Modeling the Growth of the World Economy: The Basic Overlapping Generations Model

Abstract

This chapter is devoted to a simple modeling of the growth of the world economy. To this end a log-linear, CD version of Diamond’s (American Economic Review, 55, 1126–1150, 1965) neoclassical growth model is used as basic overlapping generations (OLG) growth model. This is then extended across several further dimensions in the following chapters. As a simple but complete intertemporal general equilibrium model it comprises the optimization problems of agents (younger and older households as well as firms) and the market clearing conditions for each model period. The fundamental equation of motion of the efficiency-weighted capital intensity is derived from the first-order conditions for household’s utility and firm’s profit maximization and from labor and capital market clearing conditions. The “golden rule” of capital accumulation ensures that consumption per efficiency capita is maximized over the long run.

Appendix

1.1 Constrained Optimization

All agents in this chapter aim at optimizing their decisions to reach their goals in the best possible way. However, they are all confronted with various restrictions (constraints) – in some cases they are of a natural or technological nature, in other cases choices are limited due to available income. How can one find the optimum decision in the face of such constraints? The method of mathematical (classical) programming provides a solution. In order to formalize the decision problem we first need to define the following: What are the objectives of the different actors? Which variables are to be included in agent decision making? Which restrictions do they face?

The objectives of the agents can be formalized by use of the objective function, Z. This function assigns a real number to every decision (consisting of a list of n decision variables) made by an agent.

$$ Z:{\Re^n}\to {\Re^1} $$

(2.49)

We introduced two objective functions in the main text of this chapter: one for households whose goal is to act in such a way that their preferences, represented by a utility function, are met best, and one for firms that try to maximize their profit function.

Concerning the second and third question we know that households can determine consumption quantities and the distribution of consumption over time. We also know that producers can determine the demand for labor as well as for capital. These variables are referred to as decision (choice) variables or instrumental variables. The quantities households can consume depend, among other things, on their income. The production cost of a specific quantity of a good depends, among other things, on the technology used in the production process. Such restrictions are represented in the form of constraints.

Mathematically speaking, the decision problem is to find values for the instrumental variables which maximize the value of the objective function (profit, utility) or minimize it (cost), subject to all constraints. Formally, the optimization problem can be written as one of the following three programs:

$$ \mathrm{ Max} Z(x)\mathrm{ s}.\mathrm{ t}.:g(x)=b,\left( {\mathrm{ classical}\ \mathrm{ optimization}} \right) $$

(2.50a)

$$ \mathrm{ Max} Z(x)\mathrm{ s}.\mathrm{ t}.:g(x)\leq b,x\geq 0,\left( {\mathrm{ non}-\mathrm{ linear}\ \mathrm{ optimization}} \right) $$

(2.50b)

$$ \mathrm{ Max} Z(x)=cx\mathrm{ s}.\mathrm{ t}.:Ax\leq b,x\geq 0.\left( {\mathrm{ linear}\ \mathrm{ optimization}} \right) $$

(2.50c)

The objective function Z is a function of n variables, i.e. x is a vector of dimension n (n decision variables). The function $g(x)$ denotes m constraints; b is a column vector of dimension m.

We now turn to classical optimization and try to find a rule which allows us to unveil the optimal decision making of agents. An example of the household objective function $U(x)$ is given by Eq. 2.12; the (only) constraint $g(x)$ by Eq. 2.15.

$$ \operatorname{Max} {U_t}\left( {c_t^1,c_{t+1}^2} \right)=\ln c_t^1+\beta \ln c_{t+1}^2 $$

(2.51a)

subject to (s.t.):

$$ c_t^1+\frac{{c_{t+1}^2}}{{1+{i_{t+1 }}}}={w_t} $$

(2.51b)

The two instrumental variables in this optimization problem are $c_t^1\ \mathrm{ and}\ c_{t+1}^2$, and are the (only) variables households can determine. Due to the constraint, future consumption can (under certain conditions) be written as a function of current consumption.

$$ c_{t+1}^2=(1+{i_{t+1 }})({w_t}-c_t^1) $$

(2.52a)

