This paper attempts to explore the nature of the goods in a Two Commodity world. The analysis suggests that the only possibility that the two goods have same income elasticity is the case when both goods have unit income elasticities. Moreover, if both the goods have equal income elasticities, then these goods will belong to the same category and will be in equal relation to each other. The analysis further suggests that if one of the good has zero income elasticity, it will always be a substitute to the other good which will always be an elastic good. These results are supported by the CES and Quasi-linear utility functions.Consumer Behaviour, Demand Functions, Two Commodity, Utility Functions

Husain, Fazal

English

Research Papers in Economics

 
 
Exploring the Two Commodity World 
 
 
 
 
 
Fazal Husain 
Pakistan Institute of Development Economics 
Islamabad, Pakistan 
E-mail: fazal_husain@hotmail.com 
 
 
 
 
 
 
 
 
 
This paper attempts to explore the nature of the goods in a Two Commodity world. The 
analysis suggests that the only possibility that the two goods have same income elasticity is 
the case when both goods have unit income elasticities. Moreover, if both the goods have 
equal income elasticities, then these goods will belong to the same category and will be in 
equal relation to each other. The analysis further suggests that if one of the good has zero 
income elasticity, it will always be a substitute to the other good which will always be an 
elastic good. These results are supported by the CES and Quasi-linear utility functions. 
 
 
JEL classification: D11 
Key Words: Consumer Behaviour,  Demand Functions, Two Commodity,  Utility Functions 
 
 
Acknowledgement 
The author is grateful to Professor Eatzaz Ahmed of Quaid-i-Azam University and Mr 
Nadeem Hanif of State Bank of Pakistan for their suggestions on earlier draft. 
 I. Introduction 
  The theory of consumer behavior has been a major area of discussion in Economics 
as it is a basic building block for many economic studies. It plays a practically important 
role in Welfare Economics, International Trade, General Equilibrium, Public Finance, etc. 
For many years the economists have been trying to explain and predict the complex and 
random human behavior with regards to his consumption pattern. Because of the 
complexities and randomness in human behavior certain axioms are made to present the 
consumer preferences in a consistent way. It is assumed that the consumer has preference 
relation on X=R2+ which is rational, continuous, strictly convex and locally non-satiated. 
Given his limited income (y>0) and the (strictly positive) market prices of the commodities 
he plans the expenditure in a way that gives him the maximum possible satisfaction. 
  The traditional theory usually presents the consumer behavior by assuming the two-
commodity case. Though the assumption seems to be unrealistic, it greatly facilitates the 
analysis in drawing conclusions about the behavior of the consumer. These conclusions 
seem to be reasonable and realistic. Secondly, we can cover all the commodities used by a 
consumer in a two-commodity case by taking the first commodity as the commodity under 
analysis and the second commodity as a composite good that takes into account for all other 
commodities. In this case we can also take the second commodity as the money income that 
is not spent on the first commodity. 
  The theory also provides certain properties and conditions, which have to be 
satisfied by the optimal consumption bundles. This paper is an attempt to discuss the 
implications of these conditions on the consumption bundles in a two-commodity world. In 
particular, the objective of the paper is to explore the nature of the two goods. That is, given 
  1the axioms, properties and conditions provided by the theory, if we know the nature of one 
good can we make any inference about the nature of the other good? 
  With this objective we start by reviewing the consumption theory briefly in the next 
section. Section III presents the results regarding the nature of the goods implied by the 
conditions. These conditions will be analyzed simultaneously in Section IV and the 
suggested results will be presented. The final section contains the conclusions. 
 
II. The Theory of Consumer Behavior 
  Let us assume that there are only two goods Q1 and Q2 for consumption. The 
satisfaction derived from consuming these goods can be represented by a continuous utility 
function, that is, 
  U = f(q1, q2) 
where U is the utility or satisfaction and q1 and q2 are the amount of the goods consumed. 
The objective of the consumer is to get the maximum possible satisfaction. That is, he wants 
to achieve 
 U
* = f(q1
*, q2
*) 
where q1
* and q2
* are the optimal consumption bundles and U
* represent the maximum 
utility that the consumer can achieve. 
  In achieving maximum satisfaction, the consumer is constrained by his limited 
income and the market prices of the goods. The constraint, known as the budget constraint, 
is written as 
 P 1q1 + P2q2 = y 
  2where y is his income and P's are the prices of the two goods. The constraint simply tells us 
that the sum of the expenditures on the two goods must be equal to the consumer's income. 
  Given the budget constraint, the optimal consumption bundles can be derived by 
setting the Lagrangian, 
 L  =  f(q1, q2) + λ{y - P1q1 - P2q2} 
The first order condition for optimality requires that 
 L 1 = f1 - λP1 = 0 
 L 2 = f2 - λP2 = 0 
 L λ = y - P1q1 - P2q2 = 0 
These equations are simultaneously solved to get the demand functions for the goods, i.e., 
 q i = qi(Pi, Pj, y) 
where i=1,2; j=1,2; and i≠j. The function tells us that the demand for a good depends on its 
own price, the price of the other good, and the income of the consumer. Thus any change in 
these variables will change the quantity demanded. The degree of change, however, depends 
on the elasticities of quantity demanded with respect to prices and income. These elasticities 
are defined and classified as follows: 
 
