Pivoting the budget line

In the optimization model, you take account of situations where one price changes by pivoting the budget line.

The budget line shows the maximum you can buy at the current prices. For example, you have $100, and a tennis racket costs $40, but a tennis lesson costs $20 per hour. You could buy one racket and have three lessons for the maximum, or two rackets (one for you and one for a tennis partner) and one lesson. Or you could buy five lessons and hope your old racket holds out. All these options are possible. But if the tennis lessons go up to $25, your options for substitution change. Now the maximum number of lessons you can buy while keeping your old racket falls to four.

Figure 6-2 shows the old equilibrium (with indifference curve I₀ tangent to the budget constraint, M). To this, we add the initial price change, which is illustrated by pivoting the budget line (as discussed in Chapter 5), so that the good whose price changes (x₁) becomes more expensive, drawing in the line along the x axis (representing the ability a consumer has to purchase x₁). This is the shape of the final budget line, and the slope of this line is the same as the MRS on whichever indifference curve the consumer ends up on. We label this M^final in Figure 6-2. In the figure, x₁ and x₂ = amounts of goods 1 and 2; M = budget constraint; U = utility at optimal bundle of goods, now unavailable; and I₀ = the indifference curve at the old optimum.

Therefore, to simulate a rise in the price of good x₁, you draw in the budget line so that maximum allocation to x₁ is reduced from x₁^old to x₁^new. The optimal point on the old constraint is no longer available.

But you can’t get to that point yet! You have to unpack a couple new features first. Two effects occur when the price of a good changes:

• Substitution effect occurs when a rise in the price of a good relative to other goods leads to a substitution away from the more expensive good and towards the less expensive ones.
• Income effect reflects the fact that when prices rise, people have less income or less purchasing power to spend overall.

The next stage of working through the change in the price of one good is to work out the two effects in two steps so that you can see clearly how they operate, as the following two sections demonstrate.

Seeing substitution in practice

Economists are interested in how people make decisions when they want to consume up to their budget constraint, but a price hike means that their previous level of preferred consumption is no longer attainable. Consider this example: A representative consumer (call her Kirsty) likes to meet up with her friends each week to watch movies and eat noodles. Suppose she has a total budget of $60 for her entertainment activities. If a movie costs $10, and a bowl of hot noodles costs $5, we will suppose that Kirsty maximizes her utility — remember, she’ll spend right up to the budget constraint if she can — when she goes to see a movie four times a week and has a bowl of noodles each time. Now suppose that — horrors! — the movie price goes up to $15: the movie becomes more expensive, relative to noodles. As a result, Kirsty can’t afford her optimal bundle. Now she will want to cut back on going to the cinema and instead consume more noodles. For example, suppose Kirsty re-optimizes by going two times to the cinema and eating noodles six times a week. Economists are interested in knowing how much of the change in her consumption bundle — two fewer movies and two more bowls of noodles — was due to movies becoming more expensive relative to noodles and how much was due to the reduced purchasing power of her weekly budget?

Of course, economists would like to set up the problem so that it’s as general as possible and doesn’t rely on in-depth knowledge about a particular person and a particular good. Kirsty makes these choices automatically, and economists want to set things up in a way that makes consumer behavior understandable and to some degree predictable.

To show how substitution takes place, you need to know about the slope of the new budget line.

Start with Figure 6-2 and draw a new shadow budget line parallel to the final budget line but tangent to I₀ (though not at the original point of tangency — the slope of the new shadow budget line is different, which means that the optimal point where MRS on I₀ is equal to the slope of the new shadow budget line will be different). We do so in Figure 6-3.

The shadow budget line (M^SHADOW) shows how the effect of the change in the price of x₁ induces substitution, even in the absence of reduced purchasing power. In the figure, x₁ and x₂ = quantities of goods 1 and 2; M = budget constraint; I₀ = indifference curve is tangent to original budget line.

Note that we don’t shift the indifference curve I₀, because the intention here is to show the substitution effect while keeping the consumer’s level of welfare or happiness constant. Consumption of x₁ falls and that of x₂ rises owing to the relative price effects, but overall level of utility or satisfaction is preserved.

