Infinite-Horizon and Overlapping-Generations Models
Both models examine the dynamics of economic aggregates as determined by decisions at the microeconomic level. Both continue to take the growth rates of labor and knowledge as given, but they derive the evolution of capital stock from the interaction of maximizing households and firms in competitive markets. Thus, saving is no longer exogenous or constant.
The Ramsey-Cass-Koopmans Model shows how competitive firms rent capital, hire labor, produce and sell output, and how a fixed number of infinitely-lived households supply labor, hold capital, consume, and save. It provides a natural benchmark case by assuming homogeneity and avoiding generational links.
The Diamond Model is the same, except it assumes the continual entry of new households into the economy. This has far-reaching consequences.
The Ramsey-Cass-Koopmans Model
2.1: Assumptions
There are a large number of identical firms utilizing the production function \(Y = F(K, AL)\), and all the inputs and outputs are subject to prices in competitive markets. A (rate of tech. growth) is given, growing exogenously at rate g. The firms are owned by households, so their profits accrue to the households.
There are a large number of identical households, growing at rate n (population growth). Each member supplies 1 unit of labor, every point in time. Initial capital holdings (assumed positive) are equal to \(\frac{K(0)}{H}\) where the initial amount of capital in the economy is divided by the number of households. There is no depreciation and the household divides its income (from labor, capital, and from firm profits) at each point in time between consumption and saving to maximize its lifetime utility:
\(U = \int _{t = 0} ^ \infty e^ {- \rho t} u (C(t)) \frac{L(t)}{H})dt\)
- C(t) is the consumption of each member of the household at a time
- u(*) is the instantaneous utility function, giving each member's utility at a time
- L(t)/H is the number of members of the household (population divided by household number)
- u(C(t))L(t)/H is the household's total instantaneous utility at time t
- p is discount rate, meaning the greater p, the less the household values future consumption relative to current consumption
The instantaneous form of the function is:
\(u(C(t)) = \frac{C(t)^{1 - \theta}}{1 - \theta}, \theta > 0, \rho - n - (1 - \theta)g > 0\)
This form is needed for the economy to converge to a balanced growth path. It is known as constant-relative-risk-aversion (CRRA) utility because the coefficient of relative risk aversion is \(\theta\) (thus independent of C). Although there is no uncertainty in the model, \(\theta\) represents the willingness to shift consumption between different periods. For example, when \(\theta\) is smaller, the marginal utility falls more slowly as consumption rises, and so the household is willing to allow consumption to vary over time. If it's close to zero, utility is almost linear in C, meaning the household is willing to accept large swings in consumption to take advantage of small differences between discount rate and the rate of return on saving. Elasticity of substitution between consumption at any two points in time is \(\frac{1}{\theta^2}\)
Finally, notice that dividing \(\frac{C^{1 - \theta}}{1 - \theta}\) ensures that marginal utility of consumption is positive regardless of the value of \(\theta\) . An often useful case to consider is that as \(\theta\) approaches 1, the instantaneous utility function simplifies to \(lnC\). Finally, the assumption that \(\rho - n - (1 - \theta)g > 0\) ensures that lifetime utility does not diverge (not infinite).
2.2: The Behavior of Households and Firms
Firms: At each point in time, they employ the stocks of labor and capital, pay them their marginal products, and sell the resulting output. Since the production function has constant returns and the economy is competitive, firms earn zero profits. Without depreciation, the real rate of return on capital equals its earnings per unit time: \(r(t) = f'(k(t))\). The real wage at time t is: \(W(t) = A(t)[f(k(t)) - k(t) f'(k(t))]\). Thus, wage per unit of effective labor is: \(w(t) = f(k(t)) - k(t) f'(k(t))\)
Households' Budget Constraint: Takes paths of r (real rate of return on capital) and w (wages) as given. The budget constraint is that the present value of its lifetime consumption cannot exceed its initial wealth plus the present value of its lifetime labor income. Formally, we need to account for the fact that r may vary over time: \(R(t) = \int_{\tau = 0}^t r(\tau)d \tau\), meaning that the value of 1 unit of output at time t in terms of output at time 0 is \(e^{-R(t)}\). It's labor income at time t is \(\frac{W(t)L(t)}{H}\), likewise, expenditures are\(\frac{C(t)L(t)}{H}\). Mathematically, the budget constraint is therefore:
\(\int_{t = 0} ^ \infty e^{-R(t)} C(t) \frac{L(t)}{H} dt \leq \frac{K(0)}{H} + \int_{t = 0} ^\infty e^{-R(t)}W(t) \frac{L(t)}{H} dt\)
We can write this integral from t = 0 to t = infinity as a limit, so it's equal to:
\(\lim_{s \rightarrow \infty} [\frac{K(0)}{H} + \int_{t = 0} ^s e^{-R(t)} [W(t) - C(t)]\frac{L(t)}{H}dt] \geq 0\)
We also know that the household's capital holdings at time s are:
\(\frac{K(s)}{H} = e^{R(t)} \frac{K(0)}{H} + \int_{t=0}^s e^{R(s) - R(t)}[W(t) - C(t)]\frac{L(t)}{H}dt\)
It is helpful to remember that \(e^{R(s) - R(t)}\) shows how the value of saving changes from t to s. Simply, the budget constraint can be written as \(\lim_{s \rightarrow \infty}e^{-R(s)}\frac{K(s)}{H} \geq 0\), meaning that the present value of the household's asset holdings cannot be negative in the limit. This precludes a Ponzi game, where households could issue debt and roll it over forever.
