Assignment II

ECO 354: Public Economics

Department of Economics

Shiv Nadar University

Spring 2021

Note: This is a take home assignment on public goods. The purpose is to have

you work it through yourself as the problems are the likes of what can be expected in

your exams. However, working together and peer learning is encouraged. Academic

dishonesty (copying someone else’s solutions) will be taken very seriously. All the

best.

Question 1: Private provision of public good

Consider two agents with dierent amount of wealth wi where w1 6= w2, but with

identical Cobb-Douglas utility function given by: ui(xi;G) = Gx1

i .

(i) Find the contribution functions that constitute a Nash equilibrium.

(ii) How big does the wealth dierence between agent 1 and 2 have to be for

agent 2 to contribute zero in equilibrium?

Now consider a variation of the situation where there are n agents with dierent

initial wealth and same identical utility function given by: ui(xi;G) = Gx1

i . The

wealth distribution, however, is dierent: there is a total amount of w wealth that

is distributed unequally between k n of those agents.

(iii) How much of the public good is provided in this situation?

(iv) How does the amount of provision changes as k changes between [1; n]?

Question 2: Neutrality, and private provision of public good (30

points)

Consider two individuals contributing towards private provision of a public good,

G. Individual i has an initial endowment of wi, which she divides into her private

consumption xi and contribution to the public good, gi. Naturally, total provision

of public good is g1 + g2 = G. The utility function of individual i is given by:

ui(xi;G) = 2 ln xi + lnG.

(a) Formulate the decision making problem of an individual and nd the equation

that determines the strategic behaviour of an individual. (5 points)

(b) Solve for the Nash equilibrium. How much an individual contribute in that

equilibrium, and a s result, how much public good is provided in a market

equilibrium? (5 points)

1

2

(c) Solve for the socially optimal level of public good provision and compare that

with the Nash equilibrium market provision? Do you nd a under provision of

public goods in market equilibrium? Brie

y explain your ndings. (5 points)

Now assume that a lump-sum part of the initial endowment is mandated to con-

tribute towards the public provision. Let us suppose that individual i is mandated

to contribute i < wi, i 2 f1; 2g, towards the contribution. (d) Construct the individual problem with mandatory contributions, carefully pointing out the departures from the previous set-up, and solve for individual contributions in Nash equilibrium. Does the voluntary contributions change in this case compared to when there were NO mandated contributions? (10 points) (e) How much is total (mandated and voluntary) provision of public good in a Nash equilibrium? Does the total provision increase or decrease with the mandatory contribution policy? Can you provide some insights behind your ndings? (5 points) Instructor: Subhra K. Bhattacharya Date: March 11, 2021. Order Now

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