Or, more generally:

$$ c_{t+1}^2=h(c_t^1), $$

(2.52b)

$$ \frac{dh }{{dc_t^1}}=-\frac{{\partial g/\partial c_t^1}}{{\partial g/\partial c_{t+1}^2}}. $$

(2.52c)

The objective function can also be formulated as a function $\tilde{\varLambda}$ of a single decision variable:

$$ \tilde{\varLambda}=\ln\,\,c_t^1+\beta \ln\,\,[(1+{i_{t+1 }})({w_t}-c_t^1)]\,. $$

(2.53a)

Or, more generally:

$$ \tilde{\varLambda}=\tilde{\varLambda}(c_t^1,h(c_t^1)). $$

(2.53b)

This intermediate step simplifies the search for a value $c$ of the decision variable $c_t^1$ that maximizes the objective function $\tilde{\varLambda}$ (utility) in our decision problem. Obviously, at a maximum, the following condition has to hold:

$$ \tilde{\varLambda}(c)\geq \tilde{\varLambda}(c+\varDelta c). $$

(2.54)

If we make use of Taylor’s theorem, we can find the maximum of the (modified) objective function Eq. 2.53b. The first-order condition (FOC) of the problem is:

$$ \frac{{d\tilde{\varLambda}}}{{dc_t^1}}=0=\frac{{\partial U}}{{\partial c_t^1}}+\frac{{\partial U}}{{\partial h}}\frac{dh }{{dc_t^1}}. $$

(2.55)

On taking account of Eq. 2.52b, then Eq. 2.55 is equivalent to:

$$ \frac{{d\tilde{\varLambda}}}{{dc_t^1}}=0=\frac{{\partial U}}{{\partial c_t^1}}+\frac{{\partial U}}{{\partial h}}\left[ {-\frac{{\partial g/\partial c_t^1}}{{\partial g/\partial c_{t+1}^2}}} \right]=\frac{{\partial U}}{{\partial c_t^1}}+\left[ {-\frac{{\partial U/\partial h}}{{\partial g/\partial c_{t+1}^2}}} \right]\frac{{\partial g}}{{\partial c_t^1}}. $$

(2.56)

We denote the expression in brackets on the right-hand side by , so that the maximization problem (2.56) can be written more simply as:

$$ \frac{{d\tilde{\varLambda}}}{{dc_t^1}}=0=\frac{{\partial U}}{{\partial c_t^1}}+\lambda \frac{{\partial g}}{{\partial c_t^1}}. $$

(2.57)

This is the solution to the household’s decision problem. However, a simpler route is provided by a function that leads us directly to condition (2.57). This is:

$$ \varLambda (c_t^1,c_{t+1}^2,\lambda )=U(c_t^1,c_{t+1}^2)+\lambda (w-g(c_t^1,c_{t+1}^2)). $$

(2.58)

This function is called the Lagrangian function and the variable the Lagrangian multiplier. After calculating the first derivative with respect to the two instrumental variables, the first-order (necessary) conditions for the solution of the optimization problem follows. Thus, differentiating Eq. 2.58 with respect to results directly in the constraint. The Lagrangian function of young households has the following form:

$$ \varLambda (c_t^1,c_{t+1}^2,\lambda )=\ln c_t^1+\beta \ln c_{t+1}^2+\lambda \left[ {w-c_t^1-\frac{{c_{t+1}^2}}{{1+{i_{t+1 }}}}} \right]. $$

(2.59)

The first-order conditions (FOCs) are:

$$ \frac{{\partial \varLambda }}{{\partial c_t^1}}=\frac{1}{{c_t^1}}-\lambda =0, $$

(2.60a)

$$ \frac{{\partial \varLambda }}{{\partial c_{t+1}^2}}=\beta \frac{1}{{c_{t+1}^2}}-\frac{1}{{1+{i_{t+1 }}}}\lambda =0, $$

(2.60b)

$$ \frac{{\partial \varLambda }}{{\partial \lambda }}={w_t}-c_t^1-\frac{{c_{t+1}^2}}{{1+{i_{t+1 }}}}=0. $$

(2.60c)

If we solve condition (2.60a) for variable and substitute the solution into Eq. 2.60b then, assuming the constraint Eq. 2.60c is also taken into account, we can determine the optimal consumption in period t Eq. 2.20, the optimal consumption in period $t+1$ Eq. 2.22 and the optimal savings per capita Eq. 2.21.