Household Maximization Problem: The household wants to maximize its lifetime utility subject to its budget constraint. Like Solow, it's easier to relate each variable in respect to the quantity of effective labor, thus expressing both formulas in terms of consumption and labor income per unity of effective labor. Starting with the objective function, define c(t) as consumption per unit of effective labor, thus consumption per worker equals A(t)c(t).
The household's instantaneous utility is therefore:
\(\frac{C(t)^{1 - \theta}}{1 - \theta} = A(0)^{1 - \theta} e^{(1 - \theta)gt} \frac{c(t)^{1 - \theta}}{1 - \theta}\)
Finally, using \(B \equiv \frac{A(0)^{1 - \theta}L(0)}{H} ; \beta \equiv \rho - n - (1 - \theta)g\), we have the objective function:
\(B \int ^ \infty _{t = 0} e^{-\beta t} \frac{c(t)^{1 - \theta}}{1 - \theta} dt\)
The household's total consumption at t, \(\frac{C(t)L(t)}{H}\) equals consumption per unit of effective labor, c(t), times the household's quantity of effective labor, \(\frac{A(t)L(t)}{H}\). Similarly, total labor income at t equals the wages per unity of effective labor, w(t), times \(\frac{A(t)L(t)}{H}\).
The household's budget constraint will be (where household consumption is less than or equal to the original capital holdings plus the total labor income of the household):
\(\int ^\infty _ {t=0} e^{-R(t)}c(t) \frac{A(t)L(t)}{H}dt \leq k(0) \frac{A(0)L(0)}{H} + \int ^\infty _{t=0} e^{-R(t)} w(t) \frac{A(t) L(t)}{H} dt\)
Knowing that \(A(t)L(t) = A(0)L(0)e^{(n + g)t}\) and \(K(s) \) is proportional to \(k(s)e^{(n + g)s}\), the final form:
\(\int ^\infty _ {t=0} e^{-R(t)}c(t) e^{(n + g)t} dt \leq k(0) + \int ^\infty _{t=0} e^{-R(t)} w(t) e^{(n + g)t} dt\)
The no-Ponzi-game form of the budget constraint is:
\(lim_{s \rightarrow \infty} e^{-R(s)} e^{(n + g)s} k(s) \geq 0\)
Household Behavior: The household's problem is to choose the path of c(t) to maximize lifetime utility, subject to the budget constraint. Using the objective function and budget constraint to set up the Lagrangian:
\(L = B \int ^\infty _{t = 0} e^{- \beta t} \frac{c(t)^{1 - \theta}}{1 - \theta}dt + \lambda [k(0) + \int ^ \infty _{t = 0} e^{-R(t)} e^{(n + g)t} w(t) dt - \int ^ \infty _{t = 0} e^{-R(t)} e^{(n + g)t} c(t) dt]\)
Thus, the first-order condition for an individual c(t) is:
\(Be^{- \beta t} c(t) ^{-\theta} = \lambda e^{-R(t)} e^{(n + g)t}\)
\(\ln B - \beta t - \theta \ln c(t) = ln \lambda - R(t) + (n + g)t\)
Using the definition of R(t) as \(\int ^t _ {\tau = 0} r(\tau) d \tau\)
\(= \ln \lambda \int ^t _ {\tau = 0} r(\tau) d \tau + (n + g)t\)
Now the derivatives must be equal with respect to t (using the fact that the time derivative of the log of a variable equals its growth rate): \(- \beta - \theta \frac{\dot c(t)}{c(t)} = -r(t) + (n + g)\). Now, solving for \(\frac{\dot c(t)}{c(t)}\) and using the definition of \(\beta = \rho - n - (1 - \theta)g\), we have the Euler equation:
\(\frac{\dot c(t)}{c(t)} = \frac{r(t) - \rho - \theta g}{\theta}\)
To interpret, remember that the growth rate of C(t), which is consumption per worker, equals the growth rate of c(t) and A(t), the consumption per unit of effective labor times the amount of effectiveness:
\(\frac{\dot C(t)}{C(t)} = \frac{\dot A(t)}{A(t)} + \frac{\dot c(t)}{c(t)} \Longrightarrow g + \frac{r(t) - \rho - \theta g}{\theta} = \frac{r(t) - \rho}{\theta}\)
In words, this means that consumption per worker is rising if the real return exceeds the rate at which the household discounts future consumption, and vice versa. Since \(\theta\) stands for the marginal utility of changing consumption, if it is smaller (marginal utility changes less with the new consumption) then the responses to differences between the real interest rate and the discount rate will be larger. The Euler equation also describes how c behaves over time (that the initial c(0) should be chosen to eventually exhaust the lifetime wealth).
