-realist approach differs from standard neoclassical analysis. In section 4, I proceed to clarify the meaning of realism in economic analysis and argue that my proposed solution by no means runs into a contradiction with the infamous law of demand, as Salerno claims. I conclude in section 5.

2. THE STANDARD MICROECONOMIC ANALYSIS OF INCOME AND SUBSTITUTION EFFECTS

In modern neoclassical microeconomics consumer behavior is typically modeled by means of mathematical utility functions and budget constraints. In the standard scenario a consumer chooses between quantities of different goods,

, as a function of their unit prices,

, and the individual’s available income or budget,

. Given the utility function,

, consumer choice is thus described as a maximization problem with a side constraint:

given that:

.

Solving this maximization problem yields the so-called Marshallian demand functions (also called primal demand) for the various goods:

.

In the above optimization problem, the consumer chooses the optimal bundle of goods, that is, the bundle that maximizes the utility function, under the constraint that monetary expenses do not exceed the available budget.

An alternative way of formalizing consumer choice is the following:

given that:

,
where

is a given level of want satisfaction or utility. In this version the optimal bundle that the consumer chooses corresponds to the cheapest bundle that yields this given level of utility. Instead of maximizing utility given the costs (i.e. the budget), the consumer is minimizing costs given a certain level of utility. The solution to this minimization problem yields the so-called Hicksian demand functions (also called dual demand):

.
The conceptual difference between the Marshallian and the Hicksian demand functions is straightforward. Marshallian demand keeps nominal income

(i.e. the budget) constant, whereas Hicksian demand keeps real income

(i.e. the level of want satisfaction or utility) constant. As a result, the Marshallian demand captures both income and substitution effects, whereas Hicksian demand only captures the substitution effect. To be more precise, it captures the Hicks-substitution effect as opposed to the Slutsky-substitution effect. The latter can be analyzed on the basis of the Marshallian demand function, if the initial nominal budget is adjusted for any given price change, such that the optimal bundle, which would have been chosen at the initial price and the initial budget, just remains affordable at the new price and the adjusted budget. This yields the so-called income-compensated Marshallian demand function.1
Let us consider a simple example to push the standard analysis closer to where we want to go with it. We consider a case with two goods, one of which is money. This has some analytical convenience as the money price of money is always 1, that is, one US dollar costs one US dollar and one euro costs one euro. In standard terminology, such a good is referred to as the numéraire good. It is also convenient to include money in the two goods example, because it makes the example somewhat more general. The demand for money can be seen as a placeholder for all the other goods that the consumer might want to buy, while we focus on the demand for one specific good. Let

be that specific good. Its unit price is

. The other good,

, is money. Hence,

. We assume a standard utility function of the Cobb-Douglas form:

. For the sake of simplicity, let us say that

. The Marshallian optimization problem is thus:

given that:

.

Solving the optimization problem yields the following Marshallian demand functions:2

;

.
Assume that the budget is

and

. The quantities demanded would be

and

. The attained level of utility is

.

In contrast, the Hicksian optimization problem is:

given that:

.

Solving the optimization problem yields the following Hicksian demand function:3

;

.
For the same level of utility as attained in the Marshallian example,

, and the same price,

, we obtain exactly the same quantities demanded:

and

. However, for any other price

, given the budget

in the Marshallian case and given the utility level

in the Hicksian case, the quantities demanded of

are different:

;

.

The two functions are plotted in Figure 1.

