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Income and substitution effects

Optimal choice when faced with a budget constraint involves moving to an indifference curve that is tangential to the budget line.

But if the prices a good changes, it will change the budget line. Here we will take the opportunity to use the “two good” model to specifically focus on the effects of the price change of one good, by considering good X on the horizontal axis and “AOG” or “all other goods” on the vertical axis. This allows us to see how much someone can spend on all other goods, once they have finished spending on good X.

If the price of good X rises, then the budget line will pivot inwards around the point where it crosses the vertical axis. At the vertical intercept, the consumer is spending all his money on “all other goods” and not buying any of good X, so a price change in good X won’t change anything. But the horizontal intercept will be different, as the consumer is then spending all his money on good X, which is now more expensive, so obviously he can afford less given his budget.

Here at the original budget constraint of B1, the optimal bundle is at A, where the consumer has aX of good X and aAOG of all other goods. When the price of X increases, the budget line shifts to B2. Here the optimal bundle is at B, where the consumer is consuming less of good X and more of other goods.

Two things have gone on here. There is a substitution effect whereby the consumer is deciding to substitute some of good X for some other goods because X is now more expensive relative to other goods. There is also an income effect whereby the fact that X has got more expensive whilst other goods are unchanged in price, means that the consumer is now relatively poorer than he was before, his money doesn’t go as far.

To show how much of the overall change in spending on the different goods is due to the substitution effect, we consider what would happen if we were ‘compensating’ the consumer by giving him enough of a raise in income to allow him to stay at the same level of utility as he was on at bundle A, now that good X has become more expensive.

We can break down the substitution and income effects like this:

Here we have drawn a new budget line (the black dashed line) with the same slope as the final budget line (B2), indicating the same relative prices between X and other goods as the final relative prices after the price change. This has been moved to the point that is tangential to the indifference curve that the original bundle, A, was on. Now this budget line effectively represents an increased budget, because it is a parallel shift outwards from B2. It shows us the budget that the consumer would have needed, to stay at the original level of utility, the same indifference curve as he was on at bundle A. However A would not be the optimal choice bundle here, it would be bundle C. Bundle C gives the same level of utility as bundle A, but it would satisfy the new relative prices after the price change. This shows us the substitution effect. If the consumer was compensated by being given enough raise in income to allow him to stay at the same level of utility as he was on at bundle A, he would have responded to the change in relative prices by shifting his spending to buy less good X and more other goods, hence the move from A to C.

The rest of the effect is the income effect. The shift from C to B, where the consumer has less money to spend on good X and also less money to spend on other goods, is as a result of the income effect of the price change making him poorer overall.

Optimal choice with a budget constraint

One of the basic premises of ‘normal’ preferences in microeconomics is that we assume that more is better. In other words, if we are considering two goods, X and Y, and we have a choice between a bundle of 3X and 4Y, or 10X and 25Y, we are going to prefer the second because we get more of both.

However most of the time there is a cost involved with both of the goods, and we don’t have unlimited resources. The more we spend on good X the less we spend on good Y. So we will have a budget constraint depending on the resources available to us. We could either spend all our budget on good X, or all on good Y, or on some combination of the two.

We usually illustrate the budget constraint on a diagram along with indifference curves indicating our preferences.

Here we have two goods, X and Y, and a budget constraint indicated by the black line. All combinations of bundles between the origin and the budget line are affordable. Anything on the budget line is just affordable, it means we are using up all our budget on the combination of goods X and Y chosen. Anything inside the budget line (ie closer to the origin) is affordable and we will have some of the budget to spare. Anything outside the budget line (ie further away from the origin than the line) is not affordable.

There are four bundles shown, A, B, C and D, each on a different indifference curve. The most preferable bundle is D, because this is on the highest indifference curve, furthest away from the origin, but this bundle lies outside our budget constraint so it is unaffordable.

Bundle A is inside our budget constraint so this is not using our full resources and we can easily move to a more preferable bundle (like B or C) simply by spending more. B and C are both bundles that use up the full budget, but C is preferable to B as it is on a higher indifference curve.

C is the optimal choice when faced with the budget constraint given here, as it is the indifference curve that is tangential to the budget constraint. This means that at point C, the slope of the indifference curve, or the marginal rate of substitution, is equal to the slope of the budget line.

Marginal utilities and the marginal rate of substitution

The marginal rate of substitution is the rate at which the consumer is willing to substitute one good for another in order to retain the same level of utility.

Lets say our goods are are X and Y, and the total utility derived from having a bundle that is a combination of some X and some Y is U.

We will have a utility function of the form $U(X,Y)$.

The marginal utility we get by adding a unit more of X will be $\frac{\partial U}{\partial X} = MU_X$.

The marginal utility we get by adding a unit more of Y will be $\frac{\partial U}{\partial Y} = MU_Y$.

