Competitive Equilibrium Growth

Our main interest is not optimal growth, but equilibrium growth. Nevertheless, the Sec­ond Welfare Theorem, Theorem 5.7 of the previous chapter, implies that the optimal growth path characterized in the previous section also corresponds to an equilibrium growth path (in the sense that, it can be decentralized as a competitive equilibrium).

Suppose that each household starts with an endowment of capital stock Ko, meaning that the initial endowments are also symmetric (recall that there is a mass 1 of households and the total initial endowment of capital of the economy is Ko). The other side of the economy is populated by a large number of competitive firms, which are modeled using the aggregate production function.

The definition of a competitive equilibrium in this economy is standard. In particular, we have:

Definition 6.3. A competitive equilibrium consists of paths of consumption, capital stock, wage rates and rental rates of capital,such that the representa­

tive household maximizes its utility given initial capital stock Ko and the time path of prices {w (t),R (t)}∞o, and the time path of pricesis such that given the time path

of capital stock and laborall markets clear.

Households rent their capital to firms.

R (t) = f' (k (t)),

and thus face a gross rate of return equal to

(6.39)

1 + r (t +1) = f⁰ (k (t)) + (1 - δ)

for renting one unit of capital at time t in terms of date t + 1 goods. Notice that the gross rate of return on assets is defined as 1 + r, since r often refers to the net interest rate. In fact in the continuous time model, this is exactly what the term r will correspond to. This notation should therefore minimize confusion.

In addition, to capital income, households in this economy will receive wage income for supplying their labor at the market wage of w (t) = f (k (t)) — k (t) f⁰ (k (t)).

Now consider the maximization problem of the representative household:

subject to the flow budget constraint

(6.40) a (t +1) = (1+ r (t + 1)) a (t) — c (t) + w (t),

where a (t) denotes asset holdings at time t and as before, w (t) is the wage income of the individual (since labor supply is normalized to 1). The timing underlying the flow budget constraint (6.40) is that the individual rents his capital or asset holdings, a (t), to firms to be used as capital at time t +1. Out of the proceeds, he consumes and whatever is left, together with his wage earnings, w (t), make up his asset holdings at the next date, a (t + 1).

In addition to this flow budget constraint, we have to impose a no Ponzi game constraint to ensure that the individual asset holdings do not tend to minus infinity. Since this constraint will be discussed in detail in Chapter 8, we do not introduce it here (though note that without this constraint, there are other, superfluous solutions to the consumer maximization problem).

For now, it suffices to look at the Euler equation for the consumer maximization problem:

(6.41)

Imposing steady state implies that c (t) = c (t + 1).

(1 + r (t +1)) β = 1.

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Next, market clearing immediately implies that 1+ r (t + 1) is given by (6.39), so the capital­labor ratio of the competitive equilibrium is given by

The steady state is given by

These two equations are identical to equations (6.38) and (6.39), which characterize the solution to the optimum growth problem. In fact, a similar argument establishes that the entire competitive equilibrium path is identical to the optimal growth path. Specifically, substituting for 1 + r (t + 1) from (6.39) into (6.41), we obtain

(6.42)

which is identical to (6.37). This condition also implies that given the same initial condition, the trajectory of capital-labor ratio in the competitive equilibrium will be identical to the behavior of the capital-labor ratio in the optimal growth path (see Exercise 6.20). This is, of course, not surprising in view of the second (and first) welfare theorems we saw above.

We will discuss many of the implications of competitive equilibrium growth in the neo­classical model once we go through the continuous time version as well.

6.8.