An overlapping generations model

We will now analyze many households jointly and see how their consumption and saving behaviour a¤ects the evolution of the economy as a whole. We will get to know the Euler theorem and how it is used to sum factor incomes to yield GDP. We will also understand how the interest rate in the household’s budget constraint is related to marginal produc-tivity of capital and the depreciation rate. All this jointly yields time paths of aggregate consumption, the capital stock and GDP. We will assume an overlapping-generations structure (OLG).

A model in general equilibrium is described by fundamentals of the model and mar-ket and behavioural assumptions. Fundamentals are technologies of …rms, preferences of households and factor endowments. Adding clearing conditions for markets and behav-ioural assumptions for agents completes the description of the model.

2.4.1 Technologies

The …rms

Let there be many …rms who employ capital Kt and labour L to produce output Yt

according to the technology

Y_t= Y (K_t; L) : (2.4.1)

Assume production of the …nal good Y (:) is characterized by constant returns to scale.

We choose Yt as our numeraire good and normalize its price to unity, pt = 1: While this is not necessary and we could keep the price pt all through the model, we would see that all prices, like for example factor rewards, would be expressed relative to the price pt: Hence, as a shortcut, we set pt = 1:We now, however, need to keep in mind that all prices are henceforth expressed in units of this …nal good. With this normalization, pro…ts are given by t = Yt w^K_t Kt w^L_tL. Letting …rms act under perfect competition, the …rst-order conditions from pro…t maximization re‡ect the fact that the …rm takes all prices as parametric and set marginal productivities equal to real factor rewards,

@Y_t

@K_t = w_t^K; @Y_t

@L = w_t^L: (2.4.2)

In each period they equate t; the marginal productivity of capital, to the factor price w^K_t for capital and the marginal productivity of labour to labour’s factor reward w^L_t.

Euler’s theorem

Euler’s theorem shows that for a linear-homogeneous function f (x1; x₂; :::; x_n) the sum of partial derivatives times the variables with respect to which the derivative was computed equals the original function f (:) ;

f (x₁; x₂; :::; x_n) = @f (:)

@x₁ x₁+@f (:)

@x₂ x₂ + ::: +@f (:)

@x_n x_n: (2.4.3) Provided that the technology used by …rms to produce Yt has constant returns to scale, we obtain from this theorem that

Y_t= @Y_t

@K_tK_t+ @Y_t

@LL: (2.4.4)

Using the optimality conditions (2.4.2) of the …rm for the applied version of Euler’s theorem (2.4.4) yields

Y_t= w_t^KK_t+ w^L_tL: (2.4.5)

Total output in this economy, Yt;is identical to total factor income. This result is usually given the economic interpretation that under perfect competition all revenue in …rms is used to pay factors of production. As a consequence, pro…ts t of …rms are zero.

2.4.2 Households

Individual households

Households live again for two periods. The utility function is therefore as in (2.2.1) and given by

U_t= ln c^y_t + (1 ) ln c^o_t+1: (2.4.6) It is maximized subject to the intertemporal budget constraint

w^L_t = c^y_t + (1 + r_t+1) ¹c^o_t+1:

This constraint di¤ers slightly from (2.2.2) in that people work only in the …rst period and retire in the second. Hence, there is labour income only in the …rst period on the left-hand side. Savings from the …rst period are used to …nance consumption in the second period.

Given that the present value of lifetime wage income is w^L_t; we can conclude from (2.2.4) and (2.2.5) that individual consumption expenditure and savings are given by

c^y_t = w_t^L; c^o_t+1 = (1 ) (1 + r_t+1) w^L_t; (2.4.7)

st= w_t^L c^y_t = (1 ) w^L_t: (2.4.8)

Aggregation

We assume that in each period L individuals are born and die. Hence, the number of young and the number of old is L as well. As all individuals within a generation are identical, aggregate consumption within one generation is simply the number of, say, young times individual consumption. Aggregate consumption in t is therefore given by C_t= Lc^y_t+ Lc^o_t:Using the expressions for individual consumption from (2.4.7) and noting the index t (and not t + 1) for the old yields

Ct= Lc^y_t + Lc^o_t = w_t^L+ (1 ) (1 + rt) w_{t 1}^L L:

2.4.3 Goods market equilibrium and accumulation identity

The goods market equilibrium requires that supply equals demand, Yt = Ct+ It; where demand is given by consumption plus gross investment. Next period’s capital stock is - by an accounting identity - given by Kt+1 = I_t+(1 ) K_t:Net investment, amounting to the change in the capital stock, Kt+1 Kt, is given by gross investment Itminus depreciation K_t, where is the depreciation rate, Kt+1 K_t= I_t K_t. Replacing gross investment by the goods market equilibrium, we obtain the resource constraint

K_t+1 = (1 ) K_t+ Y_t C_t: (2.4.9)

For our OLG setup, it is useful to rewrite this constraint slightly ,

Y_t+ (1 ) K_t = C_t+ K_t+1: (2.4.10)

In this formulation, it re‡ects a “broader”goods market equilibrium where the left-hand side shows supply as current production plus capital held by the old. The old sell capital as it is of no use for them, given that they will not be able to consume anything in the next period. Demand for the aggregate good is given by aggregate consumption (i.e.

consumption of the young plus consumption of the old) plus the capital stock to be held next period by the currently young.

