1. There are two workers in the economy, “A” and “B”; “A” can go to school and “
ID: 1249231 • Letter: 1
Question
1. There are two workers in the economy, “A” and “B”; “A” can go to school and “B” cannot. Education increases future income according to the following function: I(e)= 50(e)/(1+e)Total cost per year of schooling equals 2, and thus: mc(e)=2
a. Derive the marginal income function.
b. What is the optimal schooling level for worker A ?
Suppose that educated workers also increase the productivity of a non-educated one, according to the following function: B(e)=e (I.e. the benefit for the non educated worker for each year of schooling studied by the educated worker is 1).
c. What is the optimal schooling level for worker A from the social point of view?
d. How can you achieve it?
e. Explain the difference between your answers to b. and c.
Explanation / Answer
a. Marginal income will be the derivative of the income function I(e)= 50(e)/(1+e) Use the quotient rule. MI(e)=[50(1+e) - 50e] / [1+e]^2 MI(e)=[50+50e-50e] / [1+e]^2 MI(e)=50/[1+e]^2 b. Set MI(e) = MC(e) 50/[1+e]^2=2 25=[1+e]^2 5=1+e e=4 c. From the social point of view, we have to look at the marginal social benefit MSB, which is the sum of the marginal benefits in society. B(e)=e MB(e)=1 MSB(e)=MI(e)+MB(e) Plug in your values. MSB(e)=50/[1+e]^2 + 1 Now set MSB(e)=MC(e) 50/[1+e]^2 + 1 = 2 50/[1+e]^2 = 1 50=[1+e]^2 50^0.5=1+e e=50^0.5 - 1 e=6.07 This is more than before, which makes sense with a positive externality. d. You could achieve this level by paying A $1 for every unit of education he receives. This would make A treat the benefit to B as if it were his own benefit. Thus, we would internalize the externality. e. In part B, we calculate the optimal education level from A's perspective. That is, if A gets to choose e, the optimal level will be e=4. In part d, we calculate the optimal education level to maximize social welfare from a social planner's perspective. If a social planner wants to maximize social welfare, he will choose an education level of e=6.07
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