Adding in the income effect

When a price changes, and specifically when it increases, you feel as though you are poorer even though your income has not changed. Your reduced purchasing power will also cause you to change your consumption behavior, and this change is called the income effect. The change in the price of a good relative to other prices doesn’t just affect the amount of the good consumed relative to the others: it also changes the amount consumed as a whole, because it generally makes your preferred amount unattainable. You feel as though you are poorer. Of course, the opposite is true if the price of a good decreases. Your purchasing power increases, and you feel richer.

Here’s an example of the income effect, where someone’s original best bundle becomes unaffordable and the person consumes less in total. Ian, a representative consumer, would like to spend his weekly disposable income on shoot-em-up video games and pizza. He has $50 to spend. Initially a game costs $30 and a pizza $5, so he buys one game and has pizza four times a week. But — disaster — the price of video games goes up to $35. His initial consumption bundle is no longer possible, and he now gets only three pizzas a week because he doesn’t want to substitute away his game. There is no substitution effect, but still his total consumption must fall, from one game and four pizzas to one game and three pizzas. This is the income effect coming into play.

Again, economists want to generalize away from specific numbers and form a more general picture of the income effect. Follow along now as we show you the general model that economists use. The income effect occurs because the price change (and the corresponding shift in the budget constraint) means that the level of utility associated with I₀ is no longer available. Thus, the economically rational consumer optimizes by shifting down to a new indifference curve denoted by I₁ in Figure 6-4. As we discuss in the preceding section, M^SHADOW covers the relative price changes, and so the move from M^SHADOW to M^FINAL just deals with the fact that the purchasing power of income has gone down.

In Figure 6-4 we indicate the new optimum as U* (the best available position for the consumer after accounting for income and substitution effects). U* is at a different point of substitution between x₁ and x₂, and so less x₁ is consumed, and it’s on a lower indifference curve, I₁. This is the “final” equilibrium point, where utility is as high as it can possibly be given the new constraint. In the figure, x₁ and x₂ = quantities of goods 1 and 2; M = budget constraint; I₀ = indifference curve at original optimum; I₁ = final indifference curve tangent to final budget constraint M^FINAL; U bar = consumption bundle chosen accounting for only the substitution effect, tangent to M^SHADOW; U* = final best optimum after accounting for income and substitution effects.

Different strokes: Different goods have different income and substitution effects

The relative size and, in some cases, direction of the income and substitution effects depend on the type of good you’re examining. For instance, if (from the consumer’s point of view) a good has many close substitutes, the substitution effect is likely to be large (infinitely so in the case of perfect substitutes), and the income effect negligible. But if the good has no close substitutes and the consumer continues to buy roughly the same quantity at the higher price, the income effect is likely to be much larger compared to the substitute effect.

Several factors determine the size of each effect, including the following:

• Income elasticity: Measures the responsiveness of purchasing to a change in income. Normal goods are those that have income elasticities (see Chapter 9 for details) around 1, meaning that if income rises by 10%, consumption of those goods also rises by around 10%. But some goods (such as luxuries) respond more dramatically or elastically to income changes, whereas some respond less elastically and in some cases even negatively. A good is considered inferior if consumption of it falls as you become richer. Whether a good has an income elasticity of greater or less than one or is negative depends on consumer tastes.
• Existence and closeness of substitutes: When the two goods are perfectly interchangeable from a taste perspective, the indifference curve is a straight line and so a rise in the price of one substitute results in substitution away to the cheaper option. (Given the same units of measurement, the slope of the indifference curve is 1, so unless the prices are equal, the consumer gets to the highest indifference by buying the cheaper good. When the prices are same, the indifference and the budget line coincide.)
• Ability of a consumer to switch: An implicit assumption here is that switching from one option to another is free and costless, which isn’t always true — as you know if, say, you’ve tried to switch operating systems on your computer or your cell phone plan. Economists dealing with markets in information technology have to adapt models to add in switching costs, which are derived from, among other things, the cost of learning to use a technology or of giving up a network of users.

