Slutsky equation

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* Introduction: It is mathematics that make the economist

* Basics

* Derivation of the Slutsky equation in the theory of consumer

* Derivation of the Slutsky equation to offset claims

* Slutsky equation for labor supply

* Equation Slutksy in intertemporal choice

* Generalization of the endowment effect

* Analysis of the signs of the effects

* Conclusions

* Consulted bibliography


This paper aims to expound the importance of the Slutsky equation in terms of Hicks, and show the different expressions you can take it, explaining in each case the differences and similarities that might exist.

Keywords: microeconomics, consumer theory, income effect, substitution effect, endowment effect, income effect, price effect, very typical, Giffen good, normal good, inferior good, Slutsky, Hicks.

Introduction: It is mathematics that make the economist.

I must confess that in my life I never imagined that it would conduct a case study on an equation. What is the reason that drives me to do it To understand this, first we must remember that modern science is based on a pre sense of reality in terms of “what can be calculated and measured.” The real thing will be the “calculable”.

That is why the modern scientist was assigned much importance to mathematics, “Nature is written in mathematical characters” Galileo Galilei sentenced some time ago. I believe that today few scientists would dare to make a categorical affirmation.

However, the fact that mathematics have limited usefulness and do not serve to explain all human behavior, and in our particular case the economic behavior does not mean that mathematical science is not a helpful tool to a social science such as Economy.

That’s why I feel compelled to make this monograph, simply to show that mathematics is a tool that sometimes serves to “calculate and measure” economic phenomena.

This paper aims to expound the importance of the Slutsky equation in terms of Hicks, and show the different expressions you can take it, explaining in each case the differences and similarities that might exist.


Substitution Effect and Income Effect

In economics is not only relevant to know what is the behavior of agents in certain circumstances, but also know how to vary such behavior to changes in the environment. What will be the quantity demanded of some good to changes in the price What effect will a change in wage employment on the amount offered The change my savings if you change the interest rate

To arrive at an answer is necessary to divide these two effects changes discussed below: the substitution effect and income effect.

Change in the price of goods you can see these two effects, varying both the rate at which can be exchanged (“replace”) one good for another as the total purchasing power of our income. The variation in the quantity demanded by a variation of the terms of trade between the two goods are called the substitution effect or price, while the change in demand brought about by a change in purchasing power is called income or income effect.

The Slutsky substitution effect and the Hicks.

There are two ways to view the substitution effect. If we consider the Slutsky, we are talking about the variation in demand when prices vary, keeping constant the initial purchasing power. If we consider it to Hicks, we are talking about the variation in demand when prices change, remaining at the same level of utility, ie, in the same indifference curve. In this paper we limit ourselves only to develop the analysis based on the position of Hicks.

Derivation of the Slutsky equation in consumer theory:

As said earlier, what matters is knowing what will be the new equilibrium when varying the environment is determined. Therefore, we must find the equilibrium conditions of the system and then differentiate. By doing this we will see what the change in consumer decision variables when the data range are determined, in order to continue fulfilling the conditions of equilibrium, ie to further optimize.

In this section I will simply maximizing utility subject to budget constraint, also called primal problem. From this process emerge Marshallian demand functions, expressing that the quantity demanded of a good depends on the price of the property in question and disposable income,

Let’s see it analytically:

Maximize subject to:

As a case of maximization subject to constraint, we form a Lagrangian:

The first order conditions state that the partial derivatives of the Lagrangian must be zero (say the existence of a conditional end), while the second order conditions require that the end is a maximum (and not a minimum). The development of second-order conditions omit it, since we assume to be fulfilled.

Thus the first order conditions are:

Differentiating totally the system we obtain:

This new system can be expressed as the product of two matrices. To see this, let the unknowns in the first member, leaving the second data of the changes that have occurred.

Now, we express it in matrix form:

Having expressed the system as a matrix product, we can solve it easily using Cramer’s rule, by which we can solve any of the three unknowns, either, or.

Will develop the case for being the other similar cases.

For convenience and to simplify the notation, the denominator of the above ratio will be called simply called bordered determinant, minors being complementary to that factor. The fact pose a maximization problem imply that bordered determinant is positive. This arises from the second order conditions. If the determinant is positive, we are in a maximization, if negative, a minimization.

