# Lecture 3 – Consumer choice (part 2) 1 Walrasian demand and

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```					Lecture 3 – Consumer choice (part 2)

1. Walrasian demand and indirect utility function:
Roy’s identity

2. Comparative statics

3. Slutzky decomposition

4. Welfare: consumer surplus
1. Walrasian demand and indirect utility function: Roy’s
identity
• In the previous lecture we have shown that the Hicksian
demand is the derivative vector of the expenditure
function with respect to prices (Shephard lemma)

•   Somehow in analogous way we can show the following
relationship between indirect utility function and
walrasian demand:

Roy’s identity: Suppose that u(.) is a continuous utility
function and suppose that the indirect utility function
v(p,m) is differentiable at (p*, w*)>>0, then for every
l = 1.....L the following holds:
∂v ( p*, w*) / ∂p l
xl ( p*, w*) = −
∂v ( p*, w*) / ∂w

In “words”, the walrasian demand is equal to the derivative
of the indirect utility function with respect to prices
normalized by the marginal utility of wealth.

Proof:
Consider u*=v(p*, w*), where w*= e(p*,u*). Because this
identity holds for all p, diffentiating v(.) with respect to
prices, and evaluating at p=p* we obtain:

∂v( p*,e( p*,u*) ∂v( p*,e( p*,u*) ∂e( p*,u*)
+                            =0
∂p           ∂e( p*,u*)        ∂p

Since by the Shepard lemma we know that the derivative
of the expenditure function with respect to the price p is
equal to the Hicksian h(p, u) demand, and remembering
that that w= e(p,u), substituting in the above expression
we get
∂v( p*,e( p*,u*) ∂v( p*,e( p*,u*)
+                 h( p*,u*) = 0
∂p              ∂w
Since, w*=e(p*,u*) and from the last lecture we know that
a) if the quantity x* is optimal in the expenditure
minimisation problem where u=u*, then x* is optimal in
the utility maximisation problem when w*=p*x*. Moreover,
the maximised utility level is u
b) If x* is optimal in the utility maximisation problem, then
x* is optimal in the expenditure minimisation problem
where the required level of utility is u(x*)
Then from a) and b), h(p*,u*)=x(p*,w*), which implies:
∂v( p*,e( p*,u*) ∂v( p*,e( p*,u*)
+                 x( p*, w*) = 0
∂p              ∂w
and re-arranging:

∂v( p*,e( p*,u*)
∂p
x( p*, w*) = −
∂v( p*,e( p*,u*)
∂w

2. Comparative statics

Given the UMP and EMP we have derived:

- walrasian demand, indirect utility function and
ROY identity;
- hicksian demand, expenditure function and
SHEPARD lemma;
Now, we should be able to analyze how the consumption
changes if we change price and income, i.e. we should be
able to make comparative statics exercises!
• Income changes - - Engel curve
DRAW A GRAPH!

Positive effect: normal goods
Negative effect: inferior goods

• Price changes - - price offer good

DRAW A GRAPH!

Negative effect: ordinary good
Positive effect: Giffen good

3. Slutzky decomposition
The price effect can be decomposed into two parts

- Income effect
- Substitution effect

Exercice: SHOW GRAPHICALLY THESE TWO EFFECTS

In analytical terms, these two effects can be precisely
measured using the Slutzky decomposition.
Formally, for all (p,w) and u=v(p,w) the following holds:

∂hl ( p, u ) ∂xl ( p, w) ∂xl ( p, w)
=           +            .xk ( p, w)
∂pk           ∂pk          ∂w                    (1)

Proof:
When the consumer solves the UMP or the EMP we know
that we know that for all (p,u), h(p,u)=x(p, e(p,u)) where
w=e(p, u). Hence, differentiating the hicksian demand for
good l with respect to the price pk and evaluating the
derivative at (p*, u*) we obtain:

∂hl ( p*, u *) ∂xl ( p*, e( p*, u*)) ∂xl ( p*, e( p*, u*)) ∂e( p*, u*)
=                     +
∂pk               ∂pk                   ∂e(.)            ∂pk

Remembering that the derivative of the expenditure
function with respect to the price of a good corresponds to
the hicksian demand for that good, and that w=e(p,u), we
obtain:
∂hl ( p*, u *) ∂xl ( p*, w *) ∂xl ( p*, w *)
=              +               hk ( p*, u*)
∂pk            ∂pk             ∂w
Remembering that for all (p,u), h(p,u)=x(p, e(p,u)), we
prove that:

∂hl ( p*, u *) ∂xl ( p*, w *) ∂xl ( p*, w *)
=              +               xk ( p*, w*)
∂pk            ∂pk             ∂w
If you consider all the possible price change at the same
time, you can write the slutzky substitution matrix where
each term is given by the equation (1)…

Exercice: suppose there are only 3 goods and 3 prices.
Write the slutzky substitution matrix!

Property: when the demand generated from preference
maximization, the slutzky substitution matrix is symmetric
and negative definite.

