\(\lambda = -\frac{1+r_{t+1}}{(1+\rho)C_{2t+1}}\)
\(-\frac{1}{C_{1t}} = \lambda = -\frac{1+r_{t+1}}{(1+\rho)C_{2t+1}}\)
\(C_{1t} = \frac{1+\rho}{1+r_{t+1}}C_{2t+1}\)
\(C_{2t+1} = \frac{1+r_{t+1}}{1+\rho}C_{1t}\)
Since the budget constraint is equal to:
\(C_{1t} + \frac{C_{2t+1}}{1+r_{t+1}} = Aw_t - \frac{r_{t+1} - n}{1+r_{t+1}}T\)
\(C_{1t}(1 + \frac{1}{1+\rho}) = Aw_t - \frac{r_{t+1} - n}{1+r_{t+1}}T\)
\(C_{1t} = \frac{1+\rho}{2+\rho}[A_tw_t - \frac{r_{t+1} - n}{1+r_{t+1}}T]\)
Since \(S_t = Aw_t - C_{1t} -T\) we have:
\(S_{t} = A_{wt} - \frac{1+\rho}{2+\rho}[A_tw_t - \frac{r_{t+1} - n}{1+r_{t+1}}T] - T\)
\(S_{t} = (1- \frac{1+\rho}{2+\rho})A_tw_t - (1-\frac{(1+\rho)(r-n)}{(2+\rho)(1+r)})T\)
\(S_{t} = (\frac{1}{2+\rho})A_tw_t - (\frac{(2+\rho)(1+r) - (1+\rho)(r-n)}{(2+\rho)(1+r)})T\)
To simplify, we specify that:
\(Z_t = \frac{(2+\rho)(1+r) - (1+\rho)(r-n)}{(2+\rho)(1+r)}\)
Since there is no depreciation:
\(K_{t+1} = S_tL_t\)
\(K_{t+1} = [(\frac{1}{2+\rho})A_tw_t - Z_tT]L_t\)
To get the intensive form, and since g=0, we have to divide both sides by \(A_{t+1}L_{t+1} = L_tA_t(1+n)\)
\(k_{t+1} = \frac{1}{1+n}[(\frac{1}{2+\rho})w_t - \frac{Z_tT}{A_t}]\)
Since \(f(x) = k^\alpha\) and \(w_t = f(k)-kf'(k) = (1-\alpha)k^\alpha\), we have:
\(k_{t+1} = \frac{1}{1+n}[(\frac{1}{2+\rho})(1-\alpha)k^\alpha - \frac{Z_tT}{A_t}]\)
(ii) How, if at all, does this change affect the balanced growth path value of k?
To know what is the effect of social security on the balanced growth path, we need to determine what is the sign of Z(t)
\(Z_t = \frac{(2+\rho)(1+r) - (1+\rho)(r-n)}{(2+\rho)(1+r)}\)
\(Z_t = \frac{(1+r) + (1+\rho)[(1+r)-(r-n)]}{(2+\rho)(1+r)}\)
\(Z_t = \frac{(1+r) + (1+\rho)(1+n)}{(2+\rho)(1+r)} > 0\)
Hence the \(k_{t+1}\) curve is shifting down and the balanced groth path level of capital is lower.
(iii ) If the economy is initially on a balanced growth path that is dynamically efficient, how does a marginal increase in T affect the welfare of
current and future generations? What happens if the initial balanced
growth path is dynamically inefficient?
If the economy is dynamically efficient, then a marginal increase in T will benefit the old generation in the short run at the detriment of the young generation. But it will also reduce k* further below its golden rule level and will therefore reduce the consumption and utility of future generations.
If the economy is dynamically inefficient with k* being superior than its golden rule level, then a marginal increase in T will be welfare-improving. By reducing k*, and increase in T not only benefit the old generation which receives benefits but also benefits the future generations who can enjoy higher levels of consumption. The introduction of the tax in this case would
reduce or possibly eliminate the dynamic inefficiency caused by the over-accumulation of capital.
(b) Fully funded social security. Suppose the government taxes each young person an amount T and uses the proceeds to purchase capital. Individuals
born at t therefore receive (1 + r(t+1))T when they are old.
(i ) How, if at all, does this change affect equation (2.60) giving kt+1 as a
function of kt?
\(C_{1t} = A_tw_t - S_t -T\)
\(C_{2t} = S_t(1+r_{t+1}) + (1+r_{t+1})T\)
\(S_t = \frac{C_{2t+1}}{1+r_{t+1}} - \frac{1+r_{t+1}}{1+r_{t+1}}T\)
\(C_{1t} + \frac{C_{2t+1}}{1+r_{t+1}} - T = A_tw_t - T\)