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Embed code for: Class 3 Income and Substitution Effects
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chool of Economics
LeBow College of Business ECON 610
Drexel University Professor Stehr
Class 3: Income and Substitution Effects
(Marshallian) Demand Functions
In the last class, we discussed demand functions that are obtained when a consumer maximizes his utility given prices and an income constraint.
We defined demand for as where the Ps are the prices of the ’s, and is income. These are called Marshallian demand functions after Alfred Marshall
Marshallian demand functions are homogeneous of degree zero in prices and income. This means that if you double prices and income, demands will be unchanged. More formally,
Marshallian demand functions show how changing either prices or income affects demand for a good.
Suppose we change income, .
If then we have a normal good. Most goods are normal goods.
If then we have an inferior good. These are goods we buy less of when our income goes up, e.g. ramen noodles or high fat hamburger meat.
In the two good case, can both goods be inferior?
The budget line shifts out but retains its slope as income increases:
Normal GoodInferior Good
When the price of good x decreases, we get the following effects:
The ratio of income to the price of good x, I/Px, increases which means purchasing power increases. The resulting change in x is called the income effect.
The ratio between the price of good x and price of all other goods (i.e. price of y in the two good case) decreases, so the relative expense of good x decreases. The resulting change in x is called the substitution effect.
For normal goods, the income and substitution effects work in the same direction:
For inferior goods, the income and substitution effects work in the opposite direction:
There is a special case for inferior goods. When a good is inferior and the income effect dominates (is larger in absolute value than) the substitution effect, then we have a Giffen Good. Giffen goods are very inferior.
The demand curve is related to these graphs, but plots price of x on the vertical axis instead of the quantity of y.
Hicksian Demand Curves
The Marshallian demand curve holds income constant as price changes. Utility will change along this curve because purchasing power is changing (there is an income effect).
The Hicksian (named after Hicks) compensated demand curve holds utility constant as prices change. Income is adjusted in order to hold utility constant (there’s no income effect).
Since the Hicksian demand curve compensates (or “uncompensates” by taking income away) the consumer for changing prices, it represents only substitution effects. In other words, by holding utility constant, it’s holding real income or purchasing power constant.
The Marshallian represents points on different indifference curves with the same income. The Hicksian represents points on the same indifference curve with different incomes.
There are two ways to derive a Hicksian demand function. The first is to set up a Lagrangian as an expenditure minimization problem instead of a utility maximization problem and find the minimum expenditure or income required to achieve a certain utility.
Second, use what’s called Shephard’s Lemma:
Find the Marshallian demand functions.
Write utility as a function of prices and income (the indirect utility function).
Solve for income to find the expenditure function.
The first partials of the expenditure function w.r.t. each price will be the Hicksian demand functions (this step is Shephard’s Lemma).
Example: Maximize subject to the constraint:
The dual problem involves minimizing the expenditure required to obtain a given level of utility. The solutions to this problem are the Hicksian (i.e. compensated) demand functions.
The FOCs are:
Solving the first two conditions for and setting them equal, we obtain (it’s ok that U doesn’t actually appear in the demand for y but this is the exception, not the rule):
Then plugging this into our constraint and solving for x, we obtain:
Note: we’re ignoring the case where , the case where income is insufficient to buy Recall that with quasilinear preferences, the income consumption curve requires two equations. A complete solution would require a second function, but we’ll skip it in the interests of time.
Compare the Hicksian and Marshallian demand functions using the example above: suppose Income = 4, PX = 1 and PY = 1.
Plugging in, we can calculate x, y, and U at these initial conditions:
Now, raise the price of x to 2 and calculate what happens to x using the Marshallian and Hicksian demand functions.
The Marshallian demand holds income at I = 4 and allows utility to vary. The Hicksian demand holds utility constant at U = 3 and allows income to vary.
Now, lower the price of x to and again calculate what happens to x using the Marshallian and Hicksian demand functions.
You can relate the two demand curves (drawn with PX on the vertical axis and x on the horizontal axis) to the decomposition into income and substitution effects drawn with y on the vertical axis and x on the horizontal axis.