(a) Own Price Elasticity 
 
       ∂qi    Pi  > -1 (the good is inelastic) 
eii = --- * --  = -1 (the good is unit elastic) 
       ∂pi    qi  < -1 (the good is elastic) 
 
 
(b) Cross Price Elasticity 
 
       ∂qi    Pj  > 0 (good i is substitute to good j) 
eij = --- * --  = 0 (good i is independent to good j) 
       ∂pj    qi  < 0 (good i is complement to good j) 
 
  3(c) Income Elasticity 
 
       ∂qi     y   > 1 (good i is a luxury) 
ηi = --- * --  = 1 (good i is unit income elasticity) 
       ∂y      qi  < 1 (good i is a neccessity) 
 
  These elasticities have to satisfy some conditions which are discussed below. 
 
A. Cournot Condition 
  The condition requires that 
 S 1e11 + S2e21 = -S1
 S 1e12 + S2e22 = -S2
 Where  S1 = P1q1 / y and S2 = P2q2 / y are the respective shares of each good in the 
total expenditure. The condition, which is derived from budget constraint, tells us that any 
change in the price of one good will affects the consumption of both the goods and that the 
consumption will be adjusted to satisfy the budget constraint. 
 
B. Engel Condition 
  The condition requires that  
 S 1η1 + S2η2 = 1 
where 0 < S1, S2 < 1 and S1 + S2 = 1. The condition, also derived from the budget constraint, 
tells us that the weighted sum of the income elasticities of the two goods must be equal to 
unity. The weights are the shares of the corresponding good in total expenditure. 
 
C. Homogeniety Condition 
  The condition requires that 
  4 e 11 + e12 + η1 = 0 
 e 21 + e22 + η2 = 0 
  The condition, derived from the demand functions of the two goods, says that the 
demand functions are homogeneous of degree zero in prices and income. That is, if the 
prices of the two goods and income of the consumer are changed by the same proportion, 
the consumption pattern will not be affected. 
 
III. Results Implied by the Conditions 
  The two conditions - Cournot and Engle, discussed above provide some useful 
information regarding the nature of two goods. This section presents these results. 
 The  cournot  condition can be written alternatively as 
 S 1(1+e11) + S2e21 = 0 
 S 1e12 + S2(1+e22) = 0  1 
  We can infer some useful conclusions from these equations regarding the nature of 
two goods. These are, 
(a) e11 > -1 ⇒ (1+e11) > 0 ⇒ e21 < 0 
if Good 1 is inelastic, Good 2 will be complement to Good 1 
(b) e11 = -1 ⇒ (1+e11) = 0 ⇒ e21 = 0 
if Good 1 is unit elastic, Good 2 will be independent to Good 1 
(c) e11 < -1 ⇒ (1+e11) < 0 ⇒ e21 > 0 
if Good 1 is elastic, Good 2 will be substitute to Good 1 
  The same conclusion can be derived by looking at the budget constraint. 
 P 1q1 + P2q2 = y 
  5(a) e11 > -1 ⇒ if P1 ↑ ⇒ (P1q1) ↑ ⇒ (P2q2) ↓ ⇒ q2 ↓ ⇒ e21 < 0 
  If Good 1 is inelastic the increase in its price will result an increase in its 
expenditure. Since income is constant the expenditure on Good 2 has to decrease to satisfy 
the budget constraint. But the price of Good 2 is also constant which implies that the 
consumption of Good 2 has to go down to maintain the identity. Hence the increase in the 
price of Good 1 will result in a decrease in the consumption of Good 2 suggesting that Good 
2 is complement to Good 1. 
  Similar kind of explanation can be given for the other two cases. Similarly, by 
observing the second equation we can get the picture of the other good. The results implied 
by the Cournot condition can be summarized as follows: 
 e 11 > -1 ⇔ e21 < 0,  e22 > -1 ⇔ e12 < 0 
 e 11 = -1 ⇔ e21 = 0,      e22 = -1 ⇔ e12 = 0  2 
 e 11 < -1 ⇔ e21 > 0,  e22 < -1 ⇔ e12 > 0 
Next, the Engle condition can be solved further as 
 S 2η2 = 1 - S1η1
or 
  1   -   S 1η1
  η2 =    --------  3 
    1  -  S1
 
