QUESTION
SCHOOL OF ECONOMICS
ECONOMICS 2101
MICROECONOMICS 2
S1 2012
ASSIGNMENT 2
1 2 3 4 Total
This assignment is to be handed in no later than Monday 23 April at 16:00 at the School
of Economics Assignment Box #2 (West Lobby, ASB ground floor)
Important: Please add (and sign) the cover sheet to your assignment. Not doing so will
result in a penalty of 5 points. The cover sheet can be downloaded from:
http://www.asb.unsw.edu.au/schools/economics/Documents/Economics%20Assignment%20Cover%20Sheet.pdf
Exercise 1 [8 points]: Assume that you have an income of $m and that your preferences for
good 1 and good 2 are represented by the utility function
(
)
(
and
stand for the quantities of good 1 and 2 that you consume. Assume that
is the price of
good 1 and that
is the price of good 2;
is equal to 6.
a) Determine your optimal bundle.
b) What utility level will you reach with such a bundle?
c) Suppose m = $9 and that the price of good 1 is $3. What is your optimal bundle?
d) What would be your optimal bundle if the price of good 1 rose to $5?
e) What does an “equivalent variation” mean?
f) Determine the equivalent variation corresponding to the price rise in d).
g) What does a “compensating variation” mean?
h) Determine the compensating variation corresponding to the price rise in d).
i) Are these two variations equal? Why?
1
), where
Exercise 2 [7 points]: The demand for some commodity is given by the function
(
)
and that the supply for this commodity is given by the function
(
)
.
a) Determine the market equilibrium price and quantity for this commodity.
b) The government decides to levy a $4 tax per unit on consumers. Write an equation that
relates the price paid by demanders to the price received by suppliers. Write an equation
that states that supply equals demand. What is the equilibrium price per unit paid by
consumers? What is the equilibrium price received by suppliers?
c) With the $4 tax, how many units will be traded on the market?
d) What is the government’s revenue from imposing this tax?
e) How much of this tax revenue is paid by consumers? And how much is paid by suppliers?
Exercise 3 [7 points]:
Suppose that you are the owner of some asset which is worth $200000 and that the value of this
asset may fall to $40000 because of some financial crash that is expected to occur with a
probability 10%. To reduce the loss of value you are told that you can buy an insurance at a
price of 1$ per $10 dollars worth of insurance. This asset represents all your wealth and the
utility you get from consuming it is
(
)
√
, where c stands for your wealth.
a) You are considering to buy an insurance such that if the financial crash occurs, you would
receive $K back from the insurance company. Write down an equation that states the
amount of wealth that you would have for a given K if the financial crash does not occur.
Refer to this amount as
.
b) Write down an equation that states the amount of wealth that you would have for a given K
if the financial crash does occur. Refer to this amount as
.
c) Write down an equation that relates the amounts of wealth found in a) and b).
d) Given your preferences, the probability of a crash and the price of insurance, how would
you determine the optimal amount worth of insurance, and what is it equal to?
Exercise 4 [8 points]:
Alice (A) and bob (B) each have an endowment
of two goods, say good 1 and good 2.
Thus
(
) and
(
). The prices
and
of the goods are determined
by competitive markets. Alice would like to consume the quantities (
) whereas Bob
would like to consume (
). Their preferences are given by
*
+ and
*
+
a) Write down Alice’s and Bob’s budget constraints.
b) Compute the Marginal Rate of Substitution for Alice and Bob.
c) State the optimality constraints.
d) Determine the optimal consumption bundles.
e) Compute the prices. [Hint: Write the excess demands, set p1 =1 and use the excess
demands to compute prices]
f) Explain.
SOLUTION
Solution 1:
The utility function U( x1, x2) = x1 + 2ln x2
The budget constraint function is given by the following equation
m= p1 x1+6 x2
a) The condition of optimization is:
MU x1= ∂U/∂x1
MU x2=∂U/∂x2
U = x1 + 2 ln x2
= 1
Price of good 1 is given as p1 and price of good 2 is given as $6
The condition of optimization can be rewritten as
Solving it for x2
x2= (1)
putting the value of x2 in the budget constraint we get:
m=
solving the equation for x1 we get:
x1= (m- 2p1) / p1 (2)
The values of x1 and x2 above reflects the optimal bundle.
b) The utility level is calculated by putting the value of x1 or x2 in the utility function
=
c) The optimal bundle when m= $9 and p1= $3
putting above values in (2) ,we get
x1= (9- 2*3)/3 = 1 unit
putting values in (1) above we get
x2= 3/3 =1 unit
d) when p1 = $ 5
x1= ( 9- 2*5)/ 5
= (9-10)/ 5 ; this comes out to be negative therefore when p1 increases to $5 , the consumer does not demand good 1
x2= 5 / 3
optimal bundle: (0,5/3)
e) Equivalent variation is a measure of how much more money a consumer would pay before a price increase to avert the price increase.