The marginal utility of X is also the change in total utility we get divided by the change in X, $MU_X =\frac{\Delta U}{\Delta X} = \frac{U(X+\Delta X, Y) - U(X,Y)}{\Delta X}$.

When the change in X is small then we can simply approximate to $\Delta U = MU_X \Delta X$, and by the same logic when the change in Y is small we can approximate to $\Delta U = MU_Y \Delta Y$.

If we were increasing both X and Y then the total change in utility would be $\Delta U = MU_X \Delta X + MU_Y \Delta Y$.

The concept of marginal rate of substitution is that it tells us how much we are willing to substitute of one good in order to get more of another, whilst keeping our overall utility constant. So the key thing here is that overall utility is being unchanged. This means that if for instance we are adding a unit more of X, then we are having to give up some of Y to make up for it.

So if overall utility is unchanged, $\Delta U = 0$ so $0 = MU_X \Delta X + MU_Y \Delta Y$. Hence $- MU_Y \Delta Y = MU_X \Delta X \Rightarrow \frac{\Delta Y}{\Delta X} = \frac{- MU_X}{MU_Y}$. $\frac{\Delta Y}{\Delta X}$ is $\frac{dY}{dX}$ so this gives us an expression for the marginal rate of substitution: it is simply the ratio of the marginal utilities. In the context of an indifference curve, this is the slope of the indifference curve, which makes sense as it is $\frac{dY}{dX}$.

The Marginal Rate of Substitution and gains from trade

The slope of the indifference curve at a particular point shows us the rate at which the consumer is willing to substitute one good for another in order to retain the same level of utility. This is the marginal rate of substitution, and it is the centrepiece behind ideas of trade and exchange.

To understand the idea of being ‘indifferent’ here, consider the possibility that there are two consumers, me and you, and we both have goods X and Y. I decide that I want some more of good Y, and I am willing to offer you some of my X in order to get it. How much more X you want in order to give up one unit of Y, will depend on a couple of factors – how much you like X compared to Y, and how much X and Y you already have. If you already have loads of X and not much Y, you are likely to be less keen on my offer of trading X for Y than you would be if you had loads of Y and not much X. People generally prefer exchanging the ‘good they have more of’, for the ‘good they have less of’ (apologies for the poor English). This is also known as diminishing marginal rate of substitution and is a property held by most normal convex shaped indifference curves.

The marginal rate of substitution is the amount of Y you would be willing to give up for a unit of X, in other words the change in Y over the change in X. As you will see, this changes as you move along the indifference curve, in other words as you have different combinations of goods.

At point A, you have a lot of Y and not much X, so here the MRS is very steep, you are willing to give up Y for not much X in return. But down at point B, you have a lot of X and not much Y, and here if you are to give up one of your precious units of Y, you are asking for a lot more X in compensation.

Now you can see what would happen if you were offered trade on fixed terms (ie a certain amount of Y in exchange for a certain amount of X). This is a ‘rate of exchange’. In the diagram below, the black dotted line illustrates a fixed rate of exchange which the consumer is being offered. We will consider that he starts off at bundle A.

Here the terms of trade are very good for the consumer. He can exchange some of good Y for some good X and move to bundle C, and make a simple gain, moving to a higher indifference curve representing a gain of utility. But that would not be the full extent of the possibilities of his gain from trade. He could carry on trading at that rate of exchange all the way up to point D, where he has reached the highest indifference curve possible at this rate of exchange, and the highest level of utility possible at this rate of exchange.

What is special about point D? It is the point at which the MRS for the consumer is the same as the MRS of the rate of exchange offered, the indifference curve is tangential to the rate of exchange.

This shows an important result. When a consumer holds a bundle of goods and is offered trade at any rate of exchange different to his MRS, then he can gain utility by trading at that exchange rate. Only when the rate of exchange offered is exactly equal to his MRS, will he not be able to make any gains by trading at that rate of exchange.

Shapes of indifference curves

Most indifference curves that follow normal preferences have a convex shape:

However you will come across some different shapes:

Perfect substitutes are like this:

Here the consumer only cares about the total number of X + Y that they have, not whether they are getting more of X or Y, so the indifference curves are straight lines.

Perfect complements are like this:

This is a bit like the case where X is a left shoe and Y is a right shoe. If you have 3 left shoes and 4 right shoes, then you are no better off than having 3 left shoes and 3 right shoes as you need the pair together. In fact you are no better off even if you have 3 left shoes and 100 right shoes. So having more Y does not increase your utility unless you increase X as well.

Here the consumer doesn’t like good Y, so if you are making him have more of good Y, he has to have more of good X to compensate for it, otherwise his utility level falls.

Neutral goods are like this:

Here the consumer likes good X but does not have any opinion (good or bad) about good Y, so all that matters is the amount of good X he has, his utility doesn’t increase by having more Y.

