Consider a simple life-cycle model, in which individuals live for two periods: they work in the first, saving for the second. Let C₁ denote consumption in the first period and C₂ in the second;r, the interest rate (r >0), and w, the wage. Then:

C₂= (wC₁)(1 +r). (1)

Assume savings are used to buy capital goods, which last for one year, so:

K=wC₁, (2)

where K denotes capital stock. The gross output of the economy, Q, increases with the capital stock:

Q=F(K), (3)

the interest rate is the marginal product of capital:

1 +r=F(K), (4)

and workers receive what is left over:

wL=F(K)KF(K), (5)

where L is the labor supply, which for convenience, we set equal to 1.

Individuals supply labor inelastically but decide how much to consume by maximizing U(C₁, C₂), the two-period utility function, subject to equation (1).

a) Put C₂ on the vertical axis and C₁ on the horizontal. Then plot the set of feasible combinations of C₁and C₂ using the equation (1) (and the other equations (2) to (5)which are related).

b) For each combination of C₁ and C₂ there is a value of the marginal rate of substitution(the amount of consumption while working that individuals are willing to trade-off for more consumption in retirement). In equilibrium, the marginal rate of substitution equals the interest rate.

Pick a random point on the feasibility locus and draw a line with a slope equal to 1 +r. Next, draw the indifference curve and the equilibrium point E which will be the point where the line and the indifference curve are tangent.

c) Now, explore how a pay-as-you-go Social Security system changes the equilibrium. Let T be the transfer from the younger generation to the older.

Now C₂ = (wTC₁)(1 +r) +T. Show graphically how the equilibrium may be affected (draw the new equilibrium E).

d) Explain briefly what the graph in part c) says about the difference between a pay-as-you-go and a fully funded pension plan. In practical terms, would there be a problem in switching from pay-as-you-go to a fully funded system?