The Diamond Model
2.8 The Assumptions
The difference between this model and RCK is the feature of overlapping-generations. With the turnover, it's simpler to assume time is discrete (rather than continuous). Thus all variables of the model are defined by a time period (t = 1, 2, 3...) vs. all values of t > 0.
To simplify the analysis, we assume that each individual lives for only two periods. \(L_t \) individuals are born in period t. As before, population grows at rate \(n\), thus \(L_t = (1 + n) L_t\). At time t, there are always \(L_t\) individuals in the first period of their lives and \(L_{t - 1} = \frac{L_t}{(1 + n)} \) in their second. Each individual supplies 1 unit of labor when he is young and divides the resulting labor income between first-period consumption and saving. In the second period, the individual simply consumes the saving and any interest he earns.
Let \(C_{1t}\) and \(C_{2t}\) denote the consumption of young and old individuals. Thus the utility of an individual born at t depends on \(C_{1t}\) and \(C_{2t + 1}\). Again assuming constant-relative risk-aversion utility:
\(U_t = \frac {C^{1 - \theta}_{1t}}{1 - \theta} + \frac{1}{1 + \rho} \frac{C^{1 - \theta}_{2t + 1}}{1 - \theta}\)
Assuming also that \(\theta > 0; \rho > -1\).
This is the functional form needed for balanced growth. If \(\rho > 0\), individuals place greater weight on first-period than second-period consumption and vice versa. The second assumption ensures that second-period consumption is positive.
Production is the same as before. There are many firms, each with the same production function \(Y_t = F(K_t, A_t L_t)\). Again, there are constant returns to scale, the Inada conditions are satisfied (marginal returns for capital are positive but decreasing, actually, limit of first derivative is zero as x approaches infinity but vice versa) and A grows at the exogenous rate \(g ; A_t = (1 + g)A_{t - 1}\). The real interest rate and wage per unit of effective labor are given the same way: \(r_t = f'(k_t) \) and \(w_t = f(k_t) - k_t f'(k_t)\)
In period 0, capital owned by the old and labor supplied by the young combine to produce output. Capital and labor are paid their marginal products. The old consume capital income and existing wealth then die and exit the model; the young divide their labor income, \(w_t A_t\), between consumption and saving and the savings carry forward into the next period. Thus capital stock in period t + 1, \(K_{t + 1}\) is equal to the number of individuals in the period, \(L_t\), times their savings \(w_t A_t - C_{1t}\). This capital combines with the labor supplied by the next gen. of young, and the story continues.
2.9 Household Behavior
The second-period consumption of an individual born at t is:
\(C_{2t + 1} = (1 + r_{t + 1}) (w_t A_t - C_{1t})\)
Dividing both sides by \(1 + r_{t + 1}\) gives us the individual's budget constraint:
\(C_{1t} + \frac{1}{(1 + r_{t + 1})} C_{2t + 1} = w_t A_t\)
This signifies that the present value of lifetime consumption equals the initial wealth (zero) plus present value of lifetime labor (\(w_t A_t\)).
There are two ways to solve this maximization problem: first is derivation of the Euler equation used in the RCK model, which is much easier because of discrete time. An individual will only change their present consumption if the decision raises the present value of their lifetime consumption stream. The marginal contributions of \(C_{1t}; C_{2t + 1}\) to lifetime utility are \(C_{1t} ^{- \theta}; [\frac{1}{1 + \rho}] C_{2t + 1} ^{- \theta}\). The marginal costs are obviously just the change in consumption, and the utility benefit will be this amount multiplied by the real interest rate. Thus, optimization requires: \(C_{1t} ^{- \theta} \Delta C = [\frac{1}{1 + \rho}] C_{2t + 1} ^{- \theta} (1 + r_{r + 1}) \Delta C\)
Simplifying gives us:
\(\frac{C_{2t + 1}}{C_{1t}} = [\frac{1 + r_{t + 1}}{1 + \rho}] ^{\frac{1}{\theta}}\)
This implies that whether an individual's consumption is increasing or decreasing over time depends on whether the real rate of return is greater or less than the discount rate. Again, \(\theta\) determines how much individuals' consumption varies in response to differences between \(r\) and \(\rho\).
The second way is using the Lagrangian:
\(\mathcal{L} = \frac{C_{1t}^{1 = \theta}}{1 - \theta} + \frac{1}{1 + \rho} \frac{C_{2t + 1} ^{1 - \theta}}{1 - \theta} + \lambda [A_t w_t - (C_{1t} + \frac{1}{1 + r_{t + 1}} C_{2t + 1})]\)
The first-order conditions for consumption in both periods are
\(C_{1t} ^{-\theta} = \lambda\)
\(\frac{1}{1 + \rho} C_{2t + 1}^{- \theta} = \frac{1}{1 + r_{t + 1}} \lambda\)
Substituting this gives : \(\frac{1}{1 + \rho} C_{2t + 1}^{- \theta} = \frac{1}{1 + r_{t + 1}} C_{1t} ^{- \theta}\) which eventually simplifies down to the first equation. We also can use the Lagrangian to find how the interest rate determines the fraction consumed for a period:
\(C_{1t} = [1 - s(r_{t + 1})] A_t w_t\)
This impiles that young individuals' saving is increasing in rate of return if and only if \(\theta\) is less than 1 and decreasing if greater. Intuitively, this gets at how a rise in r has both an income and substitution effect. The saving for later consumption becomes more favorable but the fact that it would yield even more second-period consumption also can decrease it. Preferences will decide which effect dominates.