Unpacking the effect of a price change in a model with two goods

In this walk-through, we consider a decrease in relative price that allows a representative consumer to afford more of the two goods. In Figure 6-6, the consumer substitutes towards the cheaper good: x_{1 OLD} = old level of consumption of good 1; x_{2 OLD} = old level of consumption of good 2; x_{1 NEW} = post change consumption of good 1; x_{2 NEW} = post change consumption of good 2 I₀ = original indifference curve; I₁ = new indifference curve; M_old = old budget constraint; M_new = new budget constraint.

Here’s the equation of a budget line (from Chapter 5):

At this point, two things have changed regarding the original optimum point: the price of good 1 and the consumer’s purchasing power have both changed. Now we consider what income M' the consumer would need, given the new price p₁' of good 1, in order to consume her original consumption bundle. The original bundle (x₁,x₂) is available and works on both M and M'.

(x₁,x₂) works on both budget lines, and so you can write two equations describing its relationship to the original budget line and the changed one:

For consistency, the equations show M as the original sum of money, M' as the changed sum, and p₁ as the original price of good x₁ and p₁' as the changed price.

Now subtract the first equation from the second, and here’s what you get:

Use the shorthand of Δ to indicate a change so that M' – M = ΔM and p₁' – p₁ = Δp₁, and substitute into the equation so that ΔM = x₁Δp₁.

Getting what you prefer

From the preceding section, you have a (slightly abstract) equation for determining what happens to income (M) and price (p₁) when a representative consumer wants to keep the quantity of x₁ he’s consuming at the same level.

Here’s an example to show how a consumer substitutes to keep utility at an optimum level: Carol has a bit of a coffee habit. Every week she allocates $10 of her disposable income to getting coffees at work. Each coffee — her choice is a vanilla latte — costs $2 (and she consumes up to her budget line, a maximum of $10, and has five coffees a week). Now suppose her coffee rises to $2.20, (from bad weather affecting coffee growers in Columbia) and she wants to keep her consumption at five delicious vanilla lattes a week. What must happen to her income to do so?

Given her preference, Carol realizes that if she wants to keep consuming x₁ equal to five lattes and knowing that the price of a latte has risen by $0.20, she can figure out the change in income that she must gain to keep the value of her consumption constant from the formula ΔM = x₁Δp₁. She does a quick piece of mental arithmetic and finds that if her income rises by $1 a week she can keep consuming five lattes a week.

Developing the substitution effect in numbers

The model in the preceding section demonstrates the changes in M that will allow Carol to keep her original consumption level the same after the change in price. But when the budget line pivots, the bundle that was optimal usually is no longer optimal, because the relative prices of the two goods have changed, making one relatively dearer than before. Thus, the consumer is most likely to optimize by substituting towards the relatively cheaper option.

Calculating the substitution effect from a demand function

Here we make an assumption about the shape of a particular demand function and take a look at the substitution effect. Adam needs to use his car for transport and buys gasoline in gallons from the local garage. His demand is described by the following function:

His allocatable income, M, is $240, and we assume that gas costs $3 a gallon. Prices for benchmark crude oil have plummeted and — somewhat implausibly — all that is being passed on to consumers, meaning Adam now pays $2 a gallon. As his microeconomist friend, you’re fascinated by this change in his fortunes, and you want to calculate how large the substitution effect is as Adam changes consumption in response to price (as his demand function tells you he will).

You know from his choices which of two feasible bundles Adam would prefer, because that was revealed when he chose. But to make a calculation, or prediction, you need an idea of the general relationship between the price of gasoline and the quantity demanded, derived by knowing how much gets bought at any particular price.

You’re in a quandary: You really want to show how much of the change in quantity consumed depends on substitution and how much on change in income, but you can’t know that until you know the relationship between price and total change in demand. As described in the earlier section “Discerning a Consumer’s Revealed Preference,” microeconomists never know what preferences exist until they’re revealed by a consumer making a choice. You can’t know until you’ve worked out that general relationship, but underlying that relationship are the substitution and income effects. As a result, you have to start by gaining intuition about how Adam may behave and then building up a consistent picture of the relationship between price and quantity.

Therefore, learning consumer theory is a little circular and requires you to work backwards in order to move forward. In this example, Adam’s demand function (see the preceding equation) relates quantity consumed (x₁) to the price of good 1(p₁) and the amount of income he had M ($240). So you can start performing your clever calculations given this simple relationship to work with.

