Chapter 5
Price, Income, and Substitution Effect
Macroeconomics For Financial Decisions
MS Finance, Spring 2008
By Kornkarun Kungpanidchakul, Ph.D.
When the price of a good changes, there is a change in the quantity demanded of good 1 due to a change in its price is referred to as the ‘price effect’. The price effect is decomposed into two sort of effects : the substitution effect and the income effect.
I Slusky Equation
The substitution effect
When the price of good 1 decreases, you have to give up less of good 2 to purchase good 1. Therefore, the rate of exchange between the two goods changes. This is what we call the substitution effect.
Slutsky isolated the change in demand due only to the change in relative prices (substitution effect) by asking “What is the change in demand when the consumer’s income is adjusted so that, at the new prices, she can only just buy the original bundle?” In other word, if the ‘real income’ or ‘purchasing power’ remains unchanged, how will the consumer adjust his consumption at the new price?
In order to find the substitution effect, we will draw the pivoted budget line where the slope of the budget line changes while its purchasing power constant (the original bundle of goods lies on the pivoted budget line).
Let m’ be the amount of money income that will just make the original consumption bundle affordable, and from (x1, x2) is affordable at both (p1,p2,m) and (p’1,p2,m’), we have
(1)
(2)
(1)-(2), we have
(3)
We can now find the pivoted budget line. It is the budget line at the new price with income changed by .
Although (x1, x2) is affordable, the optimal consumption bundle at the pivoted budget line is not (x1, x2). The optimal consumption bundle at the pivoted budget line is (x’1, x’2). The movement from (x1, x2) to (x’1, x’2) is the substitution effect.
Precisely, the substitution effect is the change in the demand for good 1 when price of good 1 changes to p’1 and the income changes to m’ simultaneously (the income changes to m’ in order to make the purchasing power constant). Therefore,
(4)
If the price of a good goes down, he substitution effect is always nonnegative and vice versa. The explanation is that the original consumption bundle (x1, x2) is always affordable on the pivoted budget line. However, the bundle (x1, x2) is not purchased on the pivoted budget line. Instead, the optimal consumption bundle of the pivoted line is (x’1, x’2). Therefore, the optimal bundle on the pivoted line must not lie underneath the original budget line which means that when price of good 1 goes down. In conclusion, the substitution effect must always be negative ()
The income effect
The income effect is the change in demand due to having more purchasing power or more real income. To find the income effect, we change the consumer’s income from m’ to m, keeping the price constant at the new price (p’1,p2).
From the picture below, the income effect is the change from the point (x’1, x’2) to (x’’1, x’’2). Precisely, the income effect is the change in the demand for good 1 when income changes from m’ to m, keeping price constant at (p’1,p2).
(5)
The total change in demand
Slutsky identity:
(6)
(7)
1. Normal goods
Higher income increases demand, so the income and substitution effects reinforce each other. So the total effect is negative for normal goods. The Law of Downward-Sloping Demand therefore always applies to normal goods.
Therefore, we have ,, and .
2. Inferior goods
For inferior goods, demand is reduced by higher income. Therefore, The substitution and income effects oppose each other when an income-inferior good’s own price changes.
Therefore, we have ,, and can be either positive or negative.
In rare cases of extreme income-inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to fall as own-price rises. Such goods are Giffen goods.
Example 1 : Suppose The consumer has an income of $200 and his utility function is in the form of U(x1, x2) = x1x2. The price of good 1 is $2 while for good 2 is $4. If the price of good 1 decreases to $1, find the (Slutsky) substitution and income effects.
II Hicks Substitution effect
The difference between Slusky and Hicks substitution effect is how we ‘compensate’ the consumer. We have learnt that slusky substitution effect compensates consumers so that the real income or purchasing power is constant. For Hicks substitution effect, it will compensate consumers so that their utilities are constant. In other words, Hicks substitution effect implies that if the level of utility remains unchanged, how will the consumer adjust his consumption at the new price?
Expenditure Minimization Problem
So far we know that the consumer will maximize his utility subject to the budget constraint. What if the problem for the consumer when the level of the utility is given? This is what we call the dual problem. Instead, the consumer will minimize his expenditure subject to a certain level of utility :
Min E = P1x1+P2x2
s.t. U(x1,x2) = U0
Under expenditure minimization problem, the demand function we get is Hicksian Demand: xh(P,U0) while the demand function from utility maximization problem is what we call Marshallian Demand: x(P,m).
Example 2 : Given the utility function and the budget constraint
a) find marshallian and hicksian demands of each good.
b) Given that the original income is $100 and the original price of good x1 and x2 is $3 and $2 respectively. Now suppose that the price of good x1 decreases to $2, find the hicksian substitution effect and income effect.
III Rate of Change
Suppose that from the price and income set (p1,p2,m), the consumer gets the utility at the level of U0. Therefore we have:
Differentiate wrt to p1, we have:
(8)
From (3), , so we have:
(9)
Slusky equation
If we evaluate at the optimal consumption bundle with the set of price and income at (p1,p2,m) and the consumer gets the utility at the level of U0 as the optimal level, then (9) can be rewritten as:
(10)
(10)
(10) is what we call Slusky Equation.