Developing the determinant of the numerator we get:

This expression can be manipulated algebraically so that it reads as follows:

In turn, we see that several terms in the numerator is equal to the additional children. Thus, we can write the expression as:


Since what we want to see is how much changes when prices change (and therefore the purchasing power of income), what we find is, for example. This ratio represents the total direct effect (it could also calculate the crossover). This total effect is broken in the two mentioned above.

The last expression (1) to arrive after that differentiate the equilibrium system allows us to appreciate both effects. Note that the derivative of, ie, make the second term of (1) becomes zero, so that:

Here is the direct expression of the total effect, where the first term expresses the substitution effect and the second represents the income effect. To say that we rely on the substitution effect requires that we keep the same indifference curve, so that:

As in the balance maximizing the ratio of marginal utilities must be equal to the ratio of prices, then we can say that:

Looking at the fully differential system we can see that

We can say that

Therefore, the third member of (1) becomes zero. If you want to see the own price substitution effect, then the second term of (1) will also be zero. Finally, the substitution effect, ie the change in quantity demanded to price itself changes so that the utility level remains unchanged, is:

To see why the income effect is the second member we must make a much simpler procedure. If we consider the income effect of variations in the quantity demanded to change the purchasing power of the total quantity demanded, then we have that all other members of (1) become zero when we do, leaving only that multiplies it is. At about what we need to multiply by the quantity demanded, which is

Thus we deduce the Slutsky effect in terms of Hicks.

As I said earlier, we can perform an entire procedure similar to that seen for finding the total cross effect.

To summarize this part, we can only express the effects, both in matrix form as in its differential form.

Total Direct Effect:

In matrix form:


Total effect Cruzado

In matrix form:


Derivation of the Slutsky equation to offset claims:

In the previous section showed the form that the Slutsky equation when we use the primal problem. However it is the only way to reach and may also be considering a dual problem, ie minimize the cost subject to achieve a given level of utility. This procedure can obtain demand functions and Hicksian compensated.

If you pay attention you can see that the compensated demand is nothing more than the substitution effect in terms of Hicks, as the utility level unchanged. Therefore we are able to write as will the Slutsky equation to offset claims. Let’s take the case of total direct effect of commodity 1. Because it simply represents the change in quantity demanded of good 1 to changes in its price so that the utility does not change, could be expressed as, since the compensated demand is subject to a fixed utility level, so that. Thus we can write the Slutsky equation as.

You can also reach this result considering the problem from another perspective. But to understand that I will perform the procedure below is necessary to remember some identities and properties between the primal and dual problem.

In the dual expenditure function exists which is the minimum level of expenditure required to achieve a given level of utility, which are based on prices:


Besides this function complies with Shepard’s lemma which states that the derivative of the expenditure for a price equal to the compensated demand of the good whose price varied. In mathematical terms: (see demonstration in APPENDIX 1).

An important relationship between primal and dual problem is the income-expenditure identity. If the maximum level of profit achieved in the primal problem is the level parameter restriction dual minimization problem, we can assert that minimized the level of spending exactly matches the consumer’s income in the primal maximization problem.

That is, the minimum expenditure necessary to achieve a given level of utility is equal to the primal income reaches this level of utility.

Applying this, we can reach the Slutsky equation as follows:

We know that when you maximize the utility also is minimizing the cost (see Apendice2). Then, the compensated demand is equal to the Marshallian demand for

Now we derive the compensated demand for a price, using the chain rule, as what was now spending and income depends on prices. I derive here about price itself.

Applying Shepard’s lemma and recalling the identity between spending and income dual primal we can restate the last equation as:

Since the optimization point (remember that when the utility is maximized by minimizing the cost), we can write the last equation, after rearrangement of terms, as follows:

That is nothing intuitively that what we said at the beginning of the paragraph.

Similarly to the previous section, the crossover effect is:

Slutsky equation for labor supply:

“The theory of wage determination in a free market is merely a special case of the general theory of value. Wages are the price of labor”

J.R. Hicks, The Theory of Wages (1932)

Until now we were deriving the Slutsky equation with an implicit assumption that the income was given exogenously.

