a) A rise in n.
\(k_{t+1} = \frac{1}{(1+n)(1+g)}\frac{1}{2+\rho}[k^\alpha-k\alpha k^{\alpha -1}]\)
\(k_{t+1} = \frac{1}{(1+n)(1+g)}\frac{1}{2+\rho}(1-\alpha)k^\alpha\)
\(\frac{\partial k_{t+1}}{\partial n} = -\frac{1}{(1+n)^2}\frac{1}{(1+g)}\frac{1}{2+\rho}(1-\alpha)k^\alpha\)
Since \(\rho > -1 \) and \( g > -1 ; \frac{\partial k_{t+1}}{\partial n} < 0\)
A rise in n leads to a downward shift in \(k_{t+1} \). the fraction of their labor income that the young
save does not depend on n. However, a growth o fpopulation leads capital to be shared among a greated number of individuals.
b) A downward shift of the production function.
\(\frac{\partial k_{t+1}}{\partial \beta} = \frac{1}{(1+n)(1+g)}\frac{1}{2+\rho}(1-\alpha)\beta k^{\alpha}\)
Since both capital and alpha are positive and since alpha is inferiror to 1, \(\frac{\partial k_{t+1}}{\partial \beta} > 0\)
This means that a downward shift in the production function -i.e. a fall in \(\beta\), will lead to a fall in \(k_{t+1} \). Intuitively, a downward shift of the production function means that less will be produced with the same quantity of capital. Since the marginal productivity of labor is equal to \((1-\alpha)\beta k^{\alpha}\) and since the quantity saved depends on the wage (which is equal to the marginal productivity of labor), a fall in beta affects negatively capital accumulation.
c)A rise in \(\alpha\)
\(k_{t+1} = \frac{1}{(1+n)(1+g)}\frac{1}{2+\rho}[k^\alpha-k\alpha k^{\alpha -1}]\)
\(\frac{\partial k_{t+1}}{\partial\alpha}=\frac{1}{(1+n)^{ }}\frac{1}{(1+g)}\frac{1}{2+\rho}[\ln(k)k^{\alpha}-k^{\alpha}\alpha\ln(k)k^{\alpha}]\)
\(\frac{\partial k_{t+1}}{\partial\alpha}=\frac{1}{(1+n)^{ }}\frac{1}{(1+g)}\frac{1}{2+\rho}[(1-\alpha)\ln(k)-1]k^{\alpha}\)
\(\frac{\partial k_{t+1}}{\partial \alpha} > 0\) only when \(ln(k) > \frac{1}{1-\alpha}\) or when \(k > e^{\frac{1}{1-\alpha}}\)
The effect of alpha on \(k_{t+1} \) depend on the level of k.
2.16) Depreciation in the Diamond model and microeconomic foundations for the Solow model.
a) How, if at all, does this change the model affect equation (2.59) giving \(k_{t+1} \) as a function of \(k_t\)?
The utility function is equal to:
\(U_t = \frac{C_{1t}^{1-\theta}}{1-\theta} + \frac{1}{1+\rho} \frac{C_{2t+1}^{1-\theta}}{1-\theta}\)
subject to the following budget constraint:
\(C_{1t} + \frac{1}{1-r_{t+1}}C_{2t+1} = A_tw_t\)
Since there is depreciation, \(r = f'(k) - \delta\), our results from the Lagrangian do not change and we get:
\(s(r)= \frac{(1+r)^{\frac{1-\theta}{\theta}}}{(1+\rho)^{\frac{1}{\theta}}+(1+r)^{\frac{(1-\theta)}{\theta}}}\)
Hence for equation (2.59), we get:
\(k_{t+1} = \frac{1}{(1+n)(1+g)}s(f'(k) -\delta)[f(k)-kf'(k)]\)
The effect of depreciation depends on how savings vary with the interest rate. In other words, it depends on the respective magnitudes of the income and substitution effects. If \(\theta < 1\), then savings will increase with the interest rate. In this case, the depreciation rate has a negative impact on savings. Symetrically, if \(\theta > 1\), then savings will decrease with an increase in the interest rate and the depreciation rate will have a positive impact on savings.
b) In the special case of log utility, Cobb-Douglas production, and \(\delta = 1\), what is the equation for \(k_{t+1}\) as a function of \(k_t\)? Compare this with the analogous expression for the discrete-time version of the Solow model with \(\delta = 1\) from part (a) of Problem 2.15.