2.4.4 The reduced form

For the …rst time in this book we have come to the point where we need to …nd what will be called a “reduced form”. Once all maximization problems are solved and all constraints and market equilibria are taken into account, the objective consists of understanding properties of the model, i.e. understanding its predictions. This is usually done by …rst simplifying the structure of the system of equations coming out of the model as much as possible. In the end, after inserting and reinserting, a system of n equations in n unknowns results. The system where n is the smallest possible is what will be called the reduced form.

Ideally, there is only one equation left and this equation gives an explicit solution of the endogenous variable. In static models, an example would be LX = L; i.e. employment in sector X is given by a utility parameter times the total exogenous labour supply L:

This would be an explicit solution. If we are left with just one equation but we obtain on an implicit solution, we would obtain something like f (LX; ; L) = 0:

Deriving the reduced form

We now derive, given the results we have obtained so far, how large the capital stock in the next period is. Splitting aggregate consumption into consumption of the young and consumption of the old and using the output-factor reward identity (2.4.5) for the resource constraint in the OLG case (2.4.10), we obtain

w_t^KK_t+ w_t^LL + (1 ) K_t= C_t^y + C_t^o+ K_t+1:

De…ning the interest rate rt as the di¤erence between factor rewards w_t^K for capital and the depreciation rate ;

r_t w_t^K ; (2.4.11)

we …nd

r_tK_t+ w_t^LL + K_t= C_t^y + C_t^o+ K_t+1:

The interest rate de…nition (2.4.11) shows the net income of capital owners per unit of capital. They earn the gross factor rewards w^K_t but, at the same time, they experience a loss from depreciation. Net income therefore only amounts to rt: As the old consume the capital stock plus interest c^o_tL = (1 + r_t)K_t, we obtain

K_t+1= w_t^LL C_t^y = s_tL: (2.4.12)

which is the aggregate version of the savings equation (2.4.8). Hence, we have found that savings st of young at t is the capital stock at t + 1:

Note that equation (2.4.12) is often present on “intuitive”grounds. The old in period t have no reason to save as they will not be able to use their savings in t + 1: Hence, only the young will save and the capital stock in t + 1; being made up from savings in the previous period, must be equal to the savings of the young.

The one-dimensional di¤erence equation

In our simple dynamic model considered here, we obtain the ideal case where we are left with only one equation that gives us the solution for one variable, the capital stock.

Inserting the individual savings equation (2.4.8) into (2.4.12) gives with the …rst-order condition (2.4.2) of the …rm

K_t+1 = (1 ) w_t^LL = (1 )@Y (K_t; L)

@L L: (2.4.13)

The …rst equality shows that a share 1 of labour income turns into capital in the next period. Interestingly, the depreciation rate does not have an impact on the capital stock in period t + 1. Economically speaking, the depreciation rate a¤ects the wealth of the old but - with logarithmic utility - not the saving of the young.

2.4.5 Properties of the reduced form

Equation (2.4.13) is a non-linear di¤erence equation in Kt: All other quantities in this equation are constant. This equation determines the entire path of capital in this dynamic economy, provided we have an initial condition K0. We have therefore indeed solved the maximization problem and reduced the general equilibrium model to one single equation.

From the path of capital, we can compute all other variables which are of interest for our economic questions.

Whenever we have reduced a model to its reduced form and have obtained one or more di¤erence equations (or di¤erential equations in continuous time), we would like to understand the properties of such a dynamic system. The procedure is in principle always the same: We …rst ask whether there is some solution where all variables (Kt in our case) are constant. This is then called a steady state analysis. Once we have understood the steady state (if there is one), we want to understand how the economy behaves out of the steady state, i.e. what its transitional dynamics are.

Steady state

In the steady state, the capital stock is constant, Kt = K_t+1 = K , and determined by

K = (1 )@Y (K ; L)

@L L: (2.4.14)

All other variables like aggregate consumption, interest rates, wages etc. are constant as well. Consumption when young and when old can di¤er, as in a setup with …nite lifetimes, the interest rate in the steady state does not need to equal the time preference rate of households.

Transitional dynamics

Dynamics of the capital stock are illustrated in …gure2.4.1. The …gure plots the capital stock in period t on the horizontal axis. The capital stock in the next period, Kt+1; is plotted on the vertical axis. The law of motion for capital from (2.4.13) then shows up as the curve in this …gure. The 45 line equates Kt+1 to Kt:

We start from our initial condition K0. Equation (2.4.13) or the curve in this …gure then determines the capital stock K1: This capital stock is then viewed as Kt so that, again, the curve gives us Kt+1;which is, given that we now started in 1; the capital stock K₂ of period 2: We can continue doing so and see graphically that the economy approaches the steady state K which we computed in (2.4.14).

N

N

K_t 0

K_t+1

K₀

- 6

- 6

K_t= K_t+1

K_t+1 = (1 )w^L(K_t)L

Figure 2.4.1 Convergence to the steady state Summary

We started with a description of technologies in (2.4.1), preferences in (2.4.6) and factor endowment given by K0: With behavioural assumptions concerning utility and pro…t maximization and perfect competition on all markets plus a description of markets in (2.4.3) and some “juggling of equations”, we ended up with a one-dimensional di¤erence equation (2.4.13) which describes the evolution of the economy over time and steady state in the long-run. Given this formal analysis of the model, we could now start answering economic questions.