To evaluate the size of Adam’s substitution effect, you go through a four-stage process:

1. Take Adam’s demand function and evaluate x₁ for M and p₁.

   You know Adam’s original demand, and so you calculate x₁ for the demand function. Plugging in 240 for M and 3 for p₁ you get 28 gallons of fuel bought per week.

2. Find the value of M' that keeps x₁ the same when p₁ changes.

   You apply the formula ΔM = x₁Δp₁. You know that x₁ equals 28 and that Δp₁ = (2 – 3) = –1, and so M must change by –$28. That means the new value of M' must be 240 – 28, which equals (ta-da!) $212.

3. Plug M' and p₁' back into the demand function to evaluate new demand.

   Going back to the original equation for Adam’s demand for gas, you put in the new numbers for M ($212) and p₁ and now calculate: X₁ = 20 + 212 / (10x 2) = 30.6 gallons.

4. Apply the formula for the substitution effect.

   You use the equation

   

   and plug in 30.6 for x₁ (p₁', M') — the new compensated demand from step 3 — and 28 for x₁ (p₁, M) – the original demand. Therefore, the substitution effect is responsible for 2.6 gallons of extra fuel being purchased per week.

If you simply plug in a value for p₁' in the original demand function, you get an answer for the total change in demand (which you can verify by plugging in the numbers and see that it comes to 4 gallons). But this way, you identify that of the 4-gallon increase in demand, the substitution effect is responsible for 2.6 gallons, or 65% of the total change in demand.

You calculate the substitution effect by stripping out the effect of income (in other words, step 2): finding the level of income that compensates for the change in price is key. If you don’t find the compensated demand, you’re looking at total change in demand and not the size of the substitution effect.

Adding up the income effect

The other effect of a change in price is to change the purchasing power of the consumer. If the price of a good changes, it’s tantamount to arguing that a consumer’s income has also changed, because the new budget line either restricts the level of demand (when the price of a good goes up) or allows consumption of goods that were previously unavailable (when the price of a good goes down).

In the preceding section, you model the change in purchasing power by shifting the budget line out. But you can also use some clever adaptations to that process to make a calculation of the income effect on Adam’s demand — and of course adapt them to other demand situations.

To do so, define the income effect as the change in demand for a good when purchasing power or income changes from M to M ′ and price is held constant at p₁'. This definition keeps the mathematical treatment in line with the graphic example in Figure 6-4, where the budget line first pivoted and then shifted.

To express the income effect mathematically, write the following:

Here p₁', the new price post-change, is held constant and you evaluate the demand for x₁ given that M changes to a new value M'.

Now plug in Adam’s numbers from the preceding section. For the expression x₁ (p₁', M) calculate his demand using the changed price of 2, but the uncompensated level of income, 240. You get this:

Evaluate demand given the compensated income from step 2 in the preceding section:

Apply the formula for the income effect, taking the second answer from the first, to get an answer of 1.4 gallons of gas.

The sum of the two effects comes to the total change in demand, and so although his demand changes by 4 gallons overall, the income effect is responsible for 1.4 gallons and the substitution effect for 2.6 gallons.

Putting the two effects together using the Slutsky equation

Put simply, the Slutsky equation says that the total change in demand is composed of an income and a substitution effect and that the two effects together must equal the total change in demand:

This equation is useful for describing how changes in demand are indicative of different types of good. Indifference curves are always downward sloping, and so the substitution effect must always turn out to be negative. But the income effect may not be, depending on how consumption of a good changes with income. A normal good has a negative income effect, and so if the price goes down and hence purchasing power or income goes up, then demand goes up. The reverse holds when price goes up and purchasing power or income falls, because then so does demand.

But not all goods are “normal.” Some are inferior in an economic sense. We don’t mean that they’re of poor quality but that they have a negative income profile — as income goes up, a person consumes less of them. Instant noodles, for instance, aren’t generally held to be a product that people consume unless they’re constrained in terms of money; as you get richer, you consume less of them. In this case, the substitution effect is negative, but the income effect is also negative. For the opposite situation, see the nearby sidebar “Giffen goods.”