We have so far analyzed maximization problems of households in partial equilibrium. In two-period models, we have analyzed how households can be aggregated and what we learn about the evolution of the economy as a whole. We will now do the same for in…nite horizon problems.
As we did in ch.2.4, we will …rst specify technologies. This shows what is technologi-cally feasible in this economy. Which goods are produced, which goods can be stored for saving purposes, is there uncertainty in the economy stemming from production processes?
Given these technologies, …rms maximize pro…ts. Second, household preferences are pre-sented and the budget constraint of households is derived from the technologies prepre-sented before. This is the reason why technologies should be presented before households are introduced: budget constraints are endogenous and depend on knowledge of what house-holds can do. Optimality conditions for househouse-holds are then derived. Finally, aggregation over households and an analysis of properties of the model using the reduced form follows.
3.6.1 Technologies
The technology is a simple Cobb-Douglas technology
Yt= AKtL1 : (3.6.1)
Capital Kt and labour L is used with a given total factor productivity level A to produce output Yt:This good can be used for consumption and investment and equilibrium on the
goods market requires
Yt = Ct+ It: (3.6.2)
Gross investment It is turned into net investment by taking depreciation into account, Kt+1 = (1 ) Kt+ It:Taking these two equations together gives the resource constraint of the economy,
Kt+1 = (1 ) Kt+ Yt Ct: (3.6.3)
As this constraint is simply a consequence of technologies and market clearing, it is identical to the one used in the OLG setup in (2.4.9).
3.6.2 Firms
Firms maximize pro…ts by employing optimal quantities of labour and capital, given the technology in (3.6.1). First-order conditions are
@Yt
@Kt = wKt ; @Yt
@L = wLt (3.6.4)
as in (2.4.2), where we have again chosen the consumption good as numeraire.
3.6.3 Households
Preferences of households are described as in the intertemporal utility function (3.1.1).
As the only way households can transfer savings from one period to the next is by buying investment goods, an individual’s wealth is given by the “number of machines” kt, she owns. Clearly, adding up all individual wealth stocks gives the total capital stock, Lkt= Kt:Wealth kt increases over time if the household spends less on consumption than what it earns through capital plus labour income, corrected for the loss in wealth each period caused by depreciation,
kt+1 kt= wKt kt kt+ wLt ct , kt+1= 1 + wKt kt+ wLt ct: If we now de…ne the interest rate to be given by
rt wtK ; (3.6.5)
we obtain our budget constraint
kt+1 = (1 + rt) kt+ wtL ct: (3.6.6) Note that the “derivation” of this budget constraint was simpli…ed in comparison to ch. 2.5.5 as the price vt of an asset is, as we measure it in units of the consumption good which is traded on the same …nal market (3.6.2), given by 1. More general budget constraints will become pretty complex as soon as the price of the asset is not normalized.
This complexity is needed when it comes e.g. to capital asset pricing - see further below in ch. 9.3. Here, however, this simple constraint is perfect for our purposes.
Given the preferences and the constraint, the Euler equation for this maximization problem is given by (see exercise 5)
u0(ct) = [1 + rt+1] u0(ct+1) : (3.6.7) Structurally, this is the same expression as in (3.4.7). The interest rate, however, refers to t + 1, due to the change in the budget constraint. Remembering that = 1= (1 + ), this shows that consumption increases as long as rt+1> .
3.6.4 Aggregation and reduced form
Aggregation
To see that individual constraints add up to the aggregate resource constraint, we simply need to take into account that individual income adds up to output, wtKKt+wLtLt= Yt. Remember that we are familiar with the latter from (2.4.4). Now start from (3.6.6) and use (3.6.5) to obtain,
Kt+1 = Lkt+1= 1 + wtK Lkt+ wtLL Ct= (1 ) Kt+ Yt Ct:
The optimal behaviour of all households taken together can be gained from (3.6.7) by summing over all households. This is done analytically correctly by …rst applying the inverse function of u0 to this equation and then summing individual consumption levels over all households (see exercise 6 for details). Applying the inverse function again gives u0(Ct) = [1 + rt+1] u0(Ct+1) ; (3.6.8) where Ct is aggregate consumption in t:
Reduced form
We now need to understand how our economy evolves in general equilibrium. Our …rst equation is (3.6.8), telling us how consumption evolves over time. This equation contains consumption and the interest rate as endogenous variables.
Our second equation is therefore the de…nition of the interest rate in (3.6.5) which we combine with the …rst-order condition of the …rm in (3.6.4) to yield
rt = @Yt
@Kt : (3.6.9)
This equation contains the interest rate and the capital stock as endogenous variables.
Our …nal equation is the resource constraint (3.6.3), which provides a link between capital and consumption. Hence, (3.6.8), (3.6.9) and (3.6.3) give a system in three equa-tions and three unknowns. When we insert the interest rate into the optimality condition for consumption, we obtain as our reduced form
u0(Ct) = h
1 + @K@Yt+1
t+1
i
u0(Ct+1) ;
Kt+1 = (1 ) Kt+ Yt Ct: (3.6.10)
This is a two-dimensional system of non-linear di¤erence equations which gives a unique solution for the time path of capital and consumption, provided we have two initial con-ditions K0 and C0.
3.6.5 Steady state and transitional dynamics
When trying to understand a system like (3.6.10), the same principles can be followed as with one-dimensional di¤erence equations. First, one tries to identify a …xed point, i.e. a steady state, and then one looks at transitional dynamics.
Steady state
In a steady state, all variables are constant. Setting Kt+1 = Kt= K and Ct+1= Ct= C; we obtain
1 = 1 + @Y
@K , @Y
@K = + ; C = Y K;
where the “, step”used the link between and from (3.1.2). In the steady state, the marginal productivity of capital is given by the time preference rate plus the depreciation rate. Consumption equals output minus depreciation, i.e. minus replacement investment.
These two equations determine two variables K and C: the …rst determines K; the second determines C:
Transitional dynamics
Understanding transitional dynamics is not as straightforward as understanding the steady state. Its analysis follows the same idea as in continuous time, however, and we will analyze transitional dynamics in detail there.
Having said this, we should acknowledge the fact that transitional dynamics in discrete time can quickly become more complex than in continuous time. As an example, chaotic behaviour can occur in one-dimensional di¤erence equations while one needs at least a three-dimensional di¤erential equation system to obtain chaotic properties in continuous time. The literature on chaos theory and textbooks on di¤erence equations provide many examples.