2.3: The Dynamics of the Economy
The most convenient way to describe the behavior of the economy is in terms of the evolution of c and k.
The Dynamics of c
Since the discount rate is equal to the rate that k changes (derivative of output), \(r(t) = f'(k(t))\), we know that \(\dot c = 0 \) when \(f'(k) = \rho + \theta g\). We denote this as k*.
The Dynamics of k
As in the Solow model, the change in k is equal to the actual investment minus the break-even investment. Since actual investment is output minus consumption, we have: \(\dot k(t) = f(k(t)) - c(t) - (n + g)k(t)\). In other words, \(\dot k = 0\) when consumption equals the difference between actual output and break-even investment. When c exceeds the level that yields \(\dot k = 0\) , k is falling; when c is less than this level, k is rising.
The Phase Diagram
K* will always be to the left of the peak of the \(\dot k = 0\) curve because \(k* \equiv f'(k*) = \rho + \theta g\) while the golden-rule level of : \(\equiv f(k_{GR}) = n + g\).
The Initial Values of c
The graph below shows how c and k must evolve over time to satisfy households; intertemporal optimization condition.
- Point A: If c(0) is above the \(\dot k = 0\) curve
- Point B: If c(0) is such that \(\dot k = 0\) but c grows without growth in k
- Point C: If the economy begins below \(\dot k = 0\), then k and c grow positively, however, k becomes negative when c increases past the golden-rule level
- Point D: If consumption is initially very low, c and k will both eventually diminish
- Point E and F: where the economy converges to the stable point of \(\dot k = 0 = \dot c = 0\)
The Saddle Path
For any positive initial level of k, there is a unique initial level of c that is consistent with the households' intertemporal optimization, the dynamics of the capital stock, the budget constraint, and the requirement that k not be negative. The function giving this initial c as a function of k is known as the saddle path.
2.4: Welfare
The first welfare theorem from micro states that if markets are competitive and complete and there are no externalities, then the decentralized equilibrium is Pareto-efficient-- meaning that it is impossible to make anyone better off without making someone worse off (zero-sum game). Since these conditions are met, we assume all households have the highest possible utility through the allocations.
2.5: The Balanced Growth Path
Once the economy has converged to Point E, its behavior is identical to the Solow economy on the balanced growth path; capital, output, and consumption per unit of effective labor are constant. Since the saving rate per unit of effective labor is output per unit of effective labor minus consumption per unit of eff. lab. divided by output per unit of eff. lab., \(\frac{(y - c)}{y}\), the saving rate is also constant. The total capital stock, total output, and total consumption grow at rate \(n + g\). Capital per worker, output per worker, and consumption per worker grow at rate \(g\). Thus, growth in the effectiveness of labor remains the only source of persistent growth in output per worker.
The Social Optimum and the Golden-Rule Level of Capital
The only notable difference between Solow and RCK models is that the balanced growth path with a capital stock above the golden-rule level is not possible in the RCP model. Since saving is derived from the behavior of households whose utility depends on their consumption, it cannot be an equilibrium for the economy to follow a path where higher consumption can be attained in every period, since if the economy were on such a path, households would reduce their saving and take advantage of this opportunity.
