by Filip Palda
At the going relative price should you increase your purchase of caviar, stay put, or decrease it? Now instead of looking at the rate at which you are obliged to trade off one product for the other, the object of interest becomes the rate at which you are willing to trade them off. Suppose an extra tin of caviar gives you fifty times more pleasure than a can of beans. This means that giving up fifty cans of beans for a tin of caviar would neither increase nor decrease your pleasure, but would leave you indifferent. This is the maximum rate at which you would be willing to trade one commodity for the other. Economists always call this the marginal rate of substitution. “Marginal” to an economists means that extra little bit you give up to get something else. In this situation the consumer is not maximizing wellbeing because she would be willing to give up more cans of beans to get caviar than the market obliges her to give up. Put differently, the maximum she would be willing to pay exceeds the minimum she is obliged to pay. Consumers call this situation “a deal”, which basically means you are paying less than you would be willing to pay to get an extra unit of some product. In this case then you give up beans and buy more caviar.
But when do you stop? Since it is the inequality between the ability to trade-off the two goods and the desire to trade them off that drives the rush to more caviar, something will eventually have to bring these two sides of the equation into line with each other, lest you depopulate the Caspian Sea of sturgeon.
It is pretty clear that the consumer cannot change the rate at which she is able to exchange the two goods for each other. That rate is given by dollar prices and is generally beyond negotiation. But what she can change is the rate at which she is willing to exchange goods, the marginal rate of substitution. That is fairly obvious on an intuitive level. If you ate nothing but caviar you would become sickened to the point at which beans would start looking tasty and you would side with Shakespeare who wrote that “the sweetest honey is loathsome in its deliciousness”.
So the trick it would appear is to increase the consumption of caviar until you become so disgusted that the rate at which you would be willing to give up beans for some extra caviar falls down to the market rate at which you are obliged to trade them off. While there is nothing wrong with this view of how consumers adjust their consumption to maximize their well-being, it lacks the precision to clarify just how truly powerful the concept of substitution is. We can do better.
The Substitution Games
LET US GO back to the point where you the consumer are willing to give up fifty cans of beans for one can of caviar but the relative price is twenty-five cans of beans to one of caviar. Suddenly you find yourself the star of an afternoon game-show called The Substitution Games. By some artifice of Hollywood the host can measure your happiness or “utility”. In your current state you are spending your entire one thousand dollar budget, there is an imbalance between relative price and marginal rate of substitution, and your utility is ten units of happiness. Your challenge is to arrange consumption so that you stay on this fixed level of utility by spending the least amount of money. The game show will pay you the amount by which you manage to reduce spending while still staying at the same level of happiness.
If you think this show is a bit strange then so did many economists, until Eugen Slutsky clarified what was going on here in his celebrated 1915 article, the remarkable story of which is fascinatingly described in the 2002 article by Chipman and Lenfant. They write poignantly that “As is now well known, Slutsky’s article is one of the most famous examples of those neglected and ignored works whose originality and importance are recognized only after similar results have been obtained by others.” (553). Despite being rediscovered late, Slutsky became acknowledged as the master of this topic.
It is easy to see how to make money in this game. That you are willing to give up fifty units of beans for one unit of caviar means the exchange leaves you at the same level of utility. So go ahead and consume one more unit of caviar and fifty fewer beans. But now your costs of maintaining this utility have fallen because by giving up fifty units of beans the market has allowed you to purchase two units of caviar (remember they cost twenty five beans each). That means you are at the same level of utility as before but with surplus caviar in your disposable budget, which the game show host credits towards the worth of your final prize.
This sounds good, so you keep buying caviar, until that is, you run into the law of diminishing marginal rate of substitution, or the convexity of indifference curves. Yes, all contestants know what that means, but viewers might want a refresher. An assumption running through economics is that the rate at which people are willing to give up of beans to get more of caviar (or any other pairing of goods) while keeping them indifferent to the exchange, is a falling function of the amount of caviar consumed.
It all accords fairly well with intuition. The indifference curve in question is the collection of all caviar-bean combinations that leave you at the same level of happiness. In this case ten units of happiness. Each different level of happiness has its own associated indifference curve.
What all this means for the game is that you are able to save less and less money by increasing your caviar consumption, because the surplus of caviar starts shrinking. It shrinks because the amount of beans you have to give up to keep you on the indifference curves starts rising.
Eventually you reach a point at which no further cost savings is possible. At that point the rates at which you are willing and able to trade off the two commodities is equal. What you have managed in this restricted game setting is to minimize the cost of attaining a certain level of happiness or “utility” given the prices you face and your relative desires for goods.
