Unformatted text preview: Intermediate Microeconomic Analysis
Demand, Supply and Market Equilibrium
Instructor: Bin Xie
Spring 2015 Demand Function
The quantity of a good or service that consumers demand
depends on price and other factors such as consumers' incomes
and the prices of related goods.
Demand Function: the mathematical relationship between
quantity demanded (Qd ), price (p ) and other factors that
inuence purchases (e.g. income Y )
Qd = D(p, ps , pc , Y )
p : price of the good or service
ps : price of a substitute good
pc : per unit price of a complementary
Y : consumers' income good Demand Function (Cont.) Usually (in this course) the demand function takes a linear
form. Example Qd = 210 − 20p + 20ps − 15pc + 3Y
Qd : Quantity of pizza demanded
p : price of pizza
ps : price of hamburger
pc : price of coke
Y : your income Demand Function (Cont.)
Graphically (twodimension), only the relationship between Q
and p is depicted, and other factors are held constant (ceteris
paribus assumption) Example ps = 3; pc = 2; Y = 20
Qd = 210 − 20p + 20 × 3 − 15 × 2 + 3 × 20 = 300 − 20p The relationship between Qd and p can be shown by the
movement along the demand curve.
The relationship between Qd and other factors (ps , pc , Y ) can
be illustrated by the shift of the demand curve.
How about the partial derivative of Qd with respect to p ? Supply Function The quantity of a good or service that rms supply depends on
price and other factors such as cost of inputs.
Supply Function: the mathematical relationship between
quantity supplied (Qs ), price (p ) and other factors such as the
cost of inputs
Qs = S(p, ph )
p : price of the good
ph : cost of inputs or service Supply Function (Cont.) Usually the supply function takes a linear form. Example Qs = 140 + 20p − 20ph
Qs : Quantity of pizza supplied
p : price of pizza
ph : price of cheese Supply Function (Cont.)
Graphically, we only depict the relationship between Qs and p ,
and hold other factors constant (ceteris paribus assumption) Example ph = 2
Qs = 140 + 20p − 20 × 2 = 100 + 20p The relationship between Qs and p can be shown by the
movement along the supply curve.
The relationship between Qs and other factors (ph ) can be
illustrated by the shift of the supply curve. Market Equilibrium
The interaction between consumers' demand curve and rms'
supply curve determines the market price and quantity of a
good or service that is bought and sold.
Price is the bridge connecting buyers and sellers in a market,
signalling both sides how many to buy/sell.
Price too high: excess supply will drive the price down;
Price too low: excess demand will push the price up.
Equilibrium price: demand will equal supply. Conclusion: when there is excess supply or excess demand,
natural market forces will push the quantity and price to
market equilibrium. Market Equilibrium (Cont.) Graphically, market equilibrium occurs where the demand and
supply curves intersect.
Mathematically, nd the price that equates the quantity
demanded Qd and the quantity supplied Qs .
Qd = Qs
300 − 20p = 100 + 20p
p = 5. Shocking the Equilibrium (Comparative Statics)
As a policy concern, often we need to study what will happen
to the equilibrium when there is an external shock.
Suppose that supply curve is aected by an exogenous variable
a: Qs = S(p, a)
Qd = D(p)
p could
implicit function of the suppyshifter a:
Qd = D(p(a)), Qs = S(p(a), a)
Demand curve is the same: At the market equilibrium, price be written as an Implicit function A function dened by an implicit equation (Qd = Qs ), by
associating one variable with the others. (Here are p and a.) Shocking the Equilibrium (Cont.)
In equilibrium, it is always established that
D(p(a)) = S(p(a), a) To study the eect of the shock a on the equilibrium status:
a, using the chain rule :
∂S(p(a), a) dp
∂S(p(a), a)
dD(p(a)) dp
=
+
dp
da
∂p
da
∂a Total derivative with respect to Rearranging the term yields: ∂S
dp
∂a
=
dD
∂S
da
−
dp
∂p . (What does this derivative mean?) Example Demand function D = −3p + 40. Supply function with an external
shock S = 5p − 3a.
dp
Find out the comparative statics .
da Elasticities The elasticity indicates how responsive one variable is to a
change of another variable.
For instance, the price elasticity of demand measures: how
sensitive the percentage change of the demand Qd is to
percentage changes in the price of that good P .
ε= dQ/Q
dQ P
percentage change in quantity
=
=
percentage change in price
dP/P
dP Q Elasticities (Cont.) Example Qd = 300 − 20p What is the demand elasticity?
On a linear demand curve, the demand elasticities are NOT
constant. Elasticities (Cont.)
Income elasticity of demand
εIncome = dQ/Q
percentage change in quantity demanded
=
percentage change in income
dY /Y crossprice elasticity of demand
εPrice = percentage change in quantity demanded
=
percentage change in price of another good dQ/Q
dPo /Po elasticity of supply
η= dQ/Q
percentage change in quantity supplied
=
percentage change in price
dP/P Eects of Tax Two Types of Sales Tax: Ad valorem Tax (Proportional) and
Unit Tax
Ad valorem Tax is more prevalent in reality. The eect of a sales tax on equilibrium price and quantity
depends on elasticities of demand and supply.
(Who gets hurt more?) The incidence of tax on
consumers/rms is only determined by the (demand and
supply) elasticities, no matter where the government collects
the tax from. Eects of Tax (Cont.) Suppose the price at market equilibrium before tax is p , a unit
tax τ is collected from the producers.
Producers only receive p−τ if the market price is still But the market equilibrium will change (exogenous
tax)
New equilibrium: D(p(τ )) = S(p(τ ) − τ ) Implicit Function! p. shock: Eects of Tax (Cont.)
How to solve implicit function?
Usually total derivative/dierential. In this case only chain rule
is needed.
Derivative with respect to τ of both sides:
dS d(p(τ ) − τ )
dS dp
dD dp
=
=
(
− 1)
dp dτ
dp
dτ
dp dτ
Rearrange:
dS
dS p
dp
η
dp
dp Q
=
=
=
dS
dD
dS p
dD p
dτ
η−ε
−
−
dp
dp
dp Q
dp Q Eects of Tax (Cont.)
dS
dp η
dp
τ=
τ=
τ
dS
dD
dτ
η−ε
−
dp
dp
dD
−
dp
−ε
dp
Tax incidence on rms: τ − τ =
τ
τ=
dS
dD
dτ
η−ε
−
dp
dp Tax incidence on consumers: Excercise: What if the tax is collected from consumers? (Hint:
consumers have to pay p + τ when the market price is still p ). Example Demand function D = −3p + 40. Supply function with an external
shock S = 5p . If the government collects a unit tax from producers
τ = 2, what are the tax incidences on consumers and producers? ...
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 Fall '10
 Raven
 Market Equilibrium, Supply And Demand, Qd, DP

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