How to OLG

Question 1

How to OLG (normal case with PAYG-system later on)

Answer 1

1. Marginal products of capital and labour
   - Note here if the economy is dynamic efficient. It is efficient if r>n+delta, the rate of return of capital is higher than the rate on social security. It is implicit assumed that the rate on social return is given by population, n, and delta is depreciation. In many cases n is zero, so you just have to find out if r is positive, then the economy is dynamic efficient.
2. Write down the individuals budget constraints and find the Euler: Note we now have two periods households.
   - Normally individuals budget constraints: c_1t + s_t = w_t and c_2t+1 = Rs_t. Find the agregate constraint, by combining the two equations with the gross rate (R).
   - Tak the Lagrangian and FOC wrt. c_1t and c_2t+1
   - Then you can write up the Euler.
   - comment on the Euler. The Euler shows is the optimal consumption path is through a households life, where consumption today equals depreciated consumption and interest tomorrow.
3. To find the optimal saving:
   - substitute the budget constraint into the Euler.
   To find the optimal capital acumulation:
   - We assume that only young people save, thus capital comes only from the young savings.
   - K_t+1/N_t+1=N_t+s_t/N_t+1=s_t/1+n=s_t, because the assignment assumed that n is zero.
   - Thus when no population growth capital accumulation in the economy is equal to savings.
4. Introduced a PAYG system and check if it is supported by the old:
   - write up the budget constraints: c_1t + s_t = (1-tax)w_t and c_2t+1 = Rs_t + tax*w_t. Find the agregate constraint, by combining the two equations with the gross rate (R).
   - Tak the Lagrangian and FOC wrt. c_1t and c_2t+1
   - Then you can write up the Euler.
   - comment on the Euler. The Euler shows is the optimal consumption path is through a households life, where consumption today equals depreciated consumption and interest tomorrow.

Maybe all of this is not necessary with Lagrangian and Euler, because the tax is not timed on consumption. Therefore we could just write up the two new budget constraints and plug them into the optimal savings and capital accumulation.

Comment on the difference from without the PAYG: When we introduce a tax we will get lower savings and capital accumulation. The old people in period t_0 will support the PAYG system, due to the current young transfer a contribution directly to the current old. The current old will get a higher consumption, and interest benefit of that the they newer had to pay a contribution.

Sidenote: In contrast the current young will diasprove the new system, due to the assumed rate of interest is higher then the rate of return of social security, when r>n. They would be better off if they could invest the contribution in the private market. If then n was growing and higher then r, the young would approve the new system.

Question 2

How to OLG (PAYG system, with migration)

The government makes a higher tax, to cover the greater number of people.

Answer 2

Same procedure up until optimal capital accumulation. the higher amount of people.

1. The new tax, how is it affected by the immigration, higher n.
   - The tax is greater than before, and the wagerate is decreased, due to higher competition on the labourmarket. The wage and then consumption will fall for the current young generation combined with higher contributions to the old.
2. Show that the disposable income of the young is lower than before immigration.
   - Argument from before.
3. Show that the current old is strictly better off. They get the same contributions as before, but they get higher interest of their capital, because when the workforce grow to and the capital is fixed it would make it more productive, until a certain amount of immigration. To many people and capital will suffer from demising marginal returns, the Pablo Pizza shop Paradox

Question 3

How to argument for dynamic efficiency

Answer 3

1. Be aware of the information in the assignment
2. Note here if the economy is dynamic efficient. It is efficient if r>n+delta, the rate of return of capital is higher than the rate on social security. It is implicit assumed that the rate on social return is given by population, n, and delta is depreciation. In many cases n is zero, so you just have to find out if r is positive, then the economy is dynamic efficient.
3. example : In contrast the current young will diasprove the new system, due to the assumed rate of interest is higher then the rate of return of social security, when r>n. They would be better off if they could invest the contribution in the private market. If then n was growing and higher then r, the young would approve the new system.
4. PAYG sytem can make the economy dynamic efficient, because it lowers capital, which will increase the interest , because we assume the same demand to a lower amount of capital, which will increase r. Maybe then making the economy efficient.

Question 4

Are the current old individuals better or worse off in OLG (PAYG system and fully funded system, with migration).

No extra tax from the government, just migration of young and no fixed amount of capital.

Answer 4

Under PAYG:

Assuming all newcomers are employed and will be paying part of their salary as a social contribution, Current old people would be better off, because they receive a higher amount of benefits.

Under fully funded:

Because you have done the calculations with PAYG, you have to do the same with the fully funded system. The same as normally

budgets constraints look like this:

c_1t + s_t =(1-tax)w

c_2t+1 = R(s_t + tax*w)

In this case, the current old would be worse off, as there are no benefits for them in the fully funded system, where you pay your own benefits.