To ensure that Eqs. 2.20, 2.21 and 2.22 constitute a maximum (and not a minimum), we have to check the second-order conditions:

$$ \frac{{{\partial^2}\varLambda }}{{\partial {{{\left( {c_t^1} \right)}}^2}}}=-\frac{1}{{{{{\left( {c_t^1} \right)}}^2}}}<0, $$

(2.61a)

$$ \frac{{{\partial^2}\varLambda }}{{\partial {{{\left( {c_{t+1}^2} \right)}}^2}}}=-\beta \frac{1}{{{{{\left( {c_{t+1}^2} \right)}}^2}}}<0. $$

(2.62b)

Both conditions are negative, satisfying the second-order conditions for a strict (local) maximum.

One last and very important question remains: What is the meaning of the Lagrange multiplier in this optimization problem?

The Lagrange multiplier reflects the sensitivity of the value of the objective function with respect to a marginal change in the constants b (cf. Eq. 2.50a) of the constraints. In the optimization problem of young households the Lagrange multiplier is equal to:

$$ {\lambda_t}=\frac{{\partial {U_t}}}{{\partial{w_t}}}. $$

(2.63)

It indicates the amount by which the optimum value of the utility function increases when disposable income rises by one unit.

1.2 Walras’ Law

Finally, we want to show that our basic growth model satisfies Walras’ law. We therefore note the budget constraints of all economic agents for any period t and express all values in monetary units (and not in terms of output units as is done in the main text). In addition, we multiply all per-capita values by the number of corresponding number of individuals. Moreover, we indicate what savings of young households are used for, i.e. to buy investment goods and old capital at the reproduction price ${P_t}$. Thus, we have:

$$ {L_t}{s_t}{P_t}={P_t}{I_t}+{P_t}(1-\delta ){K_t}. $$

(2.64)

This equality implies that the budget constraint of young households can be rewritten as follows:

$$ {P_t}{L_t}c_t^1+{P_t}{I_t}+{P_t}(1-\delta ){K_t}={W_t}{L_t}, $$

(2.65)

while the aggregate budget constraint of old households reads as follows:

$$ {P_t}{L_{t-1 }}c_t^2={Q_t}{K_t}+{P_t}(1-\delta ){K_t}. $$

(2.66)

The linear-homogeneity of the production function implies that at a maximum profits are zero:

$$ {\varPi_t}=0={P_t}Y_t-{W_t}N_t-{Q_t}K_t^d. $$

(2.67)

Adding the left-hand sides and the right-hand sides of Eqs. 2.65 and 2.66 yields:

$$ {P_t}\left[ {{L_t}c_t^1+{L_{t-1 }}c_t^2} \right]={W_t}{L_t}+{Q_t}K_t^d-{P_t}{I_t}. $$

(2.68)

Clearing of the labor and capital market ($N_t={L_t}$ and $K_t^d={K_t}$) implies:

$$ {P_t}\left[ {{L_t}c_t^1+{L_{t-1 }}c_t^2} \right]+{P_t}{I_t}={W_t}N_t+{Q_t}K_t={P_t}Y_t. $$

(2.69)

Since ${I_t}={K_{t+1 }}-(1-\delta ){K_t}$ holds, Eq. 2.69 becomes:

$$ {P_t}\left[ {{L_t}c_t^1+{L_{t-1 }}c_t^2+{K_{t+1 }}-(1-\delta ){K_t}-Y_t} \right]=0. $$

(2.70)

Since ${P_t}>0$, the sum of the terms in square brackets in Eq. 2.70 must be zero.

Thus, we have shown that the product market clears once the labor and the capital markets clear. Equation 2.31 is thus an identity, not a constraint. Thus, we cannot determine the price level in this economy; it has therefore to be set exogenously (e.g. – and as we have assumed here – it can be set equal to one).