Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 9 Outline of the course
II. Individual choice
III. Competitive markets
IV. Market failure Outline of the course
II. Individual choice
Budget constraint (Ch. 2)
Preferences (Ch. 3)
Utility (Ch. 4)
Consumer problem (Ch. 5)
Revealed preference (Ch. 7)
Slutsky equation (Ch. 8)
Endowment income effect (Ch. 9)
Intertemporal choice (Ch. 10)
Choice under uncertainty (Ch. 12) [First-midterm material ends here!]
Consumer surplus (Ch. 14)
Aggregate demand (Ch. 15)
Partial equilibrium (Ch. 16) III. Competitive markets IV. Market failure Consumer’s surplus
Want a measure of how much a person is willing to pay for something. How
much a person is willing to sacriﬁce of one thing to get something else.
Consider a discrete-good setting with quasi-linear utility: u.x1; x2 / D v.x1 / C x2; x1 D 0; 1; 2; : : : ; x2 2 RC: Normalizing the price of good 2 to p2 D 1, the consumer’s maximization
problem (given income m and price p1) is to choose x1 to maximize v.x1 /
over all x1 D 0; 1; 2; : : : . p1x1 C m Consumer’s surplus
The solution is given by reservation prices: r.1/ D v.1/ v .0/; r.2/ D v.2/ v .1/; : : : ; r.k/ D v.k/ v .k 1/; so that
x1 .p1/ D k if and only if r.k C 1/ < p1 r.k/: The reservation price for the k th unit measures consumer’s marginal
willingness to pay for an extra unit when he has k 1 units already. Natural
to assume r.1/ > r.2/ > r.2/ >
That is, consumer is willing to pay more for an extra unit when he has fewer
units. Consumer’s surplus
Add up over all different outputs to get total willingness to pay for k units.
Total willingness to pay for k units D r.1/ C r.2/ C : : : C r.k/
This is a measure of the consumer’s beneﬁt from consuming k units, since r.k/ D v.k/ v .k 1/ and therefore, r.1/ C C r.k/ D .v.1/
D v.k/ v .0// C .v.2/ v .1// C .v.k/ v .k v .0/ The sum r.1/ C C r.k/ is called the consumer’s (gross) surplus.
To get the consumer’s net surplus must subtract the total amount that the
consumer has to spend to get the beneﬁt.. 1// Continuous demand
Suppose utility has form u.x; y/ D v.x/ C y (quasilinear)
Thus, the inverse demand curve has form p .x/ D v 0.x/.
By the Fundamental Theorem of Calculus, v.x/ v .0/ D Z x
0 v .t/ dt D
0 Z x p .t/ dt :
0 This is the generalization of the discrete-goods argument.
If utility is not quasilinear, things are a bit complicated. We’ll not see this in
In general, we are interested in the change in consumer’s surplus due to
some policy (e.g., imposing a tax). Market demand
To get market demand, just add up individual demands
– add inverse demands horizontally,
– properly account for zero demands Example: linear demand Market demand
Often think of market behaving like a single individual.
This is the so-called representative consumer model.
Not valid in general, but reasonable assumption sometimes. (Common
assumption in macroeconomics.) Inverse of aggregate demand curve measures the MRS of the
representative individual. Elasticity
measures responsiveness of demand to price in percentage terms
If p.q/ is the inverse demand function, then elasticity is given by
Linear demand: q D a bp . Then,
a bp Note that in the linear demand case, D
the demand curve 1 when we are halfway down Elasticity
Now consider q D Ap b .Then,
D p bAp b
Ap b 1 D b thus elasticity is constant along this demand curve
note that log q D log A b log p , i.e., log-linear demand what does elasticity depend on? In general, how many and how close
substitutes a good has. How does revenue change when you change price?
Consider a downward sloping demand q.p/.
R D pq.p/, so
R0.p/ D q C pq 0.p/ D q.1 j.p/j/ If price increases, two effects: lower q and higher p . Who wins?
Depends on elasticity. R0.p/ > 0 if and only if j j < 1 (inelastic).
Increase in p dominates decrease in q for inelastic demand.
This is a local phenomenon! Consider linear demand case Partial equilibrium
We will maintain the setup in which preferences are quasi-linear:
u.q1; q2/ D v.q1/ C q2
He will normalize the price of good 2 to 1 and will focus on the demand
for good 1 as a function of its price p . The ﬁrst-order condition yields:
p D v 0.q/
where q D q1 hereafter. So, the marginal utility v 0.q/, as a function of
the quantity q , is the inverse demand. In general, we will be given the inverse demand function Pd .q/ directly,
or the demand function D.p/. Supply function
A supply function S.p/ measures the amount the supplier is willing to
supply at each price. We let Ps .q/ designate the inverse supply function.
Supply functions come from the ﬁrm’s proﬁt maximization. We have not
seen that yet, so we will take the supply functions as primitives. Equilibrium
Competitive market - each agent takes prices as given (outside of their
control) Foundations for competitive behavior: large markets (many small
agents) Equilibrium price - the price where desired demand equals desired suply
D.p/ D S.p/
– vertical supply - quantity determined by supply, price determined by
– horizontal supply - price determined by supply, quantity determined by
equivalent deﬁnition of equilibrium: where inverse demand crosses
inverse supply Pd .q/ D Ps .q/
a b q D.q/ D S.q/ c C d q
∴ qD ac
bCd Comparative Statics
For example, in linear model can increase or decrease the intercept of
In general, can change a parameter of the functions Pd .q/ and/or Ps .q/
and study the effect in the equilibrium price and quantity. Changing a preference parameter, for instance, affects demand because
it affects the marginal utility v 0.q/, which is the inverse demand function. Taxes
Consider quantity taxes.
If payed by consumer, then posted price is ps and consumer effectively
pays pd D ps C t
If payed by supplier, then posted price is pd and supplier effectively
receives ps D pd t equilibrium happens when
D.pd / D S.ps /
or equivalently, D.ps C t/ D S.ps / Passing along a tax
horizontal supply function
vertical supply function Deadweight loss of a tax
change in consumer’s surplus
change in producer’s surplus
deadweight loss Price
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- Spring '09