Thus we can determine the elasticity of Good 2 given the elasticity of Good 1. That is, 
  η1 < 1 ⇒ η2 > 1 
  η1 = 1 ⇒ η2 = 1  4 
  η1 > 1 ⇒ η2 < 1 
Similar conclusions can be derived alternatively. Since 
  6 S 1 = P1q1 / y 
Therefore 
 dS1   P1q1            < 0 if η1 < 1 
  ---  = ---- (η1 - 1) = 0 if η1 = 1 
  dy       y
2              > 0 if η1 > 1 
 
  This suggests that the share of a good in total expenditure is affected by the income 
elasticity of that good. We also know that 
 S 1 + S2 = 1 
  Hence if the share of a good decreases by an increase in income the share of the 
other good must increase to satisfy the identity. This means that the goods must have 
opposite elasticities. The only possible case which will leave the shares unchanged is the 
case when the two goods have unit income elasticities. 
 
III. Some Further Results   
  So far the results have been derived from the conditions separately. However, the 
combination of these conditions provides some interesting results regarding the nature of 
two goods. we proceed with the homogeneity condition by assigning different values to 
income elasticities and finding the effect on the corresponding price elasticities. The 
condition is reproduced below 
 e 11 + e12 + η1 = 0 
 e 21 + e22 + η2 = 0  5 
 
Case I: 
  Let us assume that η1 = 1. This implies that η2 = 1 from 4. Equation 5 can now be 
written as 
 (1+e11) + e12 = 0 
 e 21 + (1+e22) = 0  6 
  7We can get some useful results from equations 6 and 2. These are. 
(a) From 6  e11 > -1 ⇒ (1+e11) > 0 ⇒ e12 < 0 
    From 2  e12 < 0 ⇒ (1+e22) > 0 ⇒ e22 > -1 
    From 2  e11 > -1 ⇒ e21 < 0 
  That is, if Good 1 is inelastic it is complement to Good 2. Moreover, Good 2 will 
also be inelastic and will be complement to Good 1. 
(b) From 6  e11 = -1 ⇒ (1+e11) = 0 ⇒ e12 = 0 
    From 2  e12 = 0 ⇒ (1+e22) = 0 ⇒ e22 = -1 
    From 2  e11 = -1 ⇒ e21 = 0 
  Thus both the goods will be unit elastic and will be independent to each other. 
(c) From 6  e11 < -1 ⇒ (1+e11) < 0 ⇒ e12 > 0 
    From 2  e12 > 0 ⇒ (1+e22) < 0 ⇒ e22 < -1 
    From 2  e11 < -1 ⇒ e21 > 0 
  Thus both the goods will be elastic and will be substitute to each other. 
  Hence the above analysis suggests that if the income elasticities of the two goods are 
equal, both the goods will belong to the same category and will be in equal relation to each 
other. An example of this kind of preferences may be a CES utility function. That is, 
  U = A[αq1
-ρ + (1-α)q2
-ρ]
-1/ρ
The demand functions for the two goods are 
  
)
- 1
*
P
P (   P   +   P
y
  =   q
2
1
1 2
+ 1
1
-
1
α
α ρ
 
)
- 1
*
P
P (   P   +   P
y
  =   q
2
1
2 1
+ 1
1
α
α ρ
2  
It can be verified that income elasticities of both the goods are equal to unity. 
   η1 = η2 = 1 
 
  8The own price elasticities are 
)
- 1
*
P
P (   P   +   P
)
- 1
*
P
P (   )
+ 1
1
(   P   +   P
  -   =   e
2
1
2 1
+ 1
1
2
1
2 1
+ 1
1
11
α
α
α
α
ρ
ρ
ρ
 
 
)
- 1
*
P
P (   P   +   P
)
- 1
*
P
P (   P   +   )
+ 1
1
(   P
  -   =   e
2
1
2 1
+ 1
1
2
1
2 1
+ 1
1
22
α
α
α
α
ρ
ρ
ρ
 
 
The cross price elasticities are 
 
)
- 1
*
P
P (   P   +   P
)
- 1
*
P
P (   )
+ 1
(   P
  -   =   e
2
1
2 1
+ 1
1
2
1
2
+ 1
1
12
α
α
α
α
ρ
ρ
ρ
ρ
 