2.10 The Dynamics of the Economy
The Equation of Motion of k
As described before, capital stock in period t + 1 is the amount saved by young individuals in period t:
\(K_{t + 1} = s(r_{t + 1}) L_tA_tw_t\)
Note that saving in period t depends on labor income and return on capital expected in the next period. Dividing both sides by \(L_{t + 1} A_{t + 1} \) gives the expression for capital per unit of effective labor:
\(k_{t + 1} = \frac{1}{(1 + n)(1 + g)} s(r_{t + 1})w_t\)
Substituting to put this in terms of f(k):
\(k_{t + 1} = \frac{1}{(1 + n)(1 + g)} s(f'(k_{t + 1}))[f(k_t) - k_t f'(k_t)]\)
The Evolution of k
How does \(k_{t + 1}\) depend upon \(k\)?
Logarithmic Utility and Cobb-Douglas Production
When \(\theta\) is 1, the fraction of labor income saved is \(\frac{1}{2 + \rho}\). When using the Cobb-Douglas production function, \(f(k) = k^\alpha\) and \(f'(k) = \alpha k^{\alpha -1}\). The equation for motion of k then becomes:
\(k_{t + 1} = \frac{1}{(1 + n)(1 + g)} \frac{1}{2 _ \rho} (1 - \alpha) k_t ^\alpha\)
In the graph below, the point where \(k_{t + 1} \) intersects the 45-degree line equals \(k_t = 0\). K* is globally stable, meaning that wherever k starts it will converge to k*. The properties of the economy once it has converged to the balanced growth path are the same as those of Solow and Ramsey economies: the saving rate is constant, output per worker is growing at rate g, and the capital-output ratio is constant.
Speed of Convergence
Using the equation for \(k_{t + 1}\) as a function of \(k_t\), we can solve for the values of k* and y*. The economy is on its balanced growth path when these two are equal.
\(k_{t + 1} = \frac{1}{(1 + n)(1 + g)} \frac{1}{2 _ \rho} (1 - \alpha) k_t ^\alpha\)
Solving for k* yields:
\(k* = [\frac{1 - \alpha}{(1 + n)(1 + g)(2 + \rho)}]^{\frac{1}{1 - \alpha}}\)
Since \(y = k^\alpha\):
\(y* = [\frac{1 - \alpha}{(1 + n)(1 + g)(2 + \rho)}]^{\frac{\alpha}{1 - \alpha}}\)
This expression shows how the model's parameters affect output per unit of effective labor on the balanced growth path. To find how quickly the economy converges to the balanced growth path, we linearize around the balanced growth path. This means replacing the equation of motion for k with a first order approximation around \(k = k*\):
\(k_{t + 1} \approxeq k* + (\frac{dk_{t + 1}}{dk_t} |_{k_t = k*}) (k_t - k*)\)
Let \(\lambda\) denote \(\frac{dk_{t + 1}}{dk_t}\) evaluated at \(k = k*\). Thus: \(k_t - k* \approxeq \lambda^t(k_0 - k*)\)
The convergence is determined by \(\lambda\). If \(\lambda\) is between 0 and 1, then the system converges smoothly. If it is outside that range, there are oscillations but each period brings it closer. If \(\lambda\) is greater than 1, the system explodes. (hah)
To find \(\lambda\), we can substitute in the equation for k*:
\(\lambda \equiv \frac{dk_{t + 1}}{dk_t} |_{k_t = k*} = \alpha [\frac{1 - \alpha}{(1 + n)(1 + g)(2 + \rho)}] k*^{\alpha - 1}\)
\(= \alpha [\frac{1 - \alpha}{(1 + n)(1 + g)(2 + \rho)}] [\frac{1 - \alpha}{(1 + n)(1 + g)(2 + \rho)}] ^{\frac{\alpha - 1}{1 - \alpha}} \Longrightarrow \alpha\)
Therefore, \(\lambda\) is simply \(\alpha\), capital's share. The rate of convergence differs from the Solow model because though the saving of the young is a constant fraction of their income and their income is a constant fraction of total income, the dissaving of the old is not a constant fraction of total income. Rather, it is a fraction of output: \(\frac{K_t}{F(K_t, A_tL_t)}\) or \(\frac{k_t}{f(k_t)}\).