In reality, people get their income by selling things, whether its property or assets of their workforce. That is, on one hand demand goods and on the other side here. The difference between what they demand and what they offer is called net demand. If you offer more than what they claim then the net demand is negative. In this case are said to be net suppliers.

The goods they can offer people is not unlimited, but are rare. Even the labor force is limited. We can therefore say that the agents start with limited endowments of goods.

When prices change, prices change both the goods consumed and those who offer. That is, it adds a new effect that we were conducting the analysis. On the one hand, relative prices change, second change the purchasing power of income. But as the income is now determined endogenously, then the money income also varies with the variation in prices, resulting need to know whether the agent is net demander or supplier’s net asset in question.

Therefore it is necessary to re-find the Slutsky equation, this time taking into account this new effect, called “endowment income effect. To do this begin with the primal maximization problem.

People have a limited amount of time to spread between two activities, work or play, so where is the endowment of time (24 hours per day, for example) is the time spent on leisure and time devoted to work . The entertainment is good, while the work can be considered an “evil.” However, the work is what provides income to enjoy other goods. Therefore the objective is to maximize consumer utility from consuming leisure and to consume other goods represented by income. Clearly, the entertainment we consume not dedicated to work, so getting the demand for leisure are both getting the job offer.

Analytically, we have:

That is, the restriction means that the income that the individual will have to consume other goods come from his salary for the time worked plus an exogenously given non-labor income from, for example, relatives or rent.

To find the optimal combination of income and leisure we pose the familiar Lagrangian and then find the first order conditions:

The first order conditions are:

The second order conditions ensure the convexity of the indifference curves, which in turn ensures that the solution is a maximum not a minimum. The development of second-order conditions and assume it obviare compliance.

From the first order conditions can be found the demand for leisure, ie labor supply (remember that it is not entertainment, it’s work). The demand for leisure will depend on the prevailing wage and non-labor income given. That is, hence labor supply is:.

Let us count the equations and see what we can do mathematical manipulations to find the total effect of disaggregating into substitution effect and income-endowment income.


(1) is the equation of a contour

(2) is the equation of the budget constraint and view.

(3) is the equation of the demand for leisure

What we find now is how much will vary the demand for leisure (and thus labor supply) when prices change, in this case the only price is (note that it is both payment for labor and cost of leisure) . The overall effect should be decomposed into three such effects before.

Comenzare totally differentiating the three equations:

(1) and (arising from the conditions of 1 order), then:

; As. Remember that it is more

than zero implies that the restriction is effective, ie the solution is given on the boundary of attainable set.


Rearranging terms the expression becomes:

Since then

Rearranging terms we have:


Since then

Dividing both sides by, we have


Perhaps the member calls attention to the left of equality. Why pose the derivative so that we stay on the same indifference curve Simply because at the beginning posed the equation (1), which is the equation of an indifference curve. Therefore any analysis that raised so. That’s what allowed us to affirm that, among other things.

By rearranging (4) we have:

or what is the same:

Here are the three effects mentioned above. The total effect is equal to the substitution effect over the income effect-envelope least ordinary income effect

Slutksy equation in intertemporal choice:

So far we analyzed the Slutsky equation for the demand for goods (clothing, food, entertainment, etc.) Assuming that the individual consumes all his income in one period, leaving no room for saving for future consumption.

In this section we enter the intertemporal choice, ie, the analysis of consumer behavior regarding decisions about savings and consumption over time. This analysis is not essentially different view of individual choice above. Instead of seeking the optimum combination of goods that maximizes the agent’s utility given their preferences, look for the basket set (), composed of different goods consumed in each period that given his time preference.

In intertemporal choice there are simplifying assumptions:

* There are only two time periods, today and tomorrow.

* The income of the individual are given (these are endowments), having an income for each period, and for the periods zero (today) and one (tomorrow), respectively.

* The ability to borrow money or lend money to a nominal interest rate is the hinge between the two periods.

* The price level is constant and equal to 1 in both periods.