The Slutsky decomposition:
Start with the observation that the Marshallian and Hicksian demand functions yield the same quantity when income is at the level required to obtain the utility specified by the Hicksian demand function:
Then, take the partial derivative of with respect to
Using Shephard’s Lemma, we obtain:
Since income is the same as expenditure in the equation, , and since at the utility maximizing point, we can write:
The SE will always be negative whereas the IE is positive in the case of what’s called a normal good and negative in the case of what’s called an inferior good. The sum of the effects is negative except in the rare case of a Giffen Good. In this case the income effect is negative and larger than the SE in absolute value and the entire expression will be positive.
Historical Note: Eugen Slutsky wrote the equation slightly differently. Instead of writing , Slutsky noted that utility would be held constant when taking the derivative of x with respect to PX, and he didn’t directly distinguish between and in his notation.
The Marshallian price elasticity of demand is for a movement along the Marshallian demand curve.
It captures how percent changes in quantity demanded are related to percent changes in prices
It can be defined intuitively in percent changes or using calculus.
This expression will be negative because is negative (except in the unusual case of the Giffen good described above).
Price elasticity of demand and total revenue:
Total revenue (TR) is defined as price multiplied by quantity:
By taking the derivative of TR w.r.t. price and rearranging a little (using the rule for differentiating products), we can get an expression in terms of the elasticity.
Three cases emerge:
; demand for these goods is called elastic. An increase in price leads to a decrease in in total revenue.
; demand for these goods is called inelastic. An increase in price leads to an increase in in total revenue.
; demand for these goods is called unit elastic. An increase in price leads to no change in total revenue.
Suppose you develop an app for the iPhone and you have an equation for demand. What price should you charge to maximize your profit? Hint: since all the costs of the app are fixed and sunk, you can ignore cost (which we have yet to consider in this course) in deciding the price.
Cross price elasticity measures how responds to changes in the price of another good, y (could also be defined in terms of Hicksian demand).
Income elasticity measures how responds to changes in income.
for a normal good
for an inferior good
What are some example of good that are likely to be normal?
What are some example of goods that are likely to be inferior?
Measuring welfare changes from changes in prices--consumer surplus (CS), compensating variation (CV) and equivalent variation (EV)
When price changes, expenditure must change if we are to keep the consumer at the same utility.
For a price decrease, if we knew how much income (or expenditure) to take away from the consumer so that he would be just as well off as he was before, we would have a way of measuring, in terms of cash, how much better off the price decrease made the consumer. This amount of cash is called the compensating variation. For a price increase you can think of it as how much more income you would need to maintain your standard of living if you were to move to Manhattan.
More formally, the compensating variation is the change in expenditure required to achieve, the old utility, at the new prices, .
The compensating variation can be illustrated graphically as the change in expenditure required to stay on the old IC (and hence at same old utility) under the new prices.
Equivalently, the CV is the change in expenditure required to jump between the old and new indifference curves at the new prices.
The second terms in each of the expressions for CV are actually equal (i.e. ). That’s because both budget lines represent the same income but at different prices.
Mnemonic: the expenditure of a hypothetical budget line has subscripts on price of x and utility that don’t match. For budget lines the consumer actually experiences, the subscripts match.
We can’t observe the expenditure function, but recalling Shephard’s Lemma, we know that the Hicksian demand, is the derivative of the expenditure function w.r.t. own price, so we can integrate over price to find the CV.
Although we can in theory calculate the CV by integrating the Hicksian demand function, in the real world we observe Marshallian demand functions. This is because it’s possible to find situations where income is held constant as price varies, but it’s not possible in reality to hold someone’s utility constant by adjusting her income while we vary price.
How does the Marshallian calculation differ from its Hicksian counterpart? In the case of a normal good, the Marshallian demand curve will be shallower than the Hicksian demand curve. For a price increase, this is because the Hicksian compensates the consumer with more income to hold utility constant while the Marshallian includes no income adjustment forcing the consumer to buy even less x. Thus, for a price increase, the change in welfare under the Marshallian will be larger than the change in welfare under the Hicksian. For a price decrease, however, the opposite holds true.
The compensating variation isn’t the only way to calculate the welfare change from a price change. For a price increase, what if we were to start at the new utility and work backwards by asking how much income we would have to take away from the consumer at the old prices to make him just as bad off as he was under the price increase. This is called the Equivalent Variation (EV).
