Hawassa University

Wondogenet collage of forestry and natural resources

Advanced Microeconomics (NrEp 521)

Solution to Assignment Problems

BY:

1. Teshome Sorato

2.Aster Abera(MSc/NRE/043/11)

3. Daniel W/Meskel

4. Meselu Tegenie (MSc/NRE/046/11)

Submitted to: Alemayehu Siyum (PHD)

December , 2011

Wondogenet, Ethiopia

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1. Assume the commodity X has demand function

Q=150-5Px and supply function

SS=8+10Px

a) Calculate the equilibrium quantity and priceb) Support your finding with graphical representation

Solution:

The market equilibrium quantity is reached when SS=Q=Q*

150-5Px=8+10Px

The equilibrium price is

Then the equilibrium quantity is :

Q*=8+10( )

The equilibrium quantity is

4. The n-goods Cobb-Douglass utility function is :U(X)=Ax1

α1x2α2x3

α3…xnαn

Where A>0 and

a) Derive the Marshallian utility function1

P

P⃰*******

SS

DD

QQ⃰*⃰

SSP

P⃰*******

b) Derive the indirect utility functionc) Derive the expenditure functiond) Derive the Hicksian demand function

Solution:a) Suppose the price of good xi be pi and the consumer income be M

Then the primal problem we consider here is:

Maximize U(X)=Ax1α1x2

α2x3α3…xn

αn Where A>0 and

s.t

Now setting the Lagrangian

Finding first order partial derivatives and setting up first order conditions we have

Now

Equating the partial derivatives of the Lagrangian with respect to the x1 and xi i=2, 3, 4,…., n we get the following

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Substituting these values of xi ,i=2,3,4,….,n in to the equation

we get

Substituting this value of x1 in

We get the Marshallian demand functions

The Marshallian demand function for each good xi is given by:

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b) The indirect utility function is found by substituting the Marshallian demand expressions of each good Xi in to the direct utility function

c) The expenditure function e is the dual of the utility function and is given by:

and the total expenditure for the maximized utility is

d) Derive the Hicksian demand functionHicksian demand function is the solution of the dual problem to the direct utility maximization problem set as shown below

Minimize

Subject to:

where is the fixed utility to be achieved

Now setting the Lagrangian and the first order necessary conditions we have

Solving for we get the following

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Substituting this values of xi in

We get

Then substituting this value of x1 we get the Hicksian demand function of each good

7. Duopolists produce substitute goods Q1 and Q2 they face inverse demand function

P1 = 20 + P2-Q1 P2 = 20+ P1-Q2

Each firm has constant marginal cost MC= 20 and no fixed costs and each firm is a Cournot competitor in price not in quantity. Compute the Cournot equilibriums in this market giving the equilibrium price and output of each good?Support your answer with graphSolution Given

P1 = 20 + P2-Q1

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P2 = 20+ P1-Q2

and MC=20 the direct demand functions are

Q1 = 20 + P2-p1 (demand function for firm 1)

Q2 = 20 + P1-p2 (demand function for firm 2)

Setting this partial derivative to zeroWe get

…. (1) (the reaction function of firm 1 to firm 2)

Similarly the reaction function of firm 2 to firm 1 is

……(2)

Substituting (2) in (1) we get the Cournot equilibrium prices

And

The Cournot equilibrium quantities Can be calculated by substituting Cournot prices in to the demand functions

Q1 = 20 + p2-p1

Q2 = 20+ p1-p2

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10. Explain Pareto efficient allocation in consumption and production.Answer

An economic situation is Pareto efficient if there is no way to make some group of people better off without making some other group of people worse off.

So an economic situation is Pareto efficient in consumption when each group or individual in the economy system is maximizing its consumption without undermining the consumption of other groups in the system. Similarly an economic system is Pareto efficient in production if each firm or individual is maximizing its output without making the production of other firms in the economic system worse off.

13. A consumer has A utility function U(x, y)=xy and the budget line M = Pxx + PyyA) Formulate the Lagrange for and drive the first order condition

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The first order conditions are

B) From the first order condition find expression from the demand of commodity x and y (good x and y) Solving the first order conditions simultaneously we getExpressions for the demands of good x and y as follows

C) Find out the dual form of the U(x, y)=xy function The dual form of Maximize U(x, y)=xys.tM = Pxx + Pyy IsMinimize e(x,y)= Pxx + Pyys.t.xy= where is fixed level of utility and e(x,y) is expenditure for goods x and y

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D) Calculate the expression (X and Y) from the dual function

Solution

13. Suppose Duopolists have the following demand curve; P = 30 - Q; MC1 = 0 and MC2 = 0 and total quantity: Q = Q1 + Q2.Find the reaction curve of both firmsAssume the firm collude, find the new price ‘p’ and profit at Cournot equilibrium.SolutionGiven P=30-Q, Q=Q1+Q2 and MC1=MC2=0TR1=PQ1=(30-Q1-Q2)Q1

TR2=PQ2=(30-Q1-Q2)Q2

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If the firms collude they maximize their profit at MC=MR

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P1

P2

P1

F12=reaction curve of firm1 to 2

F21=reaction curve of firm 2to 1

Collusion point

Q2

Q1

F12=reaction curve of firm1 to 2

F21=reaction curve of firm 2to 1

Collusion point