Mathematically, the problem here can be stated as follows:

Here the constraint implies that future consumption can not be greater than the future income more savings in the period zero with the interest generated by these savings. The restriction could be manipulated mathematically expressed in terms of present consumption. It is common to write the constraints expressed in present value terms it is a good way to express the intertemporal budget constraint because it measures the future in relation to this:

To find the optimal combination between present consumption and future consumption should follow the familiar steps of maximization subject to constraint. First we form the Lagrangian:

The first order conditions are:

The second order conditions, as in the previous paragraphs, I assume to be fulfilled. The process to find the Slutsky equation in intertemporal choice is the same as in the section on labor supply.

From the equilibrium conditions can be found temporary consumer demands, which depend on the interest rate and the amount of income:.

Because we have endowments, Slutsky equation will have a revenue-allocation. As in the previous section we talked in terms of suppliers and demanders net net, here we can do the same, in terms of net savings providers (lender) or applicants net savings (borrowing).

If we remember that the savings (or debt) is we could get to “guess” by intuition as we did in the section with offset requirements, and with reference to the labor supply equation, the Slutsky equation is:

If “adivinasemos” in this way would be correct. Deduction for below.

As in the previous section, be raising three equations, which will differentiate completely, and after a little mathematical manipulation leads to the Slutsky equation as we came to see.

(1) is the equation of a contour

(2) is the budget constraint in present value terms.

(3) is the demand for consumption in period zero.


Of the first order conditions shows that. Replacing the total differential of (1) yields:


Taking the common factor we have:


As in the previous section, since the solution is given at the border and the restriction is effective. Therefore we can deduce that


The last equation can be greatly simplified, since, on the other hand we know that as we are located in the zero period, ie today, so the income already received (and therefore not a variable). That said we can rewrite the equation as:

Or what is the same

Rearranging terms we get:

Now, if we look closely at the budget constraint we see that



Since we know that and rewrite the differential of (3):

Dividing both sides by we get:


Here the member to the left of the equality has the same meaning as that seen for the labor supply. Because (1) is the equation of an indifference curve, the analysis is raised so.

Rearranging terms, the expression is as seen in other sections:

The effects (substitution, ordinary income and income-supplied) can be seen as the labor supply. The overall effect is, is ordinary income and income-allocation is.


Generalization of the endowment effect:

This issue has been mentioned in previous chapters, more specifically in section labor supply. In this section provides the conceptual basis of economic endowment effect and subsequently figured in the particular case of the demand for leisure and below for intertemporal choice.

In this part of the essay I will explain the mathematical expression of the endowment effect of a slightly more general than previously done. But before doing so I think it should, at the risk of being repetitive, re-review the nature of the effect that brings us together.

A trader endowments part with goods sold in the market prices, thus obtaining his income, which used to buy goods consumed. As we are under the assumption that there are only two goods that truly buy into the market is net demand, expressed as the difference between what we consume (gross demand) and what he has, ie. Note that while gross claims are positive, the net demand can be negative if the envelope than it consumes. That is, offers more than demand, so it is bidder equity.

The amount of assets that can be consumed are limited by their income, which is equal to the value of its endowment. Therefore.

In this equation, the envelope is determined exogenously, so changes in the envelope (involving changes in nominal income) and remained fixed prices serves as exogenous changes in income for the simple model in which revenue was determined by outside model.

Note that now the monetary income is determined endongenamente: a price change implies a change in the exchange rate between the two goods (for the substitution effect), a variation of purchasing power (income effect for the-ordinary), but also a variation the value of the endowment and therefore a change in nominal income (for the income effect-envelope).

That is, we define the income-endowment effect and the change experienced by the nominal income when prices change by the change experienced by the demand when income varies.

The overall effect is the sum of these three effects, two of which we should be familiar by now.

Total effect = substitution effect + income effect + income effect regular budget

Returning to the definition of the endowment effect, we realize that is the product of two derivatives:

Recalling that we can say that so-endowment income effect is, so that the total direct effect is:

Or what is the same:

Recall that the expression is the net demand, so the income effect (more regular strength) depends on whether the property in question is normal or less and if we are net suppliers or net applicants.

