ECON191 (Spring 2011)
23 & 25.2.2011 (Tutorial 2)
Chapter 3 Consumer Behavior
Consumer price index (CPI) and cost-of-living adjustment (COLA)

Cost-of-Living Index: ratio of the present cost of a typical bundle of consumer goods and
services compared with the cost during a base period.
Example:
Two sisters, A and B whose preferences are identical. Person A and B began their university
education in 1990 and 2000 respectively.
Person A in 1990
 Her parents gave $500 for to A to spend on food and books
 Price of food = $2, and Price of book is $20
 A’s consumption: 100 food and 15 books
 I1  $2  100  $20  15  $500
Person B in 2000
 Price of food = $2.2, and price of book is $100
 Her parents decided to give B the amount which is equivalent in buying power to the
budget given to A. How much would be the budget?
 B’s consumption: 300 food and 6 books (same utility level as A’s bundle and B
consumes more food and less books after the price change)
 I 2  $2.2  300  $100  6  $1260

The ideal cost of living index: the cost of attaining a given level of utility at current
prices relative to the cost of attaining the same utility at base prices

In this example, ideal cost of living index: $1260/$500 = 2.52 (152% in the cost of
living)
Books
25
I4
A
15
12.6
B
U1
6
I1
100
I3
I2
250 300
Food
572
1
Laspeyres price index (keep the bundle fixed as the base year bundle)

Amount of money at current year prices that an individual requires to purchase a bundle
of goods/services chosen in a base year divided by the cost of purchasing the same
bundle at base-year prices. Example: CPI (Consumer price index)
P Q  P2t Q2b
 LI  1t 1b
P1b Q1b  P2b Q2b

The Laspeyres price index assumes that consumers do not alter their consumption
patterns as prices change

The Laspeyres price index tends to overstate the true cost of living index. (Why?)
$2.2  100  $100  15 $1720
 In this example, LI =

 3.44
$2  100  $20  15
$500
Paasche index (keep the bundle fixed as the current year bundle)

Amount of money at current year prices that an individual requires to purchase a current
bundle of goods/services divided by the cost of purchasing the same bundle in a base
year
P Q  P2t Q2t
 PI  1t 1t
P1b Q1t  P2b Q2t

It assumes that individuals will buy current year bundle in the base period, however,
facing the base year prices, consumers would be able the achieve the same utility at
lower cost with another bundle

The Paasche price index tends to understate the true cost of living index. (Why?)
$2.2  300  $100  6 $1260

In this example, PI =

 1.75
$2  300  $20  6
$720
Comparison of indexes

Comparison between LI and PI

Both LI and PI are fixed weight indexes. Quantities of various goods and services in
each index remain unchanged (Verify!)
 Chain-Weighted Indexes: Cost-of-living index that accounts for changes in quantities of
goods and services
 Introduced to overcome problems that arose when long-term comparisons were make
using fixed weight price indexes
COLA
If salary is indexed to inflation, COLA recipients are always better off. Would individuals be
better off or worse off in the time of deflation if there is COLA in their salaries?
Price
I1
A
U2
U1
I4
I3
I2
Quantity
2
Example 1: (From individual demand to market demand)
Given there are 2 individuals, person A and B in the economy. Their demand functions for
good X are given by the following equations:
Person A: p = 20 – qA
Person B: p = 15 – 2qB
Rearrange the terms,
Person A: qA = 20 – P
Person B: qB = 7.5 – 0.5P
The market demand could be obtained by adding up the quantity demanded of person A and
person B at all price levels (Horizontal summation). (Q = 27.5 – 1.5P)
However, at P >15, the demand for person B is 0. Therefore the market demand will have a
kink. The equation for the market demand function will be:
P
20
p = 20 – Q
1 2
p  18  Q
3 3
Market demand
15
if P  15
if P  15
Slope = -2/3
Slope = -1
Slope = -2
7.5
DB
DA
Q
20
27.5
Example 2: (Demand and Supply, Elasticity)
Suppose the supply and demand curves for wheat are as follows:
Supply: QS  1800  240P
Demand: QD  3550  266P
(a) Find the equilibrium price and quantity of wheat
In the equilibrium, QS  QD
1800  240P  3550  266P
P  3.46 , Q  2630
(b) Find the elasticity of demand at the equilibrium
%Q Q Q Q P
3.46
E PD 


