Prices, real income, and income compensation

Why you might want to stay in your college town after you graduate

Let's say that you study in a mid-sized or small university town, where housing prices tend to be less than in the big cities. When you graduate with your degree in economics you have two job offers waiting for you. One is in the university town you now reside, while the other is in a big city and pays more. You have no strong preferences for either job or city.

The question you need to figure out then is how much extra in income you need to compensate for the higher housing prices in the capital. Intuitively you might think that you would need just enough extra income to pay for the added rent. But economic theory suggests that may not be quite right. You might be happy with a bit less.

Income compensation

When changes in prices and changes in income have the same effect:

Increase both the price for food and rent in the figure above. We see that the prices increase proportionally (25%).
Press the Optimal consumption button.
We see that the indifference curve moves inward--that is to say we are worse off. Prices have increased, while our income has not. Another way of saying this is that our real income has declined.
How much more income do we need to get back to our previous level of utility?
Press the Increase income button. We get back to our old level of utility by increasing our income by 25%, proportional to the increase in prices.
Increasing income to compensate for higher prices is called income compensation.
In this case, where all prices increased proportionally by 25%, then income also needed to increase proportionally by 25% to fully compensate for prices.

How much more income do you need to work in the big city.

How much extra do we need in income to compensate for higher rent in the big city.

Here we go back to our introductory example of deciding on whether to accept a job in our university town (for concreteness, let's say this is Trondheim, Norway), or move to a higher paying job in Oslo. Other than pay and rent, we are indifferent about both the jobs and the cities.
Let's say we move to Oslo and have to pay more in rent.
Press Increase rent and then Optimal consumption
When the price of rent increases, we choose to have "less" consumption of housing - in this context this probably means either quality or square meters.
Relative to rent, food has become cheaper, so here we have actually chosen to increase our consumption slightly.
At this point, since rent is higher but we still haven't increased our income, our welfare (or utility) is lower.
So how much income do we need to get back to our original welfare level?
Press Income compensation to see.
The budget curve moves outward until it touches the indifference curve.
Notice that we are back to our old indifference curve, but not our old optimal combination of goods.
We would actually have needed more income to get back to our old combination of goods. But the relative change in prices has led us to switch some of our consumption from housing to food. Thus we have needed less income to get us back to our initial indifference curve.

Price effects and income effects

We can use the concept of income compensation to understand the difference between a price effect and income effect when a price changes

We start at point A.
Again we start by increasing the rent by pressing Increase rent and then seeing how we change our optimal bundle by pressing Optimal consumption.
But pressing income compensation represents a little bit of magical thinking: No one is going to compensate us for living in Oslo, right?
Therefor, when you press income compensation, you come to a point (B) in parenthesis - to remind us that this is a point we may not actually experience.
But (B) is still an interesting combination. It represents how much more food you would buy when the rent increases when we take away the effect of a lower real income. We call that the price effect.
We can also see that we can move from our theoretical point (B) to our actual consumption bundle, B by reducing our income (shifting our budget curve inwards).
This we can interpret as our income effect: how much our real income is reduced when the rent increases.

Problems

In the figure below, we have a situation where rent has increased and we have moved out consumption bundle from A to B. However, our indifference curve is such that even though rent is increased, we still want the same amount (quality) of car. Since we have the same car-consumption, does that mean we don't have a price effect?

here we still have both price effect and income effect, it's just that those two effects exactly cancel each other out!

When rent increases, we would prefer to move to (B) (if only we received income compensation). But in reality we move to B because our real income has decreased. The price effect on car is positive (it has become relatively cheaper compared to housing). In reality, there is no good reason why these two effects should be exactly equal and cancel each other out - it is just a consequence of the mathematical format we used for the indifference curve.
We have a budget for food that we use exclusively on frozen pizza and the local italian pizzaria. Let's say that frozen pizza is a inferior good for you while eating at the pizzaria is a normal good. The pizzaria decides to increase their prices substantially.
- What is the price effect on frozen pizza? (positive or negative)
- What is the price effect on eating at the pizzaria?
- what is the real income effect on frozen pizza?
- What is the real income effect on eating at the pizzaria?
- With the information we have, do we know whether the total effect will be positive or negative?
We start with the price effect. When the pizzaria prices rise, then we eat more frozen pizza: (+) for frozen pizza, (-) for pizzaria.

With higher prices (for the pizzaria), we have less real income, and since the pizzaria is a normal good, we consume less (-). But frozen pizza is an inferior good and therefor we buy more frozen pizza (+).
- Frozen pizza: prisvr: (+), income (+) = +
- Pizzaria: prisvr:(-), income (-) = -
Housing and food are both normal goods. Assume that the price of food increases because of war in a major agricultural exporter. What sign do the price and income effects on food and housing have?
When the price of food increases, the relative price of housing decreases and we increase our consumption (higher quality, or space) of housing. The price effect: food (-), housing (+)

Real income decreases since prices (on food) are higher, food and housing (-)
- Food: Price: (-), income: (-) = (-)
- Housing: Price: (+), income: (-) = ?
You decide to move from the university town of Trondheim to Oslo, where food costs the same, but housing is more expensive. Your kind Aunt Olga who lives in Oslo feels sorry for you and wants to compensate you for the extra housing costs. She says she will pay you the difference between the rent of a apartment in Oslo and a similar apartment in Trondheim.

Aunt Olga has found an apartment in Oslo that is almost exactly the same as the one you have in Trondheim. You can choose to move into the apartment Olga found or find your own apartment, but in either case you get the same amount of money from Olga--the difference in rent between your apartment in Trondheim and the apartment she found for you. If you are maximizing your welfare, what should you choose?

If you move into the apartment that Aunt Olga found, you are maintaining the exact same bundle of goods as previously and maintain the same level of utility.

But if you choose your own apartment, you may choose to live in a slightly cheaper apartment-perhaps a bit smaller-and spend more on food. In this way you can end up on a higher utility level. You would probably choose to do this.