Analysis of the signs of the effects:

Signs of the effects of the Slutsky equation can draw important conclusions about the property in question, as is grouped into certain categories namely normal, inferior, typical, Giffen, substitutes, complementary.

Effect of change in own price:

Let us first see the meaning of the sign of the substitution effect. It can be said that the own price substitution effect is not always positive, ie, may be zero or negative. This statement comes from the fact that the indifference curves have negative slope. Recall that the indifference curves are negatively sloped because we work with the assumption of “unsaturation” where “more is preferred to less.”

That is, in mathematical terms, the marginal utilities of goods are positive, therefore the chain rule.

In economic terms, when altering the relative prices there is a tendency to substitute other goods for one whose price has fallen. That is, the goods in question are net substitutes. The net adjective means we are taking into account only the change in relative prices (substitution effect).

The sign of the ordinary income effect is slightly more intricate.

* When the income effect varies in the same direction as the substitution effect, ie it is said that it is normal. Normal goods are goods whose consumption increases as income increases. Observe that if.

* Economically we are saying that if the price of goods on the one hand increase their consumption because they fail to buy the goods that are now relatively more expensive, and secondly, to increase our real income due to increased consumption that it is normal. The overall effect will be negative, implying that the demand curve is downward sloping. In this case the property is called typical.

* It may be that the income effect varies inversely to the substitution effect, ie

, So that the object in question is called the bottom. Inferior goods are those whose consumption declines as income rises. This type of property is characterized by quality “second” or “lower” than other types of property. To cite one example, margarine is an inferior good, whose quality is lower than butter. When income increases consumers stop buying margarine and begins to burn fat. The fact that the income effect is positive and therefore varies in the opposite direction to the substitution effect is apparent that the overall effect remains indeterminate. The magnitude of the income effect on the importance lies partly within the budget of the goods in question. If you have a great importance, the income effect will have more weight in the total effect.

* If the property is actually lower and the absolute value of the income effect outweighs the substitution effect, then the total effect will be positive. That is, before the price decreases, quantity demanded decreases, so will the demand curve slopes upward (at least in part). Such goods are called Giffen. This phenomenon, which “breaks” with the Orthodox demand curve with negative slope is given in a few situations where the standard of living of consumers is very low and meet their needs, eg food, with a single good for general use ( bread, or potatoes). When the price falls to those goods, real income increases and allows you to eat a more varied diet, reducing the quantity demanded of good.

* If the income effect does not offset completely the substitution effect the overall effect is negative, so the demand curve slopes downward, being very typical.

The analysis of sign-endowment income effect is similar to that performed for-ordinary income effect, but with all signs changed. Namely:

* When the well is lower, the endowment income effect varies in the same direction as the substitution effect. Note that if the price falls, the value of the envelope decreases, thus decreasing the nominal income of the consumer, creating demand to increase the amount of good, because it is inferior.

* When the well is normal, the endowment income effect varies in the opposite direction to the substitution effect. When the price of good increases the value of the endowment and therefore income increases. By increasing the income increases the quantity demanded of the good that is normal.

From the analysis of signs of ordinary income and endowment effects suggests that whether the consumer is a net supplier or net demander has a great importance in determining their behavior to changes in the price with inferior goods or normal. If net demander, to behave according to the first analysis (the ordinary income effect). If net offeror under the second, for the income-endowment effect.

Effect of variation in the price of other goods:

Let the sign of the cross-substitution effect. We can say that the sign of the cross-substitution effect is always non-negative. The economic justification for both mathematical and is analogous to the own price substitution effect. If, when relative prices are altered there is a tendency to substitute other goods for one whose price has gone down, then when the price of good 1 increases the quantity demanded of good 2, then the relationship being positive, being the property in question substitutes net.

Regarding the sign of the effect of ordinary income and endowment income effect is not much more to say, since it is a case almost identical to direct. The sum of both effects will result in the product between the net demand of the good whose price varied and derived from the property in question in respect of income. Analytically, the sum of both income effects (regular and crew) in case that changed the price of good 1 and want to know the effect this variation had on good 2, is:

So the sign of the effect depends on the net demand of the good whose price varied and if good 2 is normal or lower.

