1 demand and consumer_s surplus.pdf - Intermediate...

Intermediate Microeconomics
Lecture 1: Demand and Consumer Surplus
Douglas Gale
New York University
1 The demand curve
The demand curve you studied in your Principles course is a relationship between the price
of a good and the quantity demanded by one or more consumers. A typical example is
illustrated in the fi gure below. The demand curve is downward sloping—lower prices are
associated with higher quantities—and, in this case illustrated below, the relationship is
linear—the slope of the curve is constant.
0 1 2 3 4 5 6 7
0
2
4
6
8
10
12
Quantity
Price
The demand curve
This simple representation of the demand curve depends on some very special assumptions.
In general, the quantity of the good demanded depends, not just on the price of this good, but
also on the consumer’s income and the prices of other goods. And the demand curve is not
always downward sloping. These complications are something we’ll postpone for another
day. In this lecture, we are just going to derive the simple demand curve above as the
solution to a utility maximization problem . In doing so, we will learn about the special
assumptions that lie hidden behind this innocuous-looking curve.
1

1.1 Quasi-linear utility
Consider the problem of a consumer deciding how much to spend on a single good and
how much to spend on everything else. For simplicity, we assume money and the good
are perfectly divisible, so their quantities can be represented by non-negative real numbers.
Then the outome of the consumer’s decision is a bundle (   ) , where  ≥ 0 is the amount
of the good and  ≥ 0 is the amount of money spent on everything else.
The consumer is assumed to have preferences that can be represented by a utility func-
tion that assigns a real number  (   ) to each bundle (   ) ≥ 0 . We say the consumer
weakly prefers a bundle (   ) to a bundle (  0   0 ) if  (   ) ≥  (  0   0 ) and strongly
prefers (   ) to (  0   0 ) if  (   )   (  0   0 ) .
We do not attach any meaning to the utility number assigned to a bundle (   ) ; only the
preference ordering on the set of bundles matters. The same preferences can be represented
by many di ff erent utility functions. For example, if  is any real number and  is a positive
real number, then the utility function  (   ) represents the same preferences as the utility
function
ˆ
 (   ) ≡  +  (   ) . This is because
[  +  (   )   +  ( 
0
 
0
)] ⇐⇒ [  (   )   ( 
0
 
0
)] 
In fact, if  (  ) is an increasing function of real variable, then  (  (   )) represents the
same preference ordering as the utility function  (   ) .
To make things even simpler, we are going to focus on a particular type of utility function
called a quasi-linear utility function:
 (   ) =  (  ) + 
where   0 is a constant. The utility of the bundle (   ) is the sum of two components,
 (  ) , which represents the utility of  units of the good, and  , which represents the utility
of the  units of money spent on everything else. The function is called quasi-linear because
the second component is a linear function of  . The parameter   0 represents the utility
of an additional unit of money (the