Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect
Alternative to Utility Maximization We have examined how individuals maximize their welfare by maximizing their utility. Diagrammatically, it is like moderating the individual’s indifference curve until it is just tangent to the budget constraint. The individual’s choice thus selected gives us a demand for the goods in terms of their price, and income. We call such a demand function a Marhsallian Demand function. In general mathematical form, letting x be the quantity of a good demanded, we write the Marshallian Demand as x M ≡ x M (p, y) .
Thinking about the process, we can reverse the intuition about how individuals maximize their utility. Consider the following, what if we fix the utility value at the above level, but instead vary the budget constraint? Would we attain the same choices? Well, we should, the demand thus achieved is however in terms of prices and utility, and not income. We call such a demand function, a Hicksian Demand, x H ≡ x H (p,u). This method means that the individual’s problem is instead framed as minimizing expenditure subject to a particular level of utility. Let’s examine briefly how the problem is framed,
min p1 x1 + p2 x2
subject to u(x1 ,x2 ) = u We refer to this problem as expenditure minimization. We will however not consider this, safe to note that this problem generates a parallel demand function which we refer to as Hicksian Demand, also commonly referred to as the Compensated Demand.
Decomposition of Changes in Choices induced by Price Change Let’s us examine the decomposition of a change in consumer choice as a result of a price change, something we have talked about earlier. Before we discuss this in general form, let us use the result from our earlier discussion using Cobb-Douglas utility function.
We found earlier that assuming a Cobb Douglas utility function, for the problem of max (x )α (x )β xA ,xB A B
subject to y = p A x A + pB xB the consumer’s choice may be characterized by the following demand functions
M 1 α M 1 β x A = y and xB = y . However, at the same time, note that p A (β + α ) pB (β + α ) income is actually a function of prices. Let the choices at the current prices and income
level be x A and x B . Then the fixed level of income yielding the choices can be written as y = p A x A + pB x B . We can rewrite both the demand functions as
S α p A x A + pB x B S β p A x A + pB x B x A = and xB = . ()β + α p A ()β + α pB These demand functions are now known as Slutsky Demand functions. The question now is how does a price change alter choices? We call this type of examination S comparative statics. Let the price of good A change. Differentiating x A with respect to
p A , we get the following,
1 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect
S ∂x α p x A − y α x A α y A = A = + − ∂p α + β 2 α + β p α + β 2 A ()p A A ()p A
α x A From the last equality, the first term, , is the income effect, and the second, α + β p A α y − , is called the substitution effect. Why? (Note that income effects if α + β 2 ()p A always positive, and the substitution effect is always negative.) Well, let us revert back to
M 1 α the Marshallian Demand for good A, x A = y . Differentiating the Marshallian p A (β + α ) α 1 Demand with respect to the fixed income of good A, y , we get, . That is the α + β p A M M ∂x A ∂x A first term is just x A . Similarly, the second element is just , that is it is the ∂y ∂p A y= y derivative of the marshallian demand function, evaluated at the original income level, y .
Examining the scenario of a price increase in good A, and representing the change diagrammatically, the substitution effect is the movement from point A to point B. It involves the substitution between the consumption of good A for good B on account of the price increase. Note that in Slutsky’s decomposition, we hold the income constant since in moving from point A to point B along the new budget constraint, we have ensured that the initial consumption point is still affordable. This is achieved by varying the prices of the two goods, hence the change in slope of the budget constraint.
x A The first shift in budget constraint involves pivoting on the original consumption choice, point A, ensuring that we are examining substitution at the same income level. The second shift is from the second budget line, but in a A parallel fashion since we are considering a pure income B
x A U3 C U2
U1
x B
2 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect
The movement from point B to point C involves the movement between two parallel budget constraints, and hence involves the change (more precisely a decrease in income, since the increase in the price of good A, without a commiserating change in the price of good B implies a real change in income). Note that the two effects moves in opposing directions as suggested by the decomposition of our marshallian demand function.
***There is an important point to be made here; the sequence of the decomposition must follow with substitution effect, and then the income effect. This is because the substitution effect considers the trade off between the consumption of goods holding the initial consumption choice constant.
