Question from ISI 2015 Sample Question Paper

[soumen08]

Q. Suppose the borders of a state, B, coincide with the circumference of a circle of
radius r > 0, and its population is distributed uniformly within its borders (so that the
proportion of the population living within some region of B is simply the proportion
of the state's total land mass contained in that region), with total population
normalized to 1.For any resident of B, the cost of travelling a distance  d is  kd , with  k >
0.Every resident of B is endowed with an income of 10, and is willing to spend up to
this amount to consume one unit of a good, G, which is imported from outside the
state at zero transport cost. The Finance Minister of B has imposed an entry tax at the
rate 100 t % on shipments of G brought into B. Thus, a unit of G costs  p(1+t ) inside
the borders of B, but can be purchased for just  p outside;  p(1+t ) < 10.Individual
residents of B have to decide whether to travel beyond its borders to consume the
good or to purchase it inside the state. Individuals can travel anywhere to shop and
consume, but have to return to their place of origin afterwards.
(a) Find the proportion of the population of B which will evade the entry tax by
shopping outside the state. [ 5 marks ]
(b) Find the social welfare-maximizing tax rate. Also find the necessary and
sufficient conditions for it to be identical to the revenue-maximizing tax rate.
[ 5 marks ]
(c) Assume that the revenue-maximizing tax rate is initially positive. Find the
elasticity of tax revenue with respect to the external price of G, supposing that
the Finance Minister always chooses the revenue-maximizing tax rate.
[ 10 marks ]

Now, I think I get the first part. One just has to solve

p + 2*k*d = p(1 + t)

for d and then solve for the proportion.

The second part has me floored though. What is a welfare maximizing tax? I though all taxes hurt welfare! If the government were not to tax the good, then no one will have to travel at all, and isn't that the best for society?

[Rajat]

My approach below is very crude as I have not yet covered the chapter on welfare economics (to which i believe this questions belongs).

1) Proportion : d/r[2 - d/r]    where d = pt/2k
2) Sorry. not sure about this part. how to quantify Social welfare ? let alone maximize it.
3)Revenue from all entities in the economy : R = NPt + NPG  
where N is the number of people who buy the good locally, G is the number of goods bought by one person.
Npt is the revenue earned by Government (hence called Rt)
NPG is the revenue earned by sellers of the good.

Also, we have P(1+t)G=10.

Substituting these values in the revenue formula and maximizing it, we get revenue maximizing tax rate
t = (10/p)^0.5  - 1

Elasticity of tax revenue wrt external price of G = p/Rt * dRt/dp
Rt = NPt
=>p/Rt = 1/Nt
and
=> dRt/dp = Nt
Thus elasticity is 1 !

Is this correct ?

[soumen08]

It is the second part which is complex. That is what we need an answer to. Thanks for replying though!

[Rajat]

yes. What about the third part ? have you also got the same answer

[L]

For social welfare maximizing tax rate, consumer surplus should be maximized. I am assuming the price of G to be less than 10, otherwise nobody would buy it.
Initial CS = 10-P
After tax CS = 10-[ P(1+t)(1-pt/2rk)^2 + P(pt/2rk (2-pt/2rk)) +TC]

where TC is travelling which you need to calculate by integrating from d to r,
and, respective Prices are multiplied with the proportion of population.
Then, differentiate w.r.t t and you will get welfare maximizing tax rate.
I am getting t=2rk/p, which seems weird because at this tax, d=r, so, everybody is buying from outside. I don't think the approach is wrong, maybe I have done calculation mistake. Please somebody else confirm.

Revenue maximizing is easy. t = 2rk/3p. At this tax, r=d/3.
I don't know about the necessary and sufficient conditions for revenue maximizing rate equal to welfare one.

And Revenue at revenue maximizing rate is independent of P, so elasticity should be zero.
 

[soumen08]

Whichever way you go, the tax would need to be zero for optimizing social welfare. The main reason people are travelling at all(which is a pure waste in this scenario, since there is no utility associated with it) is because there is a tax. If you dont tax at all, everyone gets the good for p, there is no money wasted on travel, and everyone is happiest.

[Anakin Skywalker]

CONTENTS DELETED

The author has deleted this message.

[Anakin Skywalker]

CONTENTS DELETED

The author has deleted this message.

[L]

I am considering that if the person goes to the border to buy G, he will come back to his place. So, your first equation will become

(r-x)2k = pt

and therefore, the difference of factor 2 in your and my solution.

[Anakin Skywalker]

CONTENTS DELETED

The author has deleted this message.