Question 2.14

\(k_{t + 1} = \frac{1}{(1 + n)(1 + g)} \frac{1}{2 + \rho} (1 - \alpha) k_t ^\alpha\)
a) A rise in population growth (n) would shift the \(k_{t + 1}\) function down, so this means a smaller \(k_{t + 1}\) for the given \(k_t\). The shift is not due to the fraction of income being saved, and the young will produce the same amount of capital, but there will be more people to spread the capital amongst. Thus capital per unit of effective labor in period t + 1 is smaller for the given \(k_t\).
b) If the parameter B added to the production function shifts down, so too will the entire function (due to constant returns to scale). Thus a lower B means a given amount of capital per unit of effective labor will produce less output per unit of effective labor. Finally, this leads to less total savings and a lower capital stock in period \(t + 1\).
c) Since alpha occurs in two places, it is easiest to determine the effect on capital by taking the partial derivative:
\(\frac{\partial k_{t + 1}}{\partial \alpha} = \frac{1}{(1 + n)(1 + g)} \frac{1}{2 + \rho} [-k_t ^\alpha + (1 - \alpha) \frac{\partial k_t ^\alpha}{\partial \alpha}]\)
Now we can define \(f(\alpha) = k_t ^\alpha\), which gives:
\(\ln f(\alpha) = \alpha \ln k_t \Longrightarrow \frac{\partial \ln (\alpha)}{\partial \alpha} = \ln k_t\)
This allows us to write:
\(\frac{\partial f(\alpha)}{\partial \alpha} = \frac{\partial f(\alpha)}{\partial \ln f(\alpha)} \frac{\partial \ln f(\alpha)}{\partial \alpha} = \frac {1}{[\frac{\partial \ln f(\alpha)}{\partial f(\alpha)}]} \frac{\partial \ln f(\alpha)}{\partial \alpha}\)
And thus finally, \(\frac{\partial f(\alpha)}{\partial \alpha} = f(\alpha) \ln k_t\).
Therefore, we have \(\frac{\partial k_t ^\alpha}{\partial \alpha} = k_t ^\alpha \ln k_t\). Substituting this into our second equation yields:
\(\frac{\partial k_{t + 1}}{\partial \alpha} = \frac{1}{(1 + n)(1 + g)} \frac{1}{2 + \rho} [k_t ^\alpha [(1 - \alpha) \ln k_t - 1]]\)
Thus we finally see that if the last argument of this equation is greater than zero, an increase in \(\alpha\) means a higher \(k_{t + 1}\) for a given capital per unit of effective labor. The \(k_{t + 1}\) will then shift up. If the last is portion, \(\ln k_t < \frac{1}{1 - \alpha}\),  then an increase will mean lower new capital per unit of effective labor. 

Question 2.16

a) Since the household's optimization problem (utility function and budget constraint) is not affected by depreciation per se, the effect of depreciation will only manifest through the real interest rate. To study this, we recall that the capital stock in \(t + 1\) equals the amount saved by the young in the previous period. Thus, \(K_{t + 1} = S_t L_t\) and \(S_t = s(r_{t + 1}) A_t w_t\). Substituting the second into the first and writing it in terms of effective labor, we have:
\(\frac{K_{t+1}}{A_{t+1} L_{t+1}} = \frac{A_t L_t}{A_{t+1} L_{t+1}} [s(r_{t + 1})w_t]\) 
Substituting in the definitions of n and g gives:
 \(\frac{K_{t+1}}{A_{t+1} L_{t+1}} = \frac{1}{(1 + n)(1 + g)} [s(r_{t + 1})w_t]\)
Finally to substitute for depreciation and wages:
\(\frac{K_{t+1}}{A_{t+1} L_{t+1}} = \frac{1}{(1 + n)(1 + g)} [s(f'(k_{t + 1} - \delta)][f(k_t - k_tf'(k_t)]\)
Comparing this to the original equation of the textbook without depreciation allows us to see that the capital stock will be affected by the way that savings varies with the real rate of interest.  
b) With logarithmic utility, the fraction of income saved doesn't depend on the interest rate but rather: \(s(r_{t+1}) = \frac{1}{2 + \rho}\)
With Cobb-Douglas and real wage we have: 
\(k_{t + 1} = [\frac{1}{1 + n(1 + g)}][\frac{1}{2 + \rho}] (1 - \alpha) k_t ^\alpha\)
The saving rate in this economy is total saving divided by total output. If we denote the economy's total savings rate as \(\hat s\), then we will have:
\(\hat s = \frac{[\frac{1}{2 + \rho}] (1 - \alpha) k_t ^\alpha}{k_t ^\alpha} = \frac{1}{2 + \rho} (1 - \alpha)\)
Thus, this is exactly the same as the function of the Solow model with depreciation at one. Solow does have some microeconomic foundations, although such an assumption about depreciation is unrealistic. 

