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Economics 2

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Question I
for optimal level of provision of public goods, the efficiency of the economic output level will occur when:
Marginal Cost=Marginal Social Benefit
Since the three consumers exist in the same market, then
MSB=(50-G)+(110-G)+(150-G)
=310-3G
MC=190
=>190=310-3G
3G=120
G=40
Why public goods may not be supplied at all because of the free-rider problem
A situation whereby an individual believes that someone else will provide the goods for them is termed as a free-rider problem .for any price lower than 310, the demand for the public good is positive. Despite this, no consumer is ready and willing to pay a price above 150. To be able to meet a price of 190 then means that two or more consumers must be willing to contribute towards the public goods. The free riders will consume the goods without paying anything in anticipation that other consumers will pay. It, therefore, becomes difficult to raise the funds to supply the public with goods hence this leads to underproduction of the good and no supply at all.
the benefit that arises from the provision of the public is given by the triangle xyz where the height xy is the difference between 310 and 190. If the public good is not supplied the benefit will be lost and therefore becomes the deadweight loss.
The area of the triangle xyz=1/2(310-190)*(40) =4800.
Question 3
MSB=30-x1G+50-x2G+20-x3G
MSB=100-G(x1+x2+x3)
The marginal cost for public good=10
Marginal rate of substitution=1
MRS1+MRS2+MRS3=1
But
MRS1=MRS2=MRS3=MRT
The quantity og consumption of public goods
UGUx=xhSumming the three quantities
UGUx1+UGUx2+UGUx3= MRS1+MRS2+MRS3
For the three individuals
G=G1+G2+G3 in the contributing individual
Utility function i=XiG
The budget constraint Xi+Gi=100
Individual I choses X1G
U=3log(100-G1)+log(G1+G2+G3)
1100-G1=1G1+G2+G33(G1+G2+G3)=100-G1
4G1=100-3(G2+G3)
G1=25-34G2+G3G2=25-34G1+G3G3=25-34G1+G2G= G1+G2+G3=75-3(G1+G2+G3)/4
2G=300/30
G=5
Question 5
G=1-1μH2i=1Hj=1Hmin⁡[MiMj]Mean level income is given by
μ=1Hi=1HMhi=1HMh=2+4+6+8+10+12+14+16+18+20
= 110µ=1/10(110)
=11
Gini index is therefore given as
G=1-1μH22H-2M1+2H-4M2+…+M10G=1-1100*11(36+64+84+96+100+96+84+64+36+40+20=0.

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34
MEDE, U(M)=lnM
For a utilitarian social welfare
HU(MEDE)=h=1hU(Mh)For the logarithmic utility function,
the income distribution,
10 log(MEDE) = log (2) + log (4) + log (6) + log (8) + log (10) + log (12) + log (14) + log (16) + log (18) + log (20),
Therefore, 10 log(M1 EDE) = 22.036,
Thus, M1 EDE = e 2.2036 = 9.0576
Atkinson inequality measure is defined by
A=1-MEDE/µ
A=1-9.0576/11
=0.17658
Rawlsian social welfare function,
The value of MEDE is equal to minimum income level for the Rawlsian social welfare. Therefore, MEDE=2 and
A=1-2/11
=0.81818
In Rawlsian social welfare function, the index of inequality obtained has a value closer to the maximum value that is 1 as compared to that obtained in utilitarian.
Question 6
Sen Poverty measure
Ps=P0(1-(1-Gp)μpzP0=head count, Gp=gini coefficient of inequality, μp=is the mean income of the poor and z= the poverty line, 6
For the first distribution, P0=6/14
μp=1Hi=1HMhi=1HMh=148μp=114*148=10.57
Gini index is therefore given as
Gp=1-1μpH22H-2M1+2H-4M2+…+M14Gp=1-110.57*142(1264)Gp=0.39The sen poverty measure is therefore given as
Ps=614(1-1-0.3910.57 6=0.2449
For the second distribution
μp=1Hi=1HMhi=1HMh=134μp=114*134=9.57
Gini index is therefore given as
Gp=1-1μpH22H-2M1+2H-4M2+…+M14Gp=1-110.57*142(1194)Gp=0.39The sen poverty measure is therefore given as
Ps=614(1-1-0.369.57 6=0.21088

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