Goals and preferences

Preferences and indifference

The marginal rate of substitution

Life is all about trade-offs. If you choose to study, then you give up time to work and earn a wage. If you are up all night at a party, you give up sleep and the next day's productivity. If you buy a new phone, you give up money in the bank.

One of the principle ideas in microeconomics is that if you are going to make good decisions, you need to be conscious of these trade-offs.

Microeconomists make use of the term the marginal rate of substitution (MRS) in order to analyse choices.

MRS: How much of one good (or service, or time, or whatever) we are willing to give up in order to get more of another good.

For example, I might insist on getting paid at least 15 Euro per hour, otherwise I would rather have an extra hour of leisure. So the marginal rate of substitution between

Indifference curve

The marginal rate of substitution between two goods is probably not constant.

If you already work a lot, you will probably put a higher price on your free-time. If you are out of work and desperate for money, you will probably place a low value on your free-time.

The way microeconomists analyse choices and marginal rates of substitution at many different levels of relative consumption are with indifference curves.

Increase apples

If the above is our indifference curve for apples and bananas, then it represents all the combinations where we are indifferent (hence the name) between our consumption and apples and bananas. No matter where we are on that curve, we are exactly equally happy.

We start with 80 apples and 168 bananas (we like fruit!). What happens with the number of bananas when we increase the number of apples (say, from 100 to 120). Is this intuitivt?

We can see that the indifference curve bends towards the origin (0 for the both the x and y axis.) This represents the idea that we like variation - a good mix of goods. We don't just want apples or bananas, but a healthy mix. So a mix of fewer apples and bananas is preferred to lots of bananas, or lots of apples.

Indifferenskurven krummer mot 0-punktet -- dette representerer at vi liker variasjon - vi vil ikke bare ha eppler eller apples, men heller en god blanding.

Utility and happiness

Everything that lies under (to the left of) the indifference curve gives lower utility (we are less happy). Everything that lies above (to the right of) the indifference curve gives more utility (we are happier.)

Increase utility Reduce utility

Utility functions

The indifference curves we have drawn are derived from what we call a Cobb-Douglas utility function.

In general, we can write a Cobb-Douglas function in the form:

$$U(x_1, x_2) = Ax_1^{1/2}x_2^b$$

We use the parameters, A=1, a=1/2 and b=1:

$$U(x_1, x_2) = x_1^ax_2$$

I use the index 1 for apples and 2 for bananas.

What does this mean? In itself, not too much. The most important part to understand is that this form gives us the bowed-form that we are able to give an economic interpretation (we like a good mix of goods.)

The choice of A doesn't influence the marginal rate of substitution (how many apples we are willing to give up to get an extra banana), so we set it to 1 and forget about it.

a and b, though, tell us about the relative preferances -- how much we like either apples or bananas, which of course will affect the marginal rate of substitution. Informally, if we like apples best, then a is bigger than b. But really it is the relative size of a vs. b that is most important. So therefor, we can just set one of the parameters (I choose b) to be equal to 1. Then we can change a in order to change the relative preferences of the two goods.

(But remember: even though we generally prefer apples more than bananas, we still prefer a good mix. We will get tired of eating only apples, even though we thing they are great.)

Draw

What happens when we increase a from .5 to .6 or .7? Why?

(It can help to know that we have locked in our utility to be a constant 1500.)

So the line that gets drawn is:

$$1500 = x_1^{1/2}x_2$$ $$x_2 =\frac{1500}{x_1^{1/2}}$$

So when we increase a, then you get more utility from apples, and therefor need fewer bananas to get the same utility.