Deriving demand function
Deriving Demand Functions - Examples1
What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x1 and x2 , with respective prices p1 and p2 , and income m.
1
Perfect Substitutes
For perfect substitutes, we have to look at respective prices. After all, if goods are perfect substitutes, then the consumer is indifferent between them, and will have no problem adjusting consumption to get the good with the lowest price.
1.1
The basic case (1:1)
For 1:1 perfect substitutes, the situation is about as plain as can be. Say p1 > p2 . The consumer will spend all their income on good 2. How do we know without doing any of that fancy math stuff? If the consumer is just as happy with a unit of good 1 as they are with a unit of good 2, and good 2 is less expensive, then they might as well use all their income on good 2 (they get more stuff that way). Similarly, if p1 < p2 , the consumer will choose only good 1. What if p1 = p2 ? Then any combination of good 1 and good 2 that uses all their budget is fine with them. So for each good, we have three possible demand functions depending on the prices. For example, demand for good 1 can be expressed as
0 if p1 > p2
m x1 (p1 .p2 , m) = if p1 < p2 p1
Any (x1 ,x2 ) that satisfies p1 x1 + p2 x2 = m if p1 = p2 and similarly for good 2 (with the inequalities reversed, of course).
1.2
A more complicated example (2:3)
Problem : Let the individual have a utility function u(x1 , x2 ) = 2x1 + 3x2 and an income of 120. They face prices p1 = 2 and p2 = 6. What is their demand for x1 ? For x2 ?
1
Disclaimer : This handout has not been reviewed by the professor. In the case of any discrepancy between this handout and lecture material, the lecture material should be considered the correct source.
Despite all efforts, typos may find their way in - please read with a wary eye.
Prepared by
What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x1 and x2 , with respective prices p1 and p2 , and income m.
1
Perfect Substitutes
For perfect substitutes, we have to look at respective prices. After all, if goods are perfect substitutes, then the consumer is indifferent between them, and will have no problem adjusting consumption to get the good with the lowest price.
1.1
The basic case (1:1)
For 1:1 perfect substitutes, the situation is about as plain as can be. Say p1 > p2 . The consumer will spend all their income on good 2. How do we know without doing any of that fancy math stuff? If the consumer is just as happy with a unit of good 1 as they are with a unit of good 2, and good 2 is less expensive, then they might as well use all their income on good 2 (they get more stuff that way). Similarly, if p1 < p2 , the consumer will choose only good 1. What if p1 = p2 ? Then any combination of good 1 and good 2 that uses all their budget is fine with them. So for each good, we have three possible demand functions depending on the prices. For example, demand for good 1 can be expressed as
0 if p1 > p2
m x1 (p1 .p2 , m) = if p1 < p2 p1
Any (x1 ,x2 ) that satisfies p1 x1 + p2 x2 = m if p1 = p2 and similarly for good 2 (with the inequalities reversed, of course).
1.2
A more complicated example (2:3)
Problem : Let the individual have a utility function u(x1 , x2 ) = 2x1 + 3x2 and an income of 120. They face prices p1 = 2 and p2 = 6. What is their demand for x1 ? For x2 ?
1
Disclaimer : This handout has not been reviewed by the professor. In the case of any discrepancy between this handout and lecture material, the lecture material should be considered the correct source.
Despite all efforts, typos may find their way in - please read with a wary eye.
Prepared by