In: Economics
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. What is the effective amount of G? Use 2 decimals.
2. What is the effective amount of the private x? Use 2 decimals.
3. Suppose person 2 pays for 30 hours of gardening. How many hours would person 1 want to pay for to maximize their utility?
4. Suppose person 1 pays for 43.33 hours of gardening. How many hours would person 2 want to pay for to maximize their utility?
Solution:
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
Further budget constraint for individual i is Px*xi + Pg*gi <= Mi that is, xi + 50*gi <= 4000 (for both persons)
Utility of person i is U(xi, G) = xi*G2
1. Effective amount of G is the efficient amount of G that say, a social planner would have wanted to be employed. Then, the total budget constraint in society = sum of both persons' Budget constraint
= (x1 + 50*g1) + (x2 + 50*g2) <= 4000 + 4000
= x1 + x2 + 50G <= 8000. Since, the utility of both persons is increasing in G, the maximization of it will occur where the consumption occurs on the budget line. So, at optimum, x1 + x2 + 50G = 8000
X = 8000 - 50G, X is total amount of private good
Further, the efficiency condition requires: sum of all individuals' marginal rate of substitution (MRS) = marginal rate of transformation (= Pg/Px)
MRSg,x = Marginal utility from G (MUg)/marginal utility from X (MUx)
Since, both persons have similar utility functions, for both of them:
MUg = = 2*xi*G
MUx = = G2
So, MRSg,x = (2*xi*G)/(G2) = 2*xi/G
Then, we have MRS of person 1 + MRS of person 2 = 50/1
2*x1/G + 2*x2/G = 50
x1 + x2 = 50*G/2
OR, X = 25*G
Using the above efficiency condition and the budget line, we have
25G = 8000 - 50G
So, G = 8000/75 = 106.67 hours
2. Effective amount of private good then is X = 25*(8000/75) = 2,666.67 units
Now, we answer questions 3 and 4 after creating a base for them ass follows:
Above we solved for what number of hours are socially efficient. Now, we find what number of hours would a person wish to employ. For this, we maximize not the total utility but only individual utility of person.
So, we maximize Ui(xi, G), subject to xi + 50gi = 4000 (budget with equality since optimization will make consumption on the budget line and not below it (binding budget constraint), due to reasoning same as mentioned above in part (1)).
Substituting for xi in utility function, we have maximization objective function to be: Ui = (4000 - 50gi)*(gi + gj); where j is the other person
We will find the best response function for each person, to see how one person's contribution takes place accordingly what other person is contributing towards.
So, Ui = (4000 - 50gj)*gi - 50gi2 + 4000gj
So, best response for an individual i can be found by finding the first order condition: = 0
= 4000 - 50gj - 100gi
With = 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
3. So, if person 2 pays for 30 hours of gardening, that is, if g2 = 30, using best response function of person 1, we can say
g1* = (80 - 30)/2 = 50/2 = 25 hours.
So, person 1 shall maximize his/her utility by contributing for 25 hours of gardening.
4. Now, in order to find the utility maximizing contribution by person 2, we use person 2's best response function. Given that g1 = 43.33 hours, we can say
g2 = (80 - 43.33)/2 = 18.335 hours