Figure 1. The uncompensated Marshallian demand curve and the compensated Hicksian demand curve

Let us now consider the effect of a price change from 

  to 

. The Marshallian demand at the new price is

and the Hicksian demand is

, which is much smaller. This is because Hicksian demand only captures the Hicks-substitution effect (in this case:

) and not the income effect. Marshallian demand captures both. The income effect of the price drop is positive because the good is normal (and not inferior).4 The Hicks-income effect in this example is thus

. The overall effect of the price drop, as captured by the uncompensated Marshallian demand function, is the sum of the substitution and income effects:

. Figure 2 illustrates the example.
Figure 2. The Hicks-substitution and income effects

The second standard version of distinguishing substitution and income effects is based on the income-compensated Marshallian demand function (this is the Slutsky approach). It is constructed in the following way. We take the same point of departure, namely,

and

. Marshallian demand is

. This means that the consumer buys 10 units of good

and keeps 50 money units in the cash balance. Any price change will now be compensated with respect to this reference bundle, in such a way that the consumer is able to acquire exactly the same bundle (

). If the price drops from

to

, the consumer would need only 40 money units (instead of 50) to buy

. Hence, the compensated budget is

(instead of

). In general, for any given price change,

, the compensated budget is

. The compensated Marshallian demand curve at the new price

is thus:

.
For the same price drop, from

to

, the income-compensated Marshallian demand function yields a different result from the Hicksian demand function:

as opposed to

.
The difference is that Hicksian demand keeps the level of utility constant, while income-compensated Marshallian demand keeps the purchasing power constant, in the sense that exactly the same bundle (and no unit more) could be bought at the new price with the compensated budget. The substitution effect on the basis of the income-compensated Marshallian demand function, the so-called Slutsky-substitution effect, is

. The corresponding income effect is of the same size:

. The overall effect is again

. Figure 3 illustrates the second way of decomposing income and substitution effects.
Figure 3. The Slutsky-substitution and income effects

This should suffice as a refresher on standard neoclassical microeconomics. In the next section, I will contrast these two approaches with causal-realist demand analysis.

3. THE DEMAND CURVE AND ITS UNDERLYING ASSUMPTIONS FROM A CAUSAL-REALIST PERSPECTIVE

Many economists in the Austrian or causal-realist tradition criticize standard microeconomics for its overly formal and mathematical style. This formal criticism is rooted in a number of conceptual disagreements. Indeed, how useful is it to describe consumer preferences by means of continuous and differentiable mathematical functions and assume infinitely divisible goods? At what point do standard microeconomists sneak in cardinal as opposed to ordinal utility? Is the theoretical concept of indifference helpful in explaining consumer choice?

For an economist who accepts Rothbard’s (2011, 304–06; 2009, 302–11) criticism of indifference analysis, it is easy to see why the Hicks-substitution and income effects are likely to be rejected. They squarely rely on the concept of indifference as the level of utility is held constant in the derivation of the Hicksian demand function.5But regardless of our stance on indifference,6 the rationale for the Hicksian decomposition is straightforward: A lower unit price for any good is always preferred to a higher unit price for that good, so the “level of utility” increases when a price decreases. In that sense real income increases. The problems concerning indifference arise when we keep the level of utility constant in order to compensate for this increase in real income.

In contrast, the Slutsky decomposition based on the distinction between uncompensated and compensated Marshallian demand does not in principle rely on the concept of indifference and can avoid any other queries that one might have with the idea of keeping the level of utility constant as the price for a good changes. In fact, it is precisely the Slutsky decomposition that is in many ways very similar, albeit not identical, to my proposed reconstruction of a causal-realist income or rather wealth effect (Israel 2018b).

Let us briefly revisit that reconstruction. I had been drawn to the subject by Professor Salerno’s (2018) paper, in which he had argued that the income effect of neoclassical microeconomics is merely an “illusion” (p. 35) stemming from a misapprehension of demand curves. The structure of Salerno’s argument was as follows:

1. In order to construct a demand curve, we have to hold constant a) the buyer’s value scale; b) the prices of all other goods; c) the buyer’s stock of money balances; and d) the purchasing power of money.

2. Given a price change along the constructed demand curve the quantity demanded of the good changes.

3. The change in demand must be interpreted entirely as a substitution effect, because the purchasing power of money is necessarily held constant when working with a given demand curve. Hence, there can be no “purchasing power effect” or in standard terminology “income effect.”

The tension lies in the fact that there can be no price change along the demand curve, when at the same time the prices of all other goods (assumption b) and the purchasing power of money (assumption d) have to remain constant (Israel 2018a).