Economists can actually calculate what this “cost function” of attaining a given level of happiness is provided they have information on prices, income, and the mathematical form of the utility function. As in so many cases of jargon, cost function is a confusing term. It should really be called a “minimum cost of attaining a certain level of utility function”. But we will stick with tradition in this case.
Winners of this round of the game get to go to the next stage. In this stage the game show host gives you the caviar you saved by minimizing your costs of attaining ten units of utility, and tells you to spend your winnings as you please. You could go ahead and consume all the extra caviar which would certainly increase your utility. But you could attain the same higher level by now going in the opposite direction from the first round and exchanging some surplus caviar for beans. You would then have extra beans to spare. In the third round you would not spend everything on beans but even things out again, but then, well, you get the message. Eventually you start acting like your own game-show host, parlaying savings from cost minimization at any given level of utility into higher levels of utility. Each time though the savings get smaller and smaller because each time it is only a fraction of the previous surplus that gets kept for the next round. When the extra savings reach zero you have both maximized utility and minimized costs.
If you understand the process of maximizing utility by substituting one good for another then you have roughly half of economics in your pocket. To maximize pleasure, or utility, a person must adjust his or her consumption so as to equate relative willingness to pay with relative price. If relative price changes, then so must consumption in order to bring equality back to both sides of the equation.
The demand curve
EXACTLY HOW CONSUMPTION reacts to changes in price depends on relative pleasure and relative opportunity. Economists call the relation between price and consumption a “demand curve”, a concept known to any who have taken a course in this subject. The demand curve is the fraternal twin of the supply curve and equal partner with it in determining something called market equilibrium. It is of vital importance to most of economics so let us examine it a bit more closely.
The demand curve is a “reaction function”. It shows how your demand for caviar reacts to a change in the pric
e of caviar, a change in the price of beans, and a change in your income. The demand curve emerges from the rule that you should always be balancing the rate at which you are willing to trade off caviar for beans (the marginal rate of substitution) with the rate at which you are able to trade them off (relative price). This rule is reactive and the reaction is based on optimization of personal happiness. You react to price and income changes so as to best enhance your wellbeing. As such the demand curve tells you how best to change your consumption when these fundamentals that hem us in force us to work within their shifting confines.
The Slutsky equation
MOST PEOPLE CAN accept as self-evident that a rise in price leads to a fall in demand and do not see quite why economists make such a fuss about what they call a demand curve. The fuss however is justified. The demand curve encapsulates two forces that act upon consumption. A substitution and an income effect. Either may work in tandem or in opposition to the other, lending to demand an air of ambiguity and also of depth. This depth is crucial in understanding the effects of certain public policies as well as the meaning of price indices. So let us try to understand the meaning of income and substitution effects.
We have just seen that for any given level of happiness or “utility” there is a minimum cost way of attaining that utility by arranging your levels of consumption of diverse goods. This minimum was attained through a clever substitution of one commodity for the other.
A corollary of this exercise is that substitution is a prophylactic but not complete defence against price increases. Suppose your income is a thousand dollars, you are consuming ten tins of caviar and the price rises to a hundred dollars. If you refuse to change your consumption of caviar your spending on beans must go to zero if you are not to violate your budget. That would present a very unbalanced consumption profile and hence a low level of utility.
By playing the Substitution Games scenario in your mind you could buy less caviar and more beans, stay on this low level of utility and have surplus beans to spare, which could then raise you to a higher level of utility, and so forth until the game had converged to the point where no more savings from substitution is possible. This new level of utility would necessarily be lower than the one before the price increase, simply because your fixed dollar budget now goes less far than it did before. But you have protected yourself to some degree.
What would happen if some benevolent soul who knew your budget and utility function calculated how much income it would take to raise you back to your old utility level and gave you the money?
Would you return to the old consumption level of caviar? The answer is no. Recall that maximizing utility is inseparable from minimizing cost. At the new prices, even on the same utility level as before the price increase you would have to shift consumption towards beans. This “substitution effect” would not be quite as large as the raw effect without the gift of the benevolent stranger. Before the gift you had two depressing effects on caviar consumption. One is the fact that even at the same utility as before, consumption would have to fall to satisfy the logic of cost minimization. The second, is that the price rise diminishes what your limited budget can achieve. You have less “real” income when a price rises. This effect on income would lead to an added fall in consumption of caviar and also in the consumption of beans, provided that both are “normal goods”. The fall in caviar consumption along the indifference curve is the substitution effect.
The total fall in consumption, less the substitution effect, is the income effect. A normal income effect is one in which consumption rises or falls as income falls. Economists believe that “inferior” goods, possessing the opposite quality are rare to non-existent.
