Download Aggregate Demand and Supply Exercise 1

MICROECONOMICS II
FIRST PART: Aggregate Demand and Supply
Exercise 1:
Consider the following demand functions:
si
0
y1 ( p ) = 
20 − 2 p si
p > 10

p ≤ 10 
y2(p)=3/p
y3(p)=1/pα
a) Find the price elasticity (ε1 (p), ε2 (p), ε3 (p)) of each of the three functions. What is
the value of the elasticities when p = 6? Explain the meaning of these numbers.
b) Obtain the aggregate market demand, y(p), assuming that there are 4
consumers with the first type of consumer demand function and 3 with the second, and
no consumer with the third.
c) Find the price elasticity of aggregate demand in question b) and evaluate it at p=
6. Interpret its value.
Exercise 2:
The demand for a consumer (A) of good X is equal to xA (p) = max {200-p, 0}.The
demand for another consumer (B) of good X is equal to xB (p) = max {90-4p, 0}.
a) What is the price elasticity of demand for each of the goods?
b) At what price is the price elasticity of A unitary? And at what price is the price
elasticity of B unitary?
c) Draw the demand curves of consumers A and B and the aggregate demand curve for
good X.
d) Find a price other than zero (0) at which there is a nonlinear and positive aggregate
demand for good X. What is the aggregate demand for prices below the point where the
slope changes? What is the aggregate demand for higher prices?
e) At what point of the aggregate demand curve is the price elasticity unitary? At what
price will the income from the sale of good X be maximized?
f) If the purpose of the sellers of good X is to maximize their income, to what
consumer/s will they sell the good?
Exercise 3:
The typical demand function of match tickets of FC Barcelona is q (p) = 200.00010.000p. The goal of managers is to maximize their income. The total capacity of Camp
Nou is 100.000 viewers.
a) Write the inverse demand function.
b) Write expressions for the Total Income (I) and the Marginal Income (Img) as
functions of the total number of tickets sold.
c) What price will generate the most income? What amount of tickets will be sold at this
price?
d) Selling the amount of the previous section, what is marginal income? What is the
price elasticity of demand for tickets? Is the stadium full?
Due to the current good luck of the team, the demand for tickets has increased and has
become q (p) = 300.000-10.000p.
e) What is the new inverse demand function?
f) Write an expression for marginal income as a function of the number of entries.
g) Ignoring stadium capacity, what price will generate the most income? How many
tickets would be sold at this price?
h) Taking now into account the capacity of the stadium, how many tickets will go
on sale to maximize revenue? At what price?
i) When the tickets are sold at the price of the previous section, what would be the
marginal income from selling an extra ticket? What is the elasticity of demand for
tickets to this combination of price and quantity?
Exercise 4:
Josep has quasi-linear preferences and loves sweets. Its inverse demand function
for sweets is p (x) = 49 - 6x where x is the number of sweets he eats. He currently
consumes 8 sweets at the price of 1 € per sweet. If the price of candy goes off at 7 € per
sweet, calculate the change in consumer surplus (Josep’s).
Exercise 5:
The bicycle industry in Barcelona is composed of 100 companies with the same cost
curve c (y) = 2 + (y2 / 2) and 80 companies with the same cost curve c (y) = y2/ 6. What
is the supply curve for this industry?
Exercise 6:
In a perfectly competitive market there are two types of firms. Type 1 firms have total
costs represented analytically by the function C (q) = 2q³-2q²+6q and type 2 firms by
the function C (q) = 8q³-4q²+2q. Determine the aggregate supply of market generated
by 8 companies of type 1 and 10 of type 2.
ADDITIONAL EXERCISES
Exercise 7
Assume C(y) = y3 – 6y2 + 18y is the long term cost function of a competitive firm.
Market demand is given by D(p) = Max{147 - 3p, 0}.
a) Find the long term supply function of this firm qi LP (p). Take into account corner
solutions (q might be 0). Draw it.
b) Assume all existent and potential firms are equal. Find the equilibrium of this
market in the long term (price, number of firms, total production and each firm’s
production).
Exercise 8:
Assume C(y) = y3 – 8y2 + 22y is the long term cost function of a competitive firm.
Market demand is given by D(p) = Max{ 118 - 3p, 0}.
a) Find the long term supply function of this firm qi LP (p). Take into account corner
solutions (q might be 0). Draw it.
b) Assume all existent and potential firms are equal. Find the equilibrium of this
market in the long term (price, number of firms, total production and each firm’s
production).
Exercise 9:
Assume C(y) = y3 – 4y2 + 18y is the long term cost function of a competitive firm.
Market demand is given by D(p) = Max{142 - 2p, 0}.
a) Find the long term supply function of this firm qi LP (p). Take into account corner
solutions (q might be 0). Draw it.
b) Assume all existent and potential firms are equal. Find the equilibrium of this
market in the long term (price, number of firms, total production and each firm’s
production).
Exercise 10:
Assume C(y) = y3 – 10y2 + 35y is the long term cost function of a competitive firm.
Market demand is given by D(p) = Max{130 - 3p, 0}.
a) Find the long term supply function of this firm qi LP (p). Take into account corner
solutions (q might be 0). Draw it.
b) Assume all existent and potential firms are equal. Find the equilibrium of this
market in the long term (price, number of firms, total production and each firm’s
production).
SOLUTION TO ADDITIONAL EXERCISES
Exercise 7
min Cme: y = 3, CMe=9
min CMg: y = 2, CMg=6
CMe = Cmg: y=0; y=3
Equilibrio: p=9, n=40, y=3, Y=120
Exercise 8
min Cme: y = 4, CMe=6
min CMg: y = 2,67, CMg=0,67
CMe = Cmg: y=0; y=4
Equilibrio: p=6, n=25, y=4, Y=100
Exercise 9
min Cme: y = 2, CMe=14
min CMg: y = 1,33, CMg=12,67
CMe = Cmg: y=0; y=2
Equilibrio: p=14, n=50, y=2, Y=100
Exercise 10
min Cme: y = 5, CMe=10
min CMg: y = 3,33, CMg=1,67
CMe = Cmg: y=0; y=5
Equilibrio: p=10, n=20, y=5, Y=130