Question

Fairyland has two citizens, Cosmo and Pixie. Each has the same utility function, U=ln(C) . Both earn $1000 in their full-time jobs, but both face
some risk of being laid off due to a potential shortage of fairy dust, in which case they will only earn $250 in an alternate part-time job. There is a 10% chance that Cosmo will be laid off and a 30% chance Pixie will be laid off.   The Governor of Fairyland, Jorgen von Strangle, is considering providing unemployment insurance. In particular, Jorgen is considering two plans: the first would pay any worker who loses his job $100; the second would pay any worker who loses his job $600. Both plans would be financed by collecting a tax from any worker who keeps his job.
a. Under each plan, how high does Jorgen have to set the tax so that the government does not lose money on the plan?
b. For the tax from part a, compute the well-being of Cosmo and Pixie under each plan.
c. For three possible policies (the two plans and the status quo of no plan), how do Cosmo and Pixie rank the policies, respectively, in terms of their own well-being.

Answer 1

Given:

Utility Function, U = ln(C)

Amount earn by Cosmo in Full time job, S(C,FT) = $1000

Amount earn by Pixie in Full time job, S(P,FT) = $1000

Amount earn by Cosmo in Part time job, S(C,PT) = $250

Amount earn by Pixie in Part time job, S(P,PT) = $250

Probability of Cosmo being laid off, P1 = 10% => Probability of Cosmo not laid off, P3 = 1-P1 = 90%

Probability of Pixie being laid off, P2 = 30% => Probability of Pixie not laid off, P4 = 1-P2 = 70%

Retirement benefit under plan 1, R1 = $100

Retirement benefit under plan 2, R2 = $600

Part (a)

Let T1, be the tax charged by Jorgen, under Plan 1

Then,

Mean Value of Benefits Paid to Laid off = Mean Value Money Collected from Employed

==> R1*P1 + R1*P2 = T1*P4 + T1*P3

==> $100*(10%) + $100*(30%) = T1*(90% + 70%)

==> $(1000 + 3000)/100 = T1*(160/100)

==> T1 = $4000/160

==> T1 = $25

Hence, Jorgen should charge tax at the rate of $25, so that government does not incur any cost.

Let T2, be the tax charged by Jorgen, under Plan 2

Then, solving as above,

T2 = $150

Part (b)

Wellbeing = Expected Utility

Also,

Expected Utility for Cosmo under plan 1, E1 = P3*ln[S(C,FT) – T1] + P1*ln[S(C,PT) + R1]

==> E1 = 90%*ln(1000 – 25) + 10%*ln(250 + 100)

==> E1 = 0.9*6.88 + 0.1*5.86

==> E1 = 6.78

Similarly,

Expected Utility for Pixie under plan 1, E2 = 6.17

Expected Utility for Cosmo under plan 2, E3 = 6.75

Expected Utility for Pixie under plan 2, E4 = 6.75

Part (c)

In case of no insurance plan,

Expected Utility for Cosmo, E5 = P3*ln[S(C,FT)] + P1*ln[S(C,PT)]

==> E5 = 90%*ln(1000) + 10%*ln(250)

==> E5 = 0.9*6.91 + 0.1*5.52

==> E5 = 6.77

Similarly,

Expected Utility for Pixie, E6 = 6.49

Now,

Cosmo will be most happy when her utility is maximized i.e. max[E1, E3, E5]

=max[6.78, 6.75, 6.77]

= 6.78

Hence, Cosmo will be most happy under Retirement benefit under plan 1

And,

Pixie will be most happy when her utility is maximized i.e. max[E2, E4, E6]

=max[6.17, 6.75, 6.49]

= 6.75

Hence, Pixie will be most happy under Retirement benefit under plan 2