  266 
 0.35
2630
%P P P P Q
(c) Find the elasticity of supply at the equilibrium
%Q Q Q Q P
3.46
E PS 


  240 
 0.32
2630
%P P P P Q
(d) Suppose that a drought causes a reduction in supply of wheat and it pushes up the
equilibrium price of wheat to $4, find the elasticity of demand at the new equilibrium
In the new equilibrium, P = $4, QD = 2486
%Q Q Q Q P
4
E PD 


  266 
 0.43
2486
%P P P P Q
3
Example 3: (Utility Maximization)
Assume that there are two goods in the world: apples and raspberries. Say that Geoffrey has a
utility function for these goods of the following type, where r denotes the quantity of
raspberries and a the quantity of apples.
U  ra
a) Draw the indifference curve that is defined by the utility function and has a utility level of
a
2500.
50
U = 2500
r
50
b) What is the marginal rate of substitution of raspberries for apples when Geoffrey
consumes 50 raspberries and 50 apples? What is the marginal rate of substitution between
these two goods when Geoffrey consumes 100 raspberries and 50 apples?
MUr a
MRS ra 
 . MRSra when Geoffrey has 50r and 50a = 1 , MRSra when Geoffrey
MU a r
has 100r and 50a = 1
2
c) If the price of raspberries is $1 per unit and the price of apples is $1 per unit and Geoffrey
has $100 to spend, what bundle of raspberries and apples would he buy? Is the marginal
rate of substitution be equal to the ratio of the prices of these goods in the optimal bundle?
If not, why not?
a
a
Optimal condition: MRSra = Pr/Pa    1 --- (1)
r
100
Budget constraint: Prr + Paa = 100
 1r + 1a = 100 --- (2)
MRS = -1
50
Slope of BL = -1
r
50
Solving (1) and (2), a* = r* = 50
This optimal bundle is an interior solution, and
marginal rate of substitution equals the price ratio.
100
d) If the unit prices of raspberries and the apples are $4 and $3, respectively, what bundle of
raspberries and apples would Geoffrey buy with his income of $100?
a
a
4
  --- (1)
r
3
33⅓
Optimal condition: MRSra = Pr/Pa  
16.67
Budget constraint: Prr + Paa = 100
 4r + 3a = 100 --- (2)
MRS = -4/3
Solving (1) and (2), a* = 16.67 and r* = 12.50
Slope of BL = -4/3
r
12.5
25
4
Example 4: (Utility Maximization)
Assume that there are two goods in the world: apples and raspberries. Say that Geoffrey has a
utility function for these goods of the following type, where r denotes the quantity of
raspberries and a the quantity of apples.
U  4r  3a
a) Draw the indifference curves that are defined by the utility function.
a
Slope of IC = 
4
3
r
b) What is the marginal rate of substitution between raspberries and the apples when
Geoffrey consumes 50 raspberries and 50 apples? What is the marginal rate o substitution
between these two goods when Geoffrey consumes 100 raspberries and 50 apples? What
do the answers to these questions imply about the type of goods the apples and raspberries
are for Geoffrey?
MRSra when Geoffrey has 50r and 50a = 4
3
MRSra when Geoffrey has 100r and 50a = 4
3
As MRS remains constant over all consumption bundles, apples and raspberries are
perfect substitutes.
c) If the price of raspberries is $1 per unit and the price of apples is $1 per unit and Geoffrey
has $100 to spend, what bundle of raspberries and apples would he buy? Would the
marginal rate of substitution be equal to the ratio of the prices of these goods in the
optimal bundle? If not, why not?
a
100
Slope of BL = -1
Slope of IC = 
4
3
r
100
Optimal bundle: (100r, 0a)
MRS is not equal to the relative price in the optimal bundle. As indifference curves are
linear functions, it ends up in corner solutions. MRS is not equal to the relative price
because in the optimal bundle, MRSra is greater than the relative price, Geoffrey should
further reduce the consumption of apple in order to equalize MRS and the relative price.
However, consumption of apple in the optimal bundle is already 0.
5
d) If the unit prices of raspberries and the apples are $4 and $3, respectively, what bundle of
raspberries and apples would Geoffrey buy with his income of $100?
a
33⅓
Slope of BL = 
4
3
Optimal bundle: any bundle on BL
(Since, BL and IC coincide)
Slope of IC = 
r
4
3
25
6