Re-arranging the terms of equation (1)

∂xl ( p, w) ∂hl ( p, u ) ∂xl ( p, w)
=            −            .xk ( p, w)
∂pk        ∂pk            ∂w

substitution effect:
∂hl ( p, u )
∂pk

income effect:

∂xl ( p, w)
.xk ( p, w)
∂w

Question: when do you have a Giffen good?
Consumer Welfare

•   Positive analysis: what are the actual consumer choices
given preferences and constraints?

•   Normative analysis: when consumer choices change,
how does consumer well-being change?

So far we just investigated consumer choice (positive
analysis), now we would like to evaluate the welfare
implication of these choices.

We have already done comparative static exercises and
we know that if prices and/or income change, then the
consumer optimal choice change

Hence, given that the consumer optimal choice changes,
we would like to know how the well-being of the consumer
changes.

The type of question we would like to answer is:

is the consumer better off or worse off as a consequence
of a price and/or income change?

Measuring consumer’s welfare

The first problem we have to solve in order to evaluate the
change of consumer’s welfare is to find a MEASURE of
consumer welfare.

•   Which MEASURE do you suggest?
Utility?
Suppose we use utility. Then, the obvious way to measure
the well-being of the consumer at two different
consumption choices is the indirect utility function

Let (p*,w*) be a given price-income pair and (p’,m’) be
another price-income pair, then the difference in utility
associated to the consumer choices for these two price-
income pairs is simply:

V(p*,w*)-V(p’,m’)

However, remember that utility is a pure ORDINAL
concept!

Problem: suppose that initially the price and income level
in an economy are (p*,w*). Suppose that following an
exogenous price shock, the level of prices increases.

•   What will happen to consumer welfare?

•   Suppose that the government wanted to keep the
consumer welfare constant. What should the
government do?

To answer this type of questions, it would be useful to
have a cardinal measure of welfare change. For example,
a measure in terms of income.

Construction of a money-metric indirect utility function

A money metric utility function tells you the wealth
required to reach a level of utility V(p,w), when the price
∧
level is any arbitrary vector p
Formally:
∧
e( p, v( p, w)
This function gives you the income necessary to achieve
∧
the utility v(p,m) when the price level is p

∧
This measure can be constructed for any p . But in
particular, since we are interested in the change of welfare
associated to a particular price change, we can write this
measure either for the initial price vector or for the final
price vector.
o       1
Let p and p be the initial and final price vector
Let u = v( p , w) and u = v( p , w)
o      o         1       1

Then we can write two measure of consumer welfare:

•   EQUIVALENT VARIATION (EV)

EV ( p o , p1 , w) = e( p o , u1 ) − e( p o , u o )

The equivalent variation tells you the amount of money
you need to give the consumer so that he is indifferent
between staying in the status quo or accepting the price
change.
o   1
Note: the term e( p , u ) gives you the income necessary to
1
guarantee the utility level u at the original prices. Loosely
speaking, this tells you how much money you should give
the consumer so as to bring him to the new level of utility
without changing prices.

•   COMPENSATING VARIATION (CV)

CV ( p o , p1 , w) = e( p1 , u1 ) − e( p1 , u o )
The compensating variation tells you the amount of money
you should give the consumer to bring him at the level of
utility he was enjoying before the price change
1   o
Note: the term e( p , u ) gives you the money necessary to
guarantee the initial level of utility at the new prices.

Question: Can you represent graphically the EV and CV?
Consumer surplus and money-metric utility measures
Another common measure of the consumer welfare is the
consumer surplus, i.e. the integral of the marshallian
demand.

DRAW A MARSHALLIAN DEMAND AND SHOW THE
CONSUMER SURPLUS!

QUESTION: how do consumer surplus, EV and CV
relate?

HINT:
• Consider the definition of EV and CV
• Apply Shepard’s lemma

RESULT?

We can re-write the EV and CV in terms of hicksian
demand:
po
EV ( p , p , w) = e( p , u ) − e( p , u ) = ∫ 1 h( p, u1 )dp
o       1       o   1            o   o
p
po
CV ( p , p , w) = e( p , u ) − e( p , u ) = ∫ 1 h( p, u o )dp
o       1       1   1        1   o
p
Now remember that from Slutzky decomposition we know
the slope of hicksian and walrasian demand….

•   DRAW ON THE SAME GRAPH HICKSIAN AND
WALRASIAN

•   SHOW CONSUMER SURPLUS, EV, CV

Consumer welfare measures and income effect

•   Remember from slutzky decomposition that the
difference in slope between hicksian and walrasian
demand function is due to the income effect

•   What happens if income effect is equal to zero?

•   Quasi-linear preferences are a special case where the
income effect, at least for “high” income level is equal to
zero

Quasi-linear function
U ( x1 , x 2 ) = x1 + u ( x 2 )
where u ( x 2 ) is a strictly concave function

Property: when the income effect is equal to zero we have
that Consumer Surplus=EV=CV
EXERCICE: show if this property holds for a quasi-linear
utility function.

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Description: Lecture 3 – Consumer choice (part 2) 1 Walrasian demand and