P a g e | 1
1. Derive the demand curve for particular goods using the concept of consumer optimization (maximizing
Utility subject to a budget constraint).
2. Explain the Income and Substitution Effects of a price change and how this varies for Normal, Inferior,
and Giffen goods.
3. Explain how a change in the price of a good affects three measures of a consumer’s wellbeing:
compensatibg variation, equivalent variation, and consumer surplus.
4. Derive market demand curves from individual demand curves.
5. How do network externalities affect market demand curves?
6. How does the labourleisure choice affect the Labour Supply curve? How do the income and substitution
effects influence the shape of the labour supply curve.
7. Explain the substitution bias in the CPI and how the CPI is constructed.
I. Deriving demand curves from the consumer optimization model.
A. Graph We can derive the demand for good X by using consumer theory. Demand curves
are downward sloping because when the price of good X decreases, the quantity demanded of X
increases (except in the case of a very rare Giffen good). When Px decreases, this changes the
amount of X we can consume given our current income, and the budget constraint pivots outward
along the xaxis. Because of the lower price of X, the consumption possibilities of X have increased. Now
the consumer must maximize utility (satisfaction) given this new budget constraint, and this almost
always leads to an increase in the amount of X consumed.
So when Px changes, the quantity of X consumed changes for two reasons:
(1) Substitution effect If Px decreases, the price of X decreases relative to the price of other
goods (namely Y), and the consumer buys more X and less Y because X has become relatively
(2) Income Effect – A decrease in Px increases the consumer’s real income. Because X is
cheaper the consumer’s purchasing power has increased. The consumer can can buy more x
and still consume the same amount of Y. Now that the consumer’s real income has increased,
the consumer will buy more X if it is a normal good and less X if it an inferior good.
The substitution effect almost always dominates the income effect, except in the case of a
Giffen good, which is why we almost always see downward sloping demand curves.
(A Giffen good is an inferior good where the income effect dominates the substitution effect,
and the demand curve is upward sloping. Ex: potatoes during the Irish Potato Famine)
This graph show how you can map out the demand curve by recording the
amount of x consumed as the price of x declines. P a g e | 2
Finding the demand curves for x and y from consumer optimization theory.
Given the consumer’s utility function and budget constraint, we can derive the consumer’s demand
for each good x and y.
The solution is to find the demand s for x and y as functions of prices and income because demand
is a function of own price, prices of related goods, and income (we ignore tastes and expectation
components of demand for now).
X* = D(Px,Py, I)
Y* = D(PY,Px, I)
Example 1: Finding the demand curves for X and Y P a g e | 3
1. Start with the utility function.
U(x,y) = x y
s.t. PxX +PyY = I
2. Find the optimal consumption bundle using the method of Ch. 4
Note whether or a corner solution is possible.
This is a CobbDouglas Utility function; so we know that the solution is interior with both
positive amounts of x and y being consumed.
Use the two equations to solve:
MUx/MUy = Px/Py
MUx = dU/dx = .5X Y .5
MUy = dU/dY = .5 X Y .5
.5 .5 .5 .5
MUx/MUy =(.5X Y )/( .5 X Y ) = Y/X = Px/Py
Y = PxX/Py
2. Budget constraint
PxX + PyY = I
PxX + Py(PxX/Py) = I
2PxX = I
X* = I/(2Px) (this is the demand for X because we have solved for X in
terms of Px, Py, and I)
Sub our solution for X into the eqn for Y.
Y = PxX/Py =[ Px(I/2Px)] / Py
Y* = I/(2Py) (the demand for Y in terms of Px,Py, and I)
Given the Demand Curve, we can calculate some elasticities:
Demand for y: Y* = I/(2Py)
∂ y Py −I Py
1. Ownprice elasticity of demand = %ΔQ /%ΔPy = × = × =
∂Py y 2Py2 y
−I × Py
2Py2 I = 1
So the price elasticity is constant and equals 1 for every Py and quantity y.
∆Y ∂y y 1 I ×
2. Income Elasticity of Deman= ∆I = ∂I × I = 2Py × y = 2Py I =
So y is a Normal good because the income elasticity is positive. Income and Y
move together. Another way to see that Y is normal i∂I , which leads to P a g e | 4
the positive income elasticity because I/y is always greater t∂In 0. ,
instead, then y would have been an inferior good.
∆Y ∂ y Px Px
3. Cross price elastici y = = × = 0 x = 0
∆Px ∂Px y y
Demand for y does not depend on the Px at all; so the derivative is 0. Y and X are
unrelated. Note that if
∂Px >0 → y and x are substitutes
∂Px <0 → y and x are complements
Decomposing the effect of a price change into the income and substitution
When the price of a good either increases or decreases, there are two effects:
1. Substitution effect
2. Income effect
To see each effect separately, the Key is to find the decomposition bundle B.
Problem 5.9 Rick purchases two goods, food and clothing. He has a diminishing marginal rate of
substitution of food for clothing. Let x denote the amo