Intuitively, it’s how much you would pay to avoid facing the higher price (e.g. how much you would pay to avoid being relocated from Philadelphia to Manhattan with no cost of living adjustment). Similar to the CV, it can be illustrated graphically.
Equivalently, the EV is the change in expenditure required to jump between the old and new indifference curves at the old prices.
Note that the first terms in each of the expressions for EV are equal (i.e. ) because both correspond to the same income under different prices and hence different utilities (they represent budget lines 0 and 1 on the graph). Written as an integral,
In general CV≠EV. Intuitively, that’s because different compensations are required under different prices. For example, we have to give you more to compensate you for moving to Manhattan than we would have to take away from you to make you as bad off in Philadelphia as you would be if you moved to Manhattan with no compensation.
Putting both Hicksian demand curves and the Marshallian on the same graph shows that the Marshallian demand curve gives a welfare change that is in between the CV and EV given by the two Hicksian demand curves. Thus, use of the Marshallian to calculate welfare changes may not be so bad. In practice, the difference between the three welfare effects may be small if the income effect for the good in question is small as it would be in the case of carrots (but not housing).
Consumer surplus—the total welfare that a consumer derives from buying a good in a market. At a given price, it’s the welfare loss that would result if the price were raised to a “choke price” where the consumer stopped buying the good. It’s also the welfare loss if the good is banned.
It’s also possible to calculate CS as the area of the triangle between demand, the vertical axis, and the line corresponding to the price paid. Or, you can write price as a function of quantity and calculate it as:
where and are the chosen quantities.
In his paper, David Bradford estimates the amount you would need to pay a pregnant woman to get her to quit smoking during pregnancy. He does this by estimating their demand for cigarettes and then using those results to estimate their consumer surplus. Since they give up welfare equal to their CS when they quit, the payment must be at least that large to be successful. See Bradford, David. 2003. “Pregnancy and the Demand for Cigarettes.” American Economic Review 93(5): 152-63.
Draw two graphs indicating the payments necessary to induce later pregnant (i.e. pregnant with second or later child) heavy and light smokers to quit.
The CV and EV are changes in Hicksian CS and thus changes in welfare under Hicksian demand curves since they are typically evaluated between two prices (they can be used to calculate total CS if one of the prices is arbitrarily high). We can also calculate a change in Marshallian CS as the change in area under the Marshallian demand curve and above the price.
Practice Question #1: A Social Security recipient has strictly convex indifference curves, is buying some x and some y and the price of x increases. Show graphically that by making her old bundle available at the new prices, the government is overcompensating her for the price change and making her better off. The US Social Security Administration makes this mistake when making cost of living adjustments (COLA) to social security benefits using the consumer price index (i.e. CPI-W).
Practice Question #2: Suppose a consumer has quasilinear preferences given by:
Furthermore, suppose so the consumer in on the horizontal portion of the income consumption curve so that
a) Find , and y if I = 4 and PX = 1.
b) Find , and y if I = 4 and PX = 2.
c) Find , and y if I = 4 and PX = .
d) Draw a graph illustrating the two demand curves at these three prices.
e) Use the expenditure function to calculate the CV of the increase in the price of x from 1 to 2.
f) Calculate the change in Marshallian CS of the increase in the price of x from 1 to 2.
g) Indicate the CV and the change in Marshallian CS on the graph.
Relationships between income effect, substitution effect, compensating variation (CV), change in consumer surplus (CS), Marshallian and Hicksian demand curves. Drawn for price increase for a normal good.
BL0 E(Px0, Py, U0)
BLHYP ; E(Px1, Py, U1)
BL2; E(Px1, Py, U0)
Area DCBAE is CV or change in Hicksian CS
Area DCAE is change in Marshallian CS
DemandMarshallian or xM
DemandHicksian or xC
4pond to the same income under different prices and hence different utilities (they represent budget lines 0 and 1 on the graph). Written as an integral,
Putting both Hicksian demand curves and the Marshallian on the same graph shows that the Marshallian demand curve gives a welfare change that is in between the CV and EV given by the two Hicksian demand curves. Thus, use of the Marshallian to calculate welfare changes may not be so bad. In practice, the difference between the three welfare effect