Question

Suppose there are two individuals who share a house and garden. They have decided to hire a gardener to take care of their front and backyard, but they will each individually decide how much to contribute to paying the gardener. The gardener works for $50 an hour. They both have an income of $4,000 and they both value a tidy garden (measured by the hours G that the gardener works) and private good. Person 1's utility function is given by u(x1,G)=x1G^2 and person 2's utility function is given by u(x2,G)=x2G^2. The price of the private good x is $1. Answer the remaining questions using this information.

1. Suppose person 2 pays for 38.89 hours of gardening. How many hours would person 1 want to pay for to maximize their utility?

2. Suppose person 1 pays for 40.37 hours of gardening. How many hours would person 2 want to pay for to maximize their utility?

3. What is a Nash Equilibrium (NE)?

A. As long as two people choose the same contribution, it's a NE.

B. Each person provides half of the effective amount of G.

C. In the NE nobody has an incentive to deviate from the chosen strategy after considering an opponent's choice

4. What is the NE contribution to the public good of person 1?

5. How many units of the public good will there be in the NE?

Answer 1

Solution:

(Partial same question already solved with same information; below is the extension)

So, we are given the following: Price of public good, Pg = $50, price of private good, Px = $1

Income of person 1, M1 = income of person 2, M2 = $4,000

Let gi be the amount of hours employed by person i, where i = {1, 2}, so G = g1 + g2

Utility of person i is U(xi, G) = xi*G²

Budget constraint for individual i is Px*xi + Pg*gi <= Mi that is, xi + 50*gi <= 4000 (for both persons)

Since, the utility is increasing in G and x for both persons, optimal or utility maximizing consumption will occur on the budget line, that is budget constraint will be binding. So, at optimum, we have: xi + 50gi = 4000

Further note that the utility function is similar for both the persons, so, we can easily generalize the entire formulation of best response function and optimal individual contribution. So, rather than solving for person 1 and 2 individually, we solve for individual i, in terms of contributions by perrson j (the other person), and lastly, substitute the values as required.

Maximizing individual utility of person i:

Maximize Ui(xi, G), subject to xi + 50gi = 4000

So, we can write: xi = 4000 - 50gi (using the constraint)

Incorporating this constraint in the objective function of person i, we get:

Ui = (4000 - 50gi)*(gi + gj); where j is the other person (as already mentioned)

That is, Ui = (4000 - 50gj)*gi - 50gi² + 4000gj

Since, for any optimization problem, we got to find the first order conditions (which here will also give us the best response functions for each person). Following the same partial derivatives methodology here:

Best response for an individual i can be found by finding the first order condition: $\partial Ui/\partial gi$ = 0

$\partial Ui/\partial gi$ = 4000 - 50gj - 100gi

With $\partial Ui/\partial gi$ = 0, we have 4000 - 50gj - 100gi = 0

So, best response function for individual i becomes: gi*(gj*) = (80 - gj*)/2

Thus, best response function for person 1 becomes: g1*(g2*) = (80 - g2*)/2

And best response function for person 2 becomes: g2*(g1*) = (80 - g1*)/2

1. With g2 = 38.89 hours, using best response function of person 1, we get

g1* = (80 - 38.89)/2 = 20.56 hours

So, in order to maximize his/her utility, person 1 would want to contribute for 20.56 hours of gardening

2. With g1 = 40.37 hours, using best response function of person 2, we get

g2* = (80 - 40.37)/2 = 19.82 hours

So, in order to maximize his/her utility, person 2 would want to contribute for 19.82 hours of gardening

3. Nash equilibrium: is the situation when each individual is playing their best strategy, that is, profit maximizing strategy, given other player's strategy. So, correct answer is (C) in NE, nobody has an incentive to deviate from chosen strategy after considering an opponent's choice, as they are choosing the strategy which is anyway best for them.

4. Under NE, we simply find it where the best response curves of the people involved in game intersect (as all are doing their best, given other's strategy).

So, finding intersection of the 2 persons' best response. We already have that:

Best response function for person 1 becomes: g1* = (80 - g2*)/2 ... (a)

And best response function for person 2 becomes: g2* = (80 - g1*)/2 ....(b)

Solving (a) and (b) simultaneously: substituting (b) in (a), we get:

g1* = (80 - ((80 - g1*)/2)/2

g1* = (160 - 80 + g1*)/4

4g1* - g1* = 80

g1* = 80/3 = 26.67 hours

Thus, NE contribution to the public good, G, of person 1 is 26.67 hours.

NOTE: By substituting this value in (b), we get g2* = (80 - (80/3))/2 = 26.67 hours. So, here we find the contribution towards the public good by both individuals to be equal, but this might not always be true (so, option A for part (3) is wrong above). Here, we have both coming out to be equal as both persons have similar payoff functions.

5. Under NE, finding the total number of units of public good, G* = g1* + g2*

As found above, g1* = g2* = 26.67 hours

So, G* = 26.67 + 26.67 = 53.33 hours.