Uses of the Slutsky Equation
NOW YOU UNDERSTAND the Slutsky equation. It decomposes the changes in demand of a consumer reacting to prices in such a way as to maximize utility into two components; an income and a substitution effect. The income effect moves consumers between indifference curves, whereas the substitution effect helps them stay on the same indifference curves. In a celebrated 1915 article Slutsky showed how to get this decomposition mathematically. It is the basis of almost all demand theory.
Take this for example. The distinction between income and substitution effects lurks behind discussions of price increases and is poorly understood by lay people and economists. Yet these concepts must be grasped in order to understand among other things the effects of many different sorts of government policy. To appreciate the importance of income effects suppose that government decided to increase taxes on the consumption of cigarettes in order to discourage smoking. The tax would surely have a substitution effect that reduces what some people smoke. However, because the revenue goes back into government coffers and is spent on behalf of all citizens, including smokers this acts as a countervailing income effect on smoking. For the tax to have its full impact on smoking government would have to burn the tax dollars it earned from its levy on smokers.
Substitution effects are also widely misunderstood and the result shows in the way governments calculate, or rather miscalculate cost-of-living indices. Michael Boskin, author of the famous report to the US president that bears his name, argued that price indices exaggerate the cost of living because they do not take into account peoples’ ability to substitute one good for another in order to fight the utility lowering effect of a price increase. The idea came out in our discussion of the cost function but is worth seeing in its most extreme guise.
Suppose caviar makes up half your budget and the price doubles. If you insist on eating the same amount of caviar as before the price increase then your cost of attaining the level of utility you had before the price increase would go up by a half. But if you could find a perfect substitute for caviar, perhaps salmon roe, which cost as much as caviar before its price increase, then you could shift all your consumption to roe and attain the same level of utility as before the price increase in caviar. A cost-of-living index would find that life had become harder, but your ability to find a substitute would mean your cost of attaining a given level of happiness had not changed. The same principle applies to a lesser degree even when perfect substitutes are not available. People can generally adapt to prices by changing what they consume. Utility can rarely be as large as it was before the price increase, but it need not be as small as it would be if they passively stuck to their initial levels of consumption before price increases.
In practical terms this means that if a government wishes to use simple cost-of-living indices as a guide for allowing the poor to continue buying the same, but now costlier basket of products, then the compensation will likely turn out to be greater than that which would be needed to return these people to the same level of utility as before the price increases.
A better strategy is to try to figure out the cost function, that is the minimum cost at which a person can attain a particular level of well-being (rather that a particular basket) by adjusting relative wants to relative opportunities for substitution. This function will tell you how much money need be given to a person to return him or her to the same level of happiness as before the price increase. Because of the prophylactic effect of substitution against price increases, the rebate will be lower than the cost-of-living increase had no substitution been carried out. This rebate will correspond to the income effect component of the Slutsky equation.
We can squeeze a final few drops of relevant insight from the Slutsky equation by examining what happens when all prices increase, including the price of labor, namely wages. This is often called “inflation”. Economists believe that inflation which is perfectly anticipated will have no effect on the decisions of consumers, firms, and workers.
Inflation is an equal percentage increase in all prices, be they the prices of goods, or the price of workers, their wages, or the price of borrowing money for risky profit ventures. If no relative prices change, there is no need for anyone to change consumption patterns. The substitution effect in consumption i
s zero. Since incomes rise at the same clip as prices, the income effect is zero. Inflation completely cancels itself out in the Slutsky equation.
Macroeconomists call this the “neutrality of money”. Microeconomists call it the “zero degree homogeneity of demand”. Only if people mistakenly believe that the rise in their salaries is greater than the rise in the price of goods they consume might we see an income effect on consumption. If people do not make such mistakes then inflation is neutral.
The relative philosophers
AS SHOWN ABOVE, substitution is intimately connected to the notion that people do as well for themselves as they can. “Maximizing utility” is the economic term for this goal of attaining peak happiness. Some very simple reasoning shows that the pursuit of such happiness under the constraint of some income level and prices leads to precise predictions about how people will behave. Yet, does attaining the goal of happiness not require economists to have some deep insight into what makes people happy?
The answer is “not too much”. Their prescription for bringing relative marginal desired tradeoffs in line with relative prices removes some aspects of explicit considerations of happiness from the analysis. An economist can say a great deal about how people will behave without knowing their absolute desires.
This is initially very difficult for students of the topic to accept but we can make headway by comparing how the depressive and the enthusiast will react to price changes. The depressive may get far less pleasure from caviar, beans, and other products than does the enthusiast. That is a reflection of absolute desires. But it is quite easy to conceive that both get the same relative pleasure from trading a bit more caviar for a few less beans. That is what economists mean when they say that for very broad categories of pleasure or “utility” functions, people are similarly inclined to trade off one good for the other when placed in similar conditions of income and price.