2.6: The Effects of a Fall in the Discount Rate
Suppose there is a fall in \(\rho\), the discount rate. Since \(\rho\) is the parameter governing the households' preferences between current and future consumption, this is analogous to a rise in the saving rate in Solow.
Qualitative Effects
Since the evolution of k is determined by technology rather than preferences, \(\rho\) enters the equation for \(\dot c\) but not for \(\dot k\), thus only the \(\dot c = 0\) line is changed (Recall that \(\frac{\dot c(t)}{c(t)} = \frac{r(t) - \rho - \theta g}{\theta}\)). The new value of k where \(\dot c = 0\) is defined by \(f'(k*) = \rho + \theta g\) , meaning that a fall in \(\rho\) raises k*. At the instant of the change, c jumps down so the economy is on the new saddle path, thereafter, c and k rise gradually to new balanced-growth path values (higher than original).
Rate of Adjustment and the Slope of the Saddle Path
A fruitful way to analyze quantitative implications for the dynamics of the economy is by replacing the nonlinear equations with linear approximations, taking the first-order Taylor approximations where k = k* and c = c*:
\(\dot c \simeq \frac{\partial \dot c}{\partial k} [k - k*] + \frac{\partial \dot c}{\partial c} [c - c*]\)
\(\dot k \simeq \frac{\partial \dot k}{\partial k} [k - k*] + \frac{\partial \dot k}{\partial c} [c - c*]\)
Recall that \(\dot c = {\frac{[f'(k) - \rho - \theta g]}{\theta}c}\) and define \(c^\sim = c - c* ; k^\sim = k - k*\). Thus, by taking derivatives and evaluating at k* and c*:
\(\dot c^\sim = \frac{f"(k*)c*}{\theta}k^\sim\)
Finally, dividing both sides by \(c^\sim\) and \(k^\sim\) yields the growth rates of \(c^\sim\) and \(k^\sim\):
\(\frac{\dot c^\sim}{c^\sim} \simeq \frac{f"(k*)c*}{\theta} \frac{k^\sim}{c^\sim}\)
\(\frac{\dot k^\sim}{k^\sim} \simeq \beta - \frac{c^\sim}{k^\sim}\)
The solution to this is a quadratic equation where \(\mu = {\frac{\beta \pm \frac{\beta^2 - 4f"(k*)c*}{\theta}}{2}}^{1/2}\)
Therefore, c and k converge to their balanced-growth-path values at rate \(\mu\).
2.7: The Effects of Government Purchases
In the United States, about 20% of total output is purchased by the government (higher in other countries).
Adding Government to the Model
Assume that government buys output at rate G(t) per unit of effective labor per unit time. The purchases are financed by lump-sum taxes; thus the government always runs a balanced budget and no utility is presumed lost. In Ch. 11, we will see that the government's choice between tax and deficit finance has no impact on any important variables. But now, investment is the difference between output and the sum of private consumption and government purchases. Thus the equation for the motion of k becomes:
\(\dot k(t) = f(k(t)) - c(t) - G(t) - (n + g)k(t)\)
A higher value of G will shift the \(\dot k = 0 \) locus down since the more goods purchased by government, the fewer can be purchased privately if k is held constant. This is expressed as a budget constraint, since part of wages must go to the lump-sum tax:
\(\int ^\infty _ {t=0} e^{-R(t)}c(t) e^{(n + g)t} dt \leq k(0) + \int ^\infty _{t=0} e^{-R(t)} [w(t) - G(t)] e^{(n + g)t} dt\)
The Effects of Permanent and Temporary Changes in Government Purchases
Simply, the \(\dot k = 0\) path will shift down by the amount of the increase in G. Since government purchases do not affect the Euler equation, the \(\dot c = 0 \) equation is unaffected. C will fall; essentially, a permanent increase in G purchases will reduce a household's lifetime wealth and they will adjust accordingly. A temporary increase in government purchases means that consumption will adjust over time to the new levels as shown in these graphs:
Empirical Application: Wars and Real Interest Rates
This analysis suggests that temporarily high government purchases cause real interest rates to rise , but permanently high purchases do not. Intuitively, when G purchases are high only temporarily, households expect their consumption to be greater in the future than in the present-- to accept this, the real interest rate must be high. Natural examples of temporary increases in G purchases can be seen in wartime. However, in the data this conclusion does not always bear out.