My suggested solution to resolve this tension is the following. A demand curve gives us the hypothetical quantities demanded of a good at different unit prices expressed in terms of money. Hence, the construction of a demand curve requires a value ranking of definite amounts of money kept (not spent) against definite quantities of the good in question acquired (bought). The subjective evaluation of money and the subjective evaluation of the good in question have to be presupposed. They have to be given and held constant for the analysis. Now, there is not much to be said about the subjective evaluation of the good to be bought. It is just what it is. However, when it comes to money, we can go a little further.

There are essentially two options: Money at a consumer’s disposal can either be spent on the good in question (option 1), or not (option 2). The value judgments that come into play are again the subjective evaluation of the good in question (option 1), about which nothing else can be said, and the subjective evaluation of the next best alternative to spending money on the good in question (option 2). This led me to argue that we have to hold constant the opportunity cost of spending a given sum of money on the good in question. And I went as far as to argue that “we cannot boil this assumption further down” (Israel 2018b, 382). But why should we, anyway? It suffices to construct a demand schedule for some good

given its unit price

as shown in Figure 4.7
Figure 4. Discrete demand function

And here we come to the first serious point of criticism raised by Salerno (2019). He does not seem to believe that my assumption is sufficient and laments my “strange reluctance to clarify the assumptions [I use] in deriving the demand curve.” This, he claims, “is inconsistent with causal-realist analysis” (p. 584). So, Salerno makes it seem as if it is an established causal-realist tradition to spell out the determinants of subjective value and the precise empirical conditions under which subjective evaluations remain constant. But is it really? Of course not. Subjective value or subjective preferences are always assumed as the starting point of the analysis. That is precisely the point of subjectivism.

This is not to say that we cannot make additional assumptions when constructing a demand curve that illustrates the relationship between the money price per unit and the quantity demanded of a certain good by a given hypothetical consumer. We can invoke all kinds of assumptions about changing side constraints. Specifically, we can make assumptions about what exactly happens when we shift the unit price along the constructed demand curve. For example, we could assume that prices of other goods change in just the way Salerno (2019) wants in his comment on my critique. But to give another and somewhat unorthodox example: We could assume that for any price change of 1 money unit along the demand curve, the air temperature changes by 10 °F in the opposite direction, that is, if the price falls by 3 (2, 1) money units, the temperature increases by 30 (20, 10) °F and so on. If we were to analyze beer consumption as in the example of my initial article (Israel 2018b), this additional assumption would lead undoubtedly8 to a flatter demand curve, that is, a larger increase in beer consumption for any given price drop than would occur without this additional assumption. A keen-witted microeconomist might then move on to construct the temperature-compensated demand curve in order to get rid of the bogus temperature effect, which is indeed merely an illusion that emerges by design.

Now, this example raises the question of what additional assumptions are analytically helpful and what assumptions are entertaining shenanigans at best. A reasonable starting point is to make as few and weak additional assumptions as possible and to keep all independent variables constant in order to gain a clear view on the one chain of cause and effect that we are interested in, namely, the effects of an exogenous price change along a given demand curve. This means that we invoke the classical ceteris paribus clause simply for the sake of analytical clarity. This means that we hold all variables constant that we cannot causally link to the exogenous price change under consideration. As an analytical point of departure, this implies that we hold all other money prices constant (and indeed the air temperature)—unless and until we can establish a causal connection to them—and this might well be possible under certain conditions (although I am not sure about the air temperature). We do not, however, assume these variables to change independently, precisely because we intend to isolate one chain of cause and effect, before embedding it into the whole picture. The latter, as Salerno rightfully points out by reference to the interdependence between markets for specific goods and the overall market for money balances, is of course the ultimate purpose of any serious economic analysis.

In order to briefly revisit my reconstruction of the income effect, let us take the reference point

and

on the demand schedule shown in Figure 4. Let us suppose that the price falls to

. The quantity demanded at this price would be

. The overall effect is thus

, that is, consumption increases by 3 units. It costs 6 money units to pay for the three additional units of the good