Let's consider a discrete version of the Solow model:
\(K_{t+1} = K_t + sY_t - \delta K_t\)
\(\frac{K_{t+1}}{A_{t+1}L_{t+1}} = \frac{K_t + sY_t - \delta K_t}{A_{t+1}L_{t+1}}\)
\(k_{t+1} = \frac{k_t + sy_t - \delta k_t}{(1+g)(1+n)}\)
\(k_{t+1} = \frac{1 - \delta}{(1+g)(1+n)}k_t + \frac{s}{(1+g)(1+n)}f(k_t)\)
Since \(\delta = 1\), we have:
\(k_{t+1} = \frac{s}{(1+g)(1+n)}f(k_t)\)
Considering the Diamond model with log utility and a Cobb-Douglas production function, the equation for \(k_{t+1}\) as a function of \(k_t\) is the following:
\(k_{t+1} = \frac{1}{(1+n)(1+g)}\frac{1}{2+\rho}(1-\alpha)k^\alpha\)
s is equal to the total savings rate. \(\frac{1}{2+\rho}\) is equal to the fraction of labor income saved by the young. It does not take into account the disvaving of the old. Since there is a 100% rate of depreciation, the old do not disave. This the total saving rate equals to:
\(s = \frac{\frac{1}{2+\rho}(1-\alpha)k^\alpha}{k^\alpha} = \frac{1-\alpha}{2+\rho}\)
\(k_{t+1} = \frac{1}{(1+n)(1+g)}\frac{1-\alpha}{2+\rho}k^\alpha\)
where \(s = \frac{1-\alpha}{2+\rho}\) and \(f(k) = k^\alpha\)
Hence, we can argue that the discrete time version of the Solow model has some micro foundations although a rate of depreciation of 100% is very irrealistic.
2.17. Social security in the Diamond model. Consider a Diamond economy where g is zero, production is Cobb-Douglas, and utility is logarithmic.
a) Pay-as-you-go social security. Suppose the government taxes each young individual an amont T and uses the proceess to pay benefits to the old individuals; thus each old person receives (1+n)T.
(i) How, if at all, does this change affect equation (2.60) giving \(k_{t+1}\) as a function of \(k_t\).
\(U_t = ln(C_{1t}) + \frac{1}{1+\rho}ln(C_{2t})\)
\(C_{1t} = Aw_t - S_t -T\)
\(C_{2t} = S_t(1+r_{t+1}) + (1+n)T\)
\(S_t = \frac{C_{2t+1}}{1+r_{t+1}} - \frac{1+n}{1+r_{t+1}}T\)
Hence:
\(\frac{C_{2t+1}}{1+r_{t+1}} - \frac{1+n}{1+r_{t+1}}T = Aw_t - T - C_{1t}\)
\(C_{1t} + \frac{C_{2t+1}}{1+r_{t+1}} = Aw_t + (\frac{1+n}{1+r_{t+1}} - 1)T\)
\(C_{1t} + \frac{C_{2t+1}}{1+r_{t+1}} = Aw_t - \frac{r_{t+1} - n}{1+r_{t+1}}T\)
Setting the Lagrangian:
\(\mathcal{L} = ln(C_{1t}) + \frac{1}{1+\rho}ln(C_{2t+1}) - \lambda[Aw_t - \frac{r_{t+1} - n}{1+r_{t+1}}T - C_{1t} - \frac{C_{2t+1}}{1+r_{t+1}}]\)
FOC:
\(\frac{\partial\mathcal{L}}{\partial C_{1t}} = 0 = \frac{1}{C_{1t}} + \lambda\)
\( \lambda = -\frac{1}{C_{1t}}\)
\(\frac{\partial\mathcal{L}}{\partial C_{2t+1}} = 0 = \frac{1}{(1+\rho)C_{2t+1}} + \lambda\frac{1}{1+r_{t+1}}\)