  
)
- 1
*
P
P (   P   +   P
)
+ 1
(   P
  -   =   e
2
1
2 1
+ 1
1
1
21
α
α
ρ
ρ
ρ
 
  It can be seen that the values of these price elasticities depend on the value of ρ (the 
substitution parameter). Hence 
(a) If -1 < ρ < 0, then e11, e22 < -1 and e12, e21 > 0, that is, both the goods will be elastic and 
will be substitute to each other. 
(b) If ρ = 0, then e11, e22 = -1 and e12, e21 = 0, that is, both the goods will be unit elastic and 
will be independent to each other. 
(c) If ρ > 0, then e11, e22 > -1 and e12, e21 < 0, that is, both the goods will be inelastic and will 
be complement to each other. 
  Hence, whatever is the value of ρ, the two goods will always belong to the same 
category and will be in equal relation to each other. 
  9Case II: 
 Now  assume  that  η1 = 0. This means that η2 > 1 from 4. From the homogenity 
condition it follows that 
 e 11 + e12 = 0 
 e 21 + (1+e22) < 0  7 
  Once again, we can get some useful results by combining equations 7 and 2. These 
are 
  From 7  e11 < 0 ⇒ e12 > 0 
  From 2  e12 > 0 ⇒ e22 < -1 
  From 2  e11 > -1 ⇒ e21 < 0, e11 = -1 ⇒ e21 = 0, e11 < -1 ⇒ e21 > 0 
 
  Hence whatever Good 1 is, it will always be a substitute to Good 2 which will 
always be an elastic good. The relationship of Good 2 to Good 1, e21, will depend on the 
nature of Good 1, e11. An example of this kind of preferences may be a Quasi-linear utility 
function. That is, 
  U = ln q1 + q2
The demand functions are 
 q 1 = P2 / P1
 q 2 = (y - P2) / P2
It can be verified that the income elasticities are 
  η1 = 0 
  η2 = y / (y - P2) > 1 
 
The own price elasticities are 
 e 11 = -1 
 e 22 = - y / (y - P2) < -1 
 
  10The cross price elasticities are 
 e 12 = 1 > 0 
 e 21 = 0 
  It can be seen that Good 1 is a substitute to Good 2 which is an elastic good. In this 
particular case, Good 1 is unit elastic and Good 2 is independent of Good 1. 
 
Case III: 
  Finally, assume that η1 < 1. This implies that η2 > 1 from 4. The homogeneity 
condition becomes 
 (1+e11) + e12 > 0 
 e 21 + (1+e22) < 0  8 
Once again we can proceed by repeating the earlier exercise, that is, combine equations 8 
and 2. 
(a) From 8  e11 < -1 ⇒ (1+e11) < 0 ⇒ e12 > 0 
    From 2  e12 > 0 ⇒ (1+e22) < 0 ⇒ e22 < -1 
    From 2  e11 < -1 ⇒ e21 > 0 
  That is, both the goods will be elastic and will be substitute to each other. 
(b) From 8  e11 = -1 ⇒ (1+e11) = 0 ⇒ e12 > 0 
    From 2  e12 > 0 ⇒ (1+e22) < 0 ⇒ e22 < -1 
    From 2  e11 = -1 ⇒ e21 = 0 
  That is, Good 1 is now unit elastic but still substitute to Good 2 whereas Good 2 is 
elastic and independent to Good 1. 
(c) From 8  e11 > -1 ⇒ (1+e11) > 0 ⇒ e12 = ? (uncertain) 
    From 2  e11 > -1 ⇒ e21 < 0  
  Hence in this case, we cannot make inferences with that much certainty as we made 
earlier. All we can say is that as long as Good 1 is either elastic or unit elastic, it will always 
be a substitute to Good 2 which will always be an elastic good and can be either independent 
  11or substitute to Good 1. But if Good 1 becomes inelastic we will be unable to make any 
conclusions. 
 
V. Conclusions 
  The purpose of the paper was to explore the nature of the goods in a Two 
Commodity world. Various properties and conditions provided by the theory guided us in 
this exercise. Some useful and interesting results were found in the process. 
  The analysis suggests that the only possibility that the two goods have same income 
elasticity is the case when both goods have unit income elasticities. Moreover, if both the 
goods have equal income elasticities, then these goods will belong to the same category and 
will be in equal relation to each other. The analysis further suggests that if one of the good 
has zero income elasticity, it will always be a substitute to the other good which will always 
be an elastic good. These results are supported by the CES and Quasi-linear utility functions. 
 
References 
 
Handerson, and Quandt. Micro Economic Theory Third Edition, McGraw Hill, 1980. 
 
Varian, H. Micro Economic Analysis, Third Edition, W.W.Norton and Company, 1992. 
 
  12

Exploring the Two Commodity World

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