IV The Price Effect when there is endowment
In this chapter, we suppose that, instead of the fixed income, consumers have endowment. The list of resource units with which a consumer starts is his endowment. We will denote the endowment with .
i) Budget constraint and budget line
Denote the endowment of good 1 and good 2 by ( ). Then the budget constraint becomes:
The above equation is represented in terms of gross demands (the amount of goods that consumers end up consuming). If we express the budget in terms of net demands (the difference between the amount of good that consumers end up consuming and the initial endowment), we have:
If net demand is positive, the consumer is a net buyer of that good. If the net demand is negative, the consumer is a net seller or net supplier.
We still have the budget line similar to the case that the income is fixed. The endowment bundle is always on the budget line as the initial endowment is always affordable from the consumer’s point of view.
The objective of a consumer is still to maximize his utility, given the level of endowment and prices. Consumers can trade their endowments with others so that they can reach a higher level of utility
ii) The movement of budget line
Changing in the endowment
Price ratio (slope)is constant. The direction of the movement depends on the change in value of the total endowment.
Changing in prices
The above picture is the example when price of good 1 changes. The price ratio will changes and the budget line will be pivoted around the endowment point as this point is always on the budget line.
iii) Consumption with Endowment
Consumer’s objective function is still to maximize is utility given the budget constraint.
Max U(x1,x2)
s.t.
Also, we still have the dual problem, the expenditure minimization problem.
Min E =
s.t.
Example 1: Suppose that the utility function is . Find the marshallian and hicksian demands for both goods.
iv)Change in consumption when price changes
Example 2 : Suppose that the consumer is a seller of good 1
and price of good 1 decreases.
If he remains a supplier, then his new consumption bundle must be on the blue part of the new budget line. So the consumer is worse off (by revealed preference). However, if the consumer decides to switch to be a buyer of good 1, it is unsure whether he will be worse of or better off.
Example 3: Suppose that the consumer is a buyer of good 1 and price of good 1 decreases.
If the consumer is originally a net buyer of good 1, when the price of good 1 decreases, he will always continue to be a net buyer of good 1. If he switches to be a supplier, his budget line is on the blue part, which lies underneath the original budget line. However, we knows that originally, ( is preferred to any point from y-intercept to the endowment point. Also, under new budget line, ( is feasible (inside the budget set). Therefore, consumers will not consume on the blue part which is less favourable than (. We can draw a conclusion that the consumer always remains a net buyer of good 1.
v) Offer curves and Demand curves
Offer curve contains all the utility-maximizing gross demands or the combinations of both goods demanded by a consumer at difference levels of prices.
If we depict the curve representing the relationship between the price and the quantity demanded of some good, we get the demand curve. The gross demand curve represents the total amount of good a consumer chooses to consume. The net demand curve represents the difference between the gross demand and the endowment of good 1 if this is positive, and zero otherwise. Finally, the net supply curve is the difference between the initial endowment and the gross demand if this term is positive, and zero otherwise.
vi) The slutsky equation revisited
Total change = substitution effect + ordinary income effect + endowment income effect
If we consider the case that endowment is involved, when the price of some good changes, there is a change in money income as well.
How to find slutsky equation?
I) Fixed money income and consider only the effect of a change in prices.
II) Find the pivoted budget line that has the same price ratio as the new budget line but passes through the original consumption bundle.
III) The movement from the original consumption bundle to the optimal consumption bundle on the pivoted budget line is substitution effect (the movement from point 1 to point 2).
IV) The movement from the optimal consumption bundle on the pivoted budget line to the optimal consumption bundle on the new budget line that has fixed income (from point 2 to point 3) is the ordinary income effect.
V) The new budget line that has a change in money income is the budget line that has the new price ration and passes through the initial endowment.
VI) The movement from the optimal consumption bundle on the new budget line that has fixed income to the new optimal consumption bundle (point 3 to point 4) is the endowment income effect.
+ endowment income effect (1)
Where endowment income effect = (2)
With , we have. Therefore,
endowment income effect = . (3)
Substitute (3) into (1), we have
(4)
(4) is the slutsky equation when a consumer has endowment.
Example 4 : From example 1, given the initial endowment is and . Suppose further that and . Then price of good 1 changes to $1, find the substitution effect, income effect and the endowment income effect.
V Intertemporal Choice
i) The Budget Constraint
Let consider the case that a consumer lives for 2 periods. The amount of money the consumer will have in each period is (m1,m2) and the amount of consumption in each period is (c1,c2). Suppose further that the prices of consumption in each period are constant at 1. First consider when there is no borrowing. Then the only way the consumer has in order to transfer money from period 1 to period 2 is by saving it without earning interest. Then we can draw the budget constraint as follows:
Now, let suppose that the consumer can borrow and lend money at the same interest rate, r. Then we can write the budget constraint as:
(1)
Or
(2)
(1) is the budget constraint in terms of future value and (2) is the budget constraint in terms of present value.
ii) Preferences for consumption
The optimal consumption bundle is still a point at which the indifference curve is tangent to the budget line. If the consumer chooses the point where c1 < m1, then he is a lender while if he chooses the point where c1 > m1, then he is a borrower.
iii) Comparative Static
Suppose that the rate of interest increases. Then the budget line goes to a steeper position. Suppose that he is a lender before, then using the revealed preference, he must remain a lender and he will be better off.
Consider the slutsky equation to decide the change in today’s consumption.
Total effect = income effect + substitution effect
Income effect: the consumer is getting richer in the future, he will increase his consumption, both today and in the future. (IE > 0)