We can describe this decomposition in a more general manner. First, as above, let M S xi ()P, y = xi (P, PX = y), where P and X are vectors of prices and goods respectively, and where i indexes the good under consideration. As before the superscript M indexes the Marshallian demand, while S indexes the Slutsky counterpart. Then the effects of a price change in good i is S M M ∂xi (P, PX ) ∂xi (P, y) ∂xi (P, y) = + x i ∂p ∂p ∂y i i M S M ∂xi ()P, y ∂xi ()P, PX ∂xi ()P, y ⇒ = − x i ∂pi ∂pi ∂y Which is the more general version of the decomposition we have before. This also tells us why we typically refer to the substitution effect as the Slutsky substitution effect, where income is held at the original choice level. From the above we can make several points about the sign/direction of impact: 1. The substitution effect of always a negative one. That is as price increases (falls) the change in quantity demanded falls (rises). 2. The income effect can be either positive or negative, that is an increase (fall) in income can lead to either an increase or fall in demand of the good. a. When the income effect of a good is positive, we call such a good a normal good. That is the greater (lower) one’s income is the greater (lower) one’s demand for the good. Further, for a normal good, the direction of both income and substitution effect reinforces each other. Example: When prices falls for a good, substitution effect means more of the good is demanded, and income effect says the effective rise in income raises the good’s consumption even further. b. When the income effect is negative, we say a good is an inferior good. For such a good, the greater the income, the lower the demand for the good, and vice versa. In such a case, the net impact of a price change is ambiguous since the ultimate change is dependent on the relative size of both effects. We will examine this further shortly.
Decomposition of the Hicksian Demand As noted before, the Hicksian Demand express the individual’s choice in terms of the H H price of the good and the utility level attained, xi ≡ xi (p1 , p2 ,u), in a two good
3 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect problem. The question then is if there is an equivalent to Slutsky substitution effect for the Hicksian demand. Well, there is! The prove of which is outside the scope of this course, but it relies on the following, ∂x H (P,u) ∂x S (P, PX = y) i = i ∂pi ∂pi Which says that the Slutsky substitution and the Hicksian substitution effect are the same. The entire decomposition can be written as M H M ∂xi (P, y) ∂xi (P, PX ) ∂xi (P, y) = − xi ∂pi ∂pi ∂y Note the slight difference also in the income effect in this decomposition, since we are not considering the income effect from the original level of income. This is because the substitution effect holds the utility and not income constant. The diagrammatic representation is similar to the above, with the exception being that the substitution effect holds the initial utility constant on which the choice is on.
x A The first shift in budget constraint involves rotating around the original indifference curve, U2, ensuring that we are examining substitution at the same level of utility. The second shift is from the second budget line, but in a A parallel fashion since we are considering a pure income B
x A U2 C
U1
x B
Note that the sole difference in both the mathematical representation, and the diagrammatic representation is the substitution effect. Again, note the sequence is first substitution, then income effect.
Types of goods, and their characterization in terms of Income and Substitution effect. There are essentially 3 types of goods, 1. Normal Goods, 2. Inferior Goods, 3. Giffen Goods
We will use the ideas we learn above to characterize them. A diagram for a normal good is as follows: Suppose the price of good B rises.
4 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect
x A
B
A
U3 C U2
U1 Substitution Effect x B Income Effect
As can be seen above, and as noted before, both effects reinforce each other for the demand of good B, leading eventually from point A to point C, a fall in quantity consumed of good B.
However, if good B is an inferior good, the income effect would be negative, that is under the same scenario of an increase in price of good B, substitution would reduce demand for good B, but income effect would increase the demand for good B since the increase in price of good B reduces the real income.
x A
B
A
U3 C U2 U1 Substitution Effect x B Income Effect
In the diagram above, the price increase still lead to a decrease in the quantity of good B consumed, which means that the net effect is still a negative one. Put another way,
5 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect although the income effect raised the consumption of the inferior good B, the substitution effect is still stronger such that in the final scene, demand still fell.
It is however possible to that the income effect is stronger than the substitution effect,
x A
B
A
U3
U2
C U1 Substitution Effect x B Income Effect
And when that happens, we refer to such a good a Giffen Good. Note that Giffen good must be a Inferior Good, but an inferior good is not a Giffen Good. The decomposition can be similarly performed for a Hicksian Substitution effect.
6 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect
Substitution and Income Effects under other forms of Indifference Curves we have discussed thus far We now examine the impact when we have a Leontieff Utility function: Because this utility function assumes that goods are perfect complements, there is only one combination for every income level, which then means that there is no substitution effect as our diagram notes.
x A
A U3
U2 B
x Income Effect B
Similarly, for a perfectly linear utility function, there would not be any income effect, but have instead a substitution effect only. You should verify this yourself.
Another interesting variation is that of a quasilinear utility function, and when the price of the numeriare good changes. The diagram below depicts a decrease in price of a numeriare good, and it effects. Just like the linear case, the effect is one that is purely generated by the substitution effect.
7 Intermediate Microeconomic Theory: ECON 251:21 Substitution and Income Effect
x A
U3
U2
U1
x Substitution Effect B
8