Question 2.17

a) The utility function is given by: \(\ln C_1 + [\frac{1}{1 + \rho}] \ln C_2\)
i)  Adding the Social Security tax, the constraint changes to: 
\(C_{1, t} + S_t = A w_t - T\)
\(C_{2, t} = (1 + r_{t + 1})S_t + (1 + n)T\)
Solving for savings gives:
\(S_t = \frac{C_{2, t + 1}}{1 + r_{t + 1}} - \frac{(1 + n)}{(1 + r_{t + 1})} T\)
Substitute savings into the first equation:
\(C_{1, t} + \frac{C_{2, t + 1}}{1 + r_{t + 1}} = Aw_t -T \frac{(1 + n)}{(1 + r_{t + 1})} T\)
Finding the intertemporal budget constraint:
\(C_{1, t} + \frac{C_{2, t + 1}}{1 + r_{t + 1}} = Aw_t - \frac{(r_{t + 1}- n)}{(1 + r_{t + 1})} T\)
Since with logarithmic utility the individual will consume \(\frac{(1 + \rho)}{(2 + \rho)}\) of lifetime wealth, we have:
\(C_{1, t} = (\frac{1 + \rho}{2 + \rho}) [A w_t - (\frac{(r_{t + 1}- n)}{(1 + r_{t + 1})}) T\)
Solving for saving per person with substitution: 
\(S_t = A w_t - (\frac{1 + \rho}{2 + \rho}) [A w_t - (\frac{r_{t + 1} - n}{1 + r_{t + 1}}) T] - T\)
\(S_t = [1 - (\frac{1 + \rho}{2 + \rho})]A w_t - [1 - (\frac{1 + \rho}{2 + \rho}) (\frac{r_{t + 1} - n}{1 + r_{t + 1}})] - T\)
If we replace the last argument in the brackets with \(Z_t\), we will have:
 \(S_t = [(\frac{1}{2 + \rho})]A w_t - Z_t T\)
And we know that the capital stock in the second period is equal to saving of the young in the first, thus:
\(k_{t + 1} = [\frac{1}{1 + n}] [(\frac{1}{2 + \rho})w_t - \frac{Z_t T}{A}]\)
Substituting in what we know from Cobb-Douglas about the real wage:
\(k_{t + 1} = [\frac{1}{1 + n}] [(\frac{1}{2 + \rho})(1 - \alpha) k_t ^\alpha - \frac{Z_t T}{A}]\)
ii) To see the effect this has on the balanced growth path value of capital, we have to observe the sign of \(Z_t\). If it is positive, the introduction of the tax will shift down the k curve and reduce the balanced growth path value. We can simplify to see this:
\(Z_t = \frac{(1 + r_{t + 1}) + (1 + \rho)(1 + n)}{(2 + \rho)(1 + r_{t + 1})} > 0\)
iii) If the economy was initially dynamically efficient, a marginal increase in T would result in a gain to the old generation that would receive the extra benefits. However it would reduce capital further and leave posterity worse off. If the original level of capital was above the Golden Rule the older generation would again gain. But now, it would be welfare-improving for posterity as well.  The tax could alleviate inefficiency from the over-accumulation of capital. 
b) 
i) \(C_{2, t+ 1} = (1 + r_{t + 1})S_t + (1 + n)T\)
Since the rate of return on social security is the same as that on saving, we can derive the intertemporal budget constraint:
\(C_{1, t} + \frac{C_{2, t + 1}}{1 + r_{t + 1}} = A w_t\)
Solving this yields the usual Euler equation:
\(C_{2, t + 1} = [\frac{1}{1 + \rho}](1 + r_{t + 1}) C_{1, t}\)
Substituting this into the budget constraint and solving for saving per person we have:
\(S_t = [\frac{1}{2 + \rho}] Aw_t - T\)
Thus the social security tax cases a one-for-one reduction in private saving. Examining the capital stock:
\(K_{t + 1} = S_t L_t + T L_t\)
Converting per person and simplifying gives us: 
\(k_{t + 1} = [\frac{1}{1 + n}][\frac{1}{2 + \rho}] (1 - \alpha) k_t ^\alpha\)
Therefore, the fully-funded social security system has no effect on the relationship between capital stock in successive periods. 
ii)  Since there is no effect on the relationship between the capital stock in the successive periods, the balanced growth path value is the same as before the fully funded social security system.   Essentially, the total investment and saving is still the same, but government is doing some of it for the young.