In turn, the sum of the three effects gives us information about how are the goods among themselves. If when the price of good 1 increases the demand for good 2, then the relevant goods are gross substitutes. The adjective refers to gross’re taking into account all existing effects. Example of gross substitute goods may be beef and chicken. When the price of beef, chicken demand increases, that is, replace the meat with chicken. If when the price of good 1 decreases the demand for good 2, then the relevant goods are gross complements. Example of complementary goods can be gross and ink printers.


* For a better understanding will begin with the dual problem statement up to the first order conditions

As a case of optimization (minimization) subject to restriction, can be easily solved with the method of Lagrange multipliers. First we form the Lagrangian:

The first order conditions are:

In the first two equations shows that

Already raised the first-order conditions we are able to demonstrate Shepard’s lemma.

Now, as then:

Pay special attention to the last term:

Since we are working with then we know so, so, thus being shown Shepard’s lemma.

* Proof of Lemma Shepard: The first order conditions of the dual problem are:

The first order conditions of the primal problem are:

Because both dual conditions as conditions require that the optimal primal is in the point of contact between the TMS and the budget line, we can say that when utility is maximized while minimizing the expense, and vice versa. In the words of famed economist Paul Samuelson: “… utility (if the costs are given) only takes the maximum when you reach a certain level in the cheapest way, ie, when the costs are minimal for any level utility. If not, the consumer could reach the same level with some money left over and use that rest on the acquisition of more goods, could be achieved, then a highest utility level … ”

Graphically you can see in the following way:

* “At the point of optimization,”

* Explanation: Throughout the monograph, and for convenience, always figured the total direct effect. But I think it’s important to at least leave are recorded as the equations in more general terms, both the direct and total effect of the cross, including the endowment effect.

Total direct effect:

Total effect Crusader:

Also I need to clarify that in different parts has been used indifferently to represent the notation or income.

Finally, and not least, the monograph has been limited to the simplified case of choice between two goods. If you would like to have more goods into consideration several aspects should be reviewed, as no direct substitution effect positivity and negativity of the substitution effect is not crossed.


To conclude this work I would like abusing a couple of quotes from the book of Max Hermann mentioned in the bibliography. In a chapter on mathematics as an auxiliary method to the economy, the author insists on paying attention to that “… the result of a mathematical deduction should be interpreted with economic criteria, since their interpretation would be purely formal logic can lead conclusions that are inconsistent with reality … ” and “… fortunately the economy is very far from becoming exact science’, because if it did, it would mean total loss of character of social science by eliminating the human factor.”

I fully agree with the author’s claims and hope that this work, despite their level of mathematical abstraction, is considered as a mathematical tool to serve the economy, never backward.

Another thing I tried to stress at work is the importance of “instinct”, “intuition” or “guessing” when it comes to deductions, whether mathematics as they were in this work, or economic. This matter seems to me valuable obituary included a study of Ragnar Frisch, dedicated to Joseph Schumpeter, where referring to the position of the learned face-econometrics science that represents a combination of economic theory, statistics and mathematics, we find these phrases:

“Mathematics – to more sophisticated forms of mathematics – are a necessary tool, but no more than a tool. No class of mathematical techniques, however sophisticated it may be, can ever replace the intuition, that inexplicable function that occurs in brain of a great intellect at the same time, understand mathematics and economic theory in a more orthodox and has lived long enough (or rather, with sufficient intensity) and to accumulate human experience and sense to the facts. ”


* Reuben H. Pardo, La Posciencia: scientific knowledge at the end of modernity, Ed Byblos.

* Hermann Max, Economic research, methodology and technique, 1 edition, 1963, Fondo de Cultura Economica.

* Hal R. Vary, Intermediate Microeconomics, 3 rd edition, Antoni Bosch.

* Fernandez de Castro, Juan Tugores, Principles of Microeconomics, 2nd edition, 1992, McGraw-Hill

* Paul Samuelson, Foundations of Economic Analysis, Ed El Ateneo

* PowerPoint presentations given by the Chair.


Leandro Ezequiel Brufman

Subject: Microeconomics I

Universidad Nacional del Sur (UNS)

Year: 2004