g) Compensating variation refers to the amount of additional money an agent would need to reach its initial utility after a change in prices, or a change in product quality, or the introduction of new products
Answer 2:
a) D(p)= 1789 – 160 p
S (p) = 189 + 240 p
Market equilibrium condition: D= S
1789- 160 p = 189+ 240p
1600= 400p
p(equilibrium)= $4
q(equilibrium)= 1149 units ( by putting the value of p in the supply function)
b) tax= $4 per unit
equation that relates price paid by demanders to the price received by the suppliers :
pd – ps = 4
equation that states that demand equals supply:
D(pd) = S( ps)
D(ps + t) = S(ps ) or
D(pd ) = S(pd – t)
1789- 160 (p+4) = 189+ 240 p
Solving it for p , we get
p=$ 2.4 (price received by the producer)
pd= 2.4+4
= $6.4( price paid by the consumer)
c) equilibrium quantity= 189 + 240 (2.4)
= 765 units
d) tax revenue= 4* 765
= $ 3060
e) amount of tax that the supplier pays per unit = 4- 2.4
= $1.6
Total tax paid by suppliers= 1.6 * 765 =$ 1224
Amount of tax paid by consumers= 6.4 -4= $2.4
Total tax paid by consumer = 2.4 * 765 =$ 1836
Answer 3:
m= $200,000 (wealth)
= 10% (probability of occurrence)
L = 160,000 (damage)
I can buy the insurance at a price of 1$ per $10 dollars worth of insurance which means I get $10 by paying $1 which implies in order to recover $1 from the insurance company I need to pay $1/10 as premium
= $1/10 ( $1 of insurance can be bought at a price of )
u(c ) =
a) Cna= m-
=200,000- K/10
b) Ca = m – L – K + K
= m – L + (1- )K.
= 40,000 + (1- 0.1) K
= 40,000+ 0.9 K
c) Cna = m – [/(1- )][Ca – (m-L)]
= Ca
C na = Ca
Cna =Ca
d) utility function u(c ) = √c
expected utility of the insurance having two sets of bundles ie (Ca, Cna) is given by
EU(C) = a Ca1/2 + (1 – a) Cna1/2
condition of optimality :
max EU(C) = a Ca1/2 + (1 – a) Cna1/2
= 0.10 ( 40000 + 0.9 K )1/2 + 0.9 (200,000 -0.1 K)1/2
subject to budget constraint
Cna =Ca
Now putting (first order derivative = 0 for maximizing a function)
0.10*1/2 ( 40000+ 0.9 K) -1/2 *0.9 + 0.9 * ½ (200000- 0.1K)-1/2 * (-0.1) = 0
0.045 (40000+ 0.9 K) -1/2 – 0.045 (200000- 0.1K)-1/2 = 0
Squaring both the sides and solving for K
K= 200000- 40000 = $ 160,000
Answer 4:
a)The utility function of Alice :
UA = min [ C1A, C2A/ 2]
Budget Constraint :
WA = p1C1A + p2 C2A/2
The utility function of Bob:
UB = min [ C1B, C2B/ 2]
Budget Constraint :
WB= p1C1B + p2 C2B/2
b) MRS= MU1/ MU 2
Alice: MU1A= ∂UA/∂C1A
=1
MU2A = ∂UA/∂C2A
= ½ = 0.5
MRS= 1/ 0.5 =2
Bob: MU1B= ∂UB/∂C1B
=1
MU2B = ∂UB/∂C2B
= ½ = 0.5
MRS= 1/ 0.5 =2
c) optimality conditions :
MRS = p1/ p2 and
p1/p2 > MRS
max U (C1, C2)
subject to
W>= p1C1 + p2 C2/2
d) Alice:
condition of optimality:
MRS= p1/ p2
2= p1/ p2
p1=2p2
y=(a/b) x
where x is the consumption of good x
y is the consumption of good y
a is the coefficient of good x
b is the coefficient of good y
C2A = 1/ 0.5 C1A
Putting the values in budget constraint
W = 2 p2 C1A + 2 C1A p2 /2
W=3 p2 C1A
C1A = W/ 3p2
C2A = 2 (W / 3p2)
Similarly for Bob the optimal consumption bundles would be same as their utility functions are same
C1B= W/ 3p2
C2B= 2 (W / 3p2)
e) excess demand:
z1(p1,p2) = C1A + C1B ‐ (W1 A+ W1B)
= (W1 A+ W1B)
= (p1W1A + p2 W2A/2)/ 3p2 + ( p1W1B + p2 W2B/2) / 3p2 – (W1 A+ W1B)
Putting p1=1 and equating the above equation to 0
(W1A + 0.5p2 W2A)/ 3p2 + ( W1B + 0.5p2 W2B) / 3p2 – (W1 A+ W1B) =0
Solving it for p2
(0.5 W2A + 0.5 W2B -3W1A -3W1B) p2 = – W1A– W1B
p2 =
= W1A+W1B/ (3W1A +3W1B -0.5 W2A – 0.5 W2B)
Similarly for good 2 we can compute the prices in the same manner
excess demand:
z2(p1,p2) = C2A + C2B ‐ (W2 A+ W2B)
= (W1 A+ W1B)
= 2(p1W1A + p2 W2A/2) / 3p2 + 2( p1W1B + p2 W2B/2)/ 3p2 (W1 A+ W1B)
Putting p1=1 and equating the above equation to 0
2(W1A + p2 W2A/2) / 3p2 + 2( W1B + p2 W2B/2)/ 3p2 (W1 A+ W1B) = 0
(2W1A + p2 W2A) / 3p2 + (2 W1B + p2 W2B)/ 3p2 (W1 A+ W1B) = 0
Solving the equation for p2
p2 = -( 2W1A +2 W1B) / (W1 A+ W1B -W2A – W2B)
f) The excess demand function is given by the difference between the total demand and total supply
Z1(p1,p2) = C1A + C1B ‐ (W1 A+ W1B)
The term highlighted in blue depicts the total demand of good 1 and the term highlighted in yellow depicts the total supply of good 1.
Price of good 1 is exogenous and the demand function is equated to zero after putting the values of C1A,
C 1B , W1A, W1B in the equation the respective price is computed.
JF72
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