Consumer preferences and consumption bundles

This is one of the cornerstones of microeconomics, and is not usually taught (as far as I am aware) on the A level syllabus, so this is one of the biggest things you will need to get used to when starting micro at undergraduate level.

A lot of concepts in microeconomics are about choice, and the simplest way to model them is to consider the choice between two goods. We can call the goods X and Y. You can actually use this model quite creatively by having one of the ‘goods’ representing ‘all other goods’ so the two good model is more useful than you might first think, but for now just think of X and Y.

You can represent on a graph, different ‘bundles’ of X and Y, for instance if X was apples and Y was oranges, then 3X, 4Y is a bundle (3 apples and 4 oranges), 5X, 2Y is a bundle (5 apples, 2 oranges). We can then compare bundles in terms of whether one is preferable to another. It would be fairly logical to say that if you had a choice of two bundles, one being 3 apples and 4 oranges, and the other being 4 apples and 5 oranges, then (assuming you liked both apples and oranges) the second bundle would be preferable to the first as you have more of each. But what about if the choice was 3 apples and 4 oranges, or 5 apples and 2 oranges? Which is better? You have 7 pieces of fruit overall but people would give different answers depending on how much they like apples and how much they like oranges. This is basically the concept of preferences in microeconomics, comparing different bundles of goods to see what is preferred to another, depending on the differing tastes and preferences of an individual.

We usually make a few assumptions about preferences:
– they are complete: you can compare any bundle, ie if bundle A and bundle B are different combinations of good X and good Y, then either A is better than B, or B is better than A, or the consumer is ‘indifferent’ between the two bundles (they are the same)
– they are reflexive: any bundle is at least as good as itself
– they are transitive: if you have bundles A, B and C, and A is better than B while B is better than C, then A is better than C

The key one here is the assumption of transitive preferences, it might not be realistic in every situation, but it is the way preferences are usually modelled in microeconomics.

Indifference curves give us a way of graphically representing preferences. Indifference curves show us the full set of bundles that correspond to one level of utility. Utility is like ‘pleasure’, it describes how much you want something. Eg if you are as happy having 3 apples and 4 oranges, as you are having 5 applies and 2 oranges, then you are indifferent between them, so they would be along the same indifference curve.

This is a typical set of indifference curves:

I have drawn five indifference curves here, but in reality there are an infinite number of the curves, each representing a different total level of utility. Here I have just labelled them U1, U2, U3, U4 and U5, in ascending order of utility so any bundle on a higher indifference curve has more utility and is preferable. Higher means further away from the origin. The further you are from the origin of the graph, the higher the indifference curve and so the more preferable the bundle.

There are three bundles there, A, B and C, each representing a different amount of good X and good Y. So which is preferable?

Bundle A and bundle C are on the same indifference curve, corresponding to the level of utility U3. Although bundle A has more good Y and bundle C has more good X, the overall value of each bundle is the same to this consumer, given the shape of his indifference curves. If for instance he really liked good X and wasn’t bothered about good Y, then the indifference curves would be at a very steep angle, so you wouldn’t have to increase good X by much to go to a higher indifference curve but you would have to increase good Y by a lot to go to a higher indifference curve. But here the shape indicates he likes the two goods roughly the same, but when he has a lot of good X, he gets more value from increasing good Y than he does increasing good X, and the same when he has a lot of good Y, he prefers to get some more good X than good Y. This is fairly realistic to the shapes of most of our indifference curves, because even if you liked both apples and oranges around the same when you didn’t have any, if you had 100 apples, you would probably decide that an orange was preferable to another apple!

Bundle B is on a higher indifference curve, corresponding to the level of utility U4, so in this diagram, B is the preferred bundle.

Utility is difficult to measure, we see it as an ordinal measure rather than cardinal measure. This means that whilst we can say that a certain bundle gives us more utility than another, we can’t say ‘by how much’ it is better than another. It’s like if I ask you would you prefer a free gift of going on holiday to Barcelona or have free tickets to watch a film at the cinema. You would probably regard the holiday as the better gift, but if you were asked the question “by how much do you prefer the holiday to the cinema tickets” then it would be impossible to answer accurately, you couldn’t say “I rate it 17 times higher” or something along those lines. You could say that 17 sets of cinema tickets would give you the same amount of utility as the holiday in Barcelona, but that would be a slightly different concept than saying you regard one holiday 17 times better than one set of tickets. This is because you do not usually get the same amount of utility from each amount of the same good. Think for instance of ice cream, on a hot day when you really felt like an ice cream, you might get a lot of pleasure from having it, but having another one might not necessarily double your pleasure, and having ten in a row definitely wouldn’t increase your pleasure by a factor of ten – you might have had well more than enough after two or three and any further ice creams would actually reduce your level of utility.