MIT 14.01 2.Preferences and Utility Functions

Supply and Demand

Supply and Demand Diagrams

• Demand Curve:
  • measures willingness of consumers to buy the good
  • comes from how consumers make choices
• Supply Curve:
  • measures the willingness of producers to sell the good
  • how firms make production decisions
• Intersection of supply and demand curve is market equilibrium.
• Supply and demand curves can shit when there are
  • shocks to the ability of producers to supply
  • shocks in consumer tastes
  • shocks to the price of complement / substitute goods. A rise in the price of a substitute good for good X raises the demand for the X.
• Interventions in the market can lead to disequilibrium:
  • for example imposing a minimum wage means that more people will want to work than employers want to hire at the minimum wage. This creates unemployment.
• The cost of these interventions is found in reduced efficiency ( trades that are not made ); there may be benefits in greater equity.

Preferences and Utility Functions

Model of consumer decision making, is a model of utility maximization. And this model have two components:

• Preferences ( consumer preferences )
  • what people want
• budget constraint
  • what they can afford

And we are going to put these two things together, we are going to maximize people’s happiness or their choice — or their happiness given their preferences, subject to the budget constraint they face.

We are going to do this in three steps,

Step one is about preferences, how do we model people’s tastes.

Step two is we will talk about how we translate this to utility function, how we mathematically represent people’s preferences in utility function.

And the budget constraint that people face.

Preferences Assumptions

1. Completeness
   • You have preferences of any set of goods you might choose from.
2. Transitivity
   • If you prefer A to B and B to C, you prefer A to C
3. Non satiation ( more is better )

Indifference Curves

So what an indifference curve represents, is all combinations of consumption among which you are different.

Points A and B lie on one indifference curve, implying that the consumer is equally satisfied with either combination. For example, they might be indifferent between having a higher quantity of pizza and a lower quantity of cookies (point A) versus having a lower quantity of pizza and a higher quantity of cookies (point B).

Point C lies on a separate indifference curve that is far to the origin, suggesting a higher level of overall satisfaction compared to the curve that contains points A and B.

Properties of Indifference Curves

1. Consumers prefer higher indifference curves
   • more is better
   • people more happy on c than any of a or b
2. Indifferent curves are downward sloping
   • Same curve means people are equally satisfied with the points ( outcomes ) on the same curve, but it is not the case if it is upward.
   • Figure 2-2 says that people are equally happy when they get 1,1 and 2,2 that is violates non-satiation we talked earlier, that more is better, no one wants less if they could have more.
3. Indifferent curves never crossing
   • different curve means different preferences, they can not cross. if they cross means people happy on them. that wont happen
4. Only one IC thought every bundle
   • You can’t have two indifference curves going through same bundle
   • and that’s because of completeness.

Utility Functions

Utility function is a scaling factor.

How do we mathematically represent your feelings about pizza versus cookies ? Imagine that all you care about in the world is pizza and cookies, how do we mathematically represent that ?

For example we can write down the utility function as :

u = \sqrt{P \times C}

What dose this mean? what is utility ? Well utility dosen’t actually mean anything. There’s not really a thing out there called utiles. Utility is not a cardinal concept, it’s only an ordinal concept. You can not say my utility is x percent higher than your utility, but you can rank them. So we assume that utility can be ranked so to allow you to tank choices.

Once you have more than one dimensional choices, you need something to combine them, so that is what utility function dose, it allows you essentially to weight the different elements of your consumption bundle, so you can rank them when it comes time to choose.

Margin Utility

A key concept is MU ( Margin utility ) , Margin Utility is just a derivative of the utility function with respect to one of the elements. So margin utility for cookies, of cookies is the utility of the next cookie, given how many cookies you’ve had.

The key feature of the utility function is the feature diminishing marginal utility. Margin utility will fall as you have more of a good. The more of a good you’ve had, the less happiness you’ll derive from the next unit.

• First cookie gives you the utility increment of 1.4, from utility of 0 to 1.4
• next cooke gives you utility increment of 0.59
• you go from utility of 1.41 to utility of 2
• next increment is 0.45
• each additional cookie, makes you less happier
• When you are hungrier, the cookie is better ( first cookie )
• the more you have, the less satisfaction and happiness you can get from
• but you will still want the more the better ( if you dont use, you can give away but you still want more )

Marginal Rate of Substitution (MSR)

The slope of the indifference curves are the graphical representation of what comes out of utility function. And indeed the slope of the indifference curve we are going t call the marginal rate of substitution the rate essentially at which you are willing to substitute one good for the other. The rate at whcih you are willing to substitute cookies for pizza, is your marginal rate of substitution.

And we define MSR as the slope of the indifference curve:

MRS = \frac{\Delta P}{\Delta C} = - \frac{MU_C}{MU_P}

Margin utilities are the negative function of the quantities, the more you have of a thing, the less you want the next unit of it.

Indifference curves are convex to the origin. They are not concave.

this is a violation.

Real-life examples

The prices of different sizes of goods, in a convenience store, say. Take Starbucks, you can get a tall iced coffee for 2.25 , or the next size, whatever the hell they call it, bigger, you can get for 70 more cents, so 2.25 and you can double it for 70 more cents. Or take McDonald’s a small drink is $1.22 at the local McDonald’s but for 50 more cents, you can double the size. What’s going on here?

Why did they give you twice as much liquid, or if you go for ice cream it’s the same thing, why do they give you twice as much for much less than twice as much money ?

The point is it’s all about diminishing marginal utility. When you come in to McDonald’s on a hot day, you are desperate for that soda, but you are not as desperate have twice as much soda, you’d like it, you probably want to pay more for it, but you dont like it nearly as much as that first bit of soda. So those prices simply reflects the market’s reaction to understanding diminishing marginal utility.