GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.

2.3 Equilibrium in the Input Markets

Now we define the effective quantity of labor employed in period t as L_t^e =
min(L_t^d , Nt). Then unemployment in period t is given by Nt - L_t^e and the rate
of employment in period t, u_t, by N^tNL^t.

We define the aggregate saving supply in period t-1, S_t^s_-1, as:
~ _         .. . . _ ~ _ . .. _ . ..

St-1 = Sw,t-1((^{1 - τ}t-1)^wt-1)Lt-1 + Su,t-1(st-1 )(Nt-1 ^- Lt-1).    (5)

Recall that aggregate saving in period t-1 is equal to the supply of capital in
period t, K_t^s. Thus, we have K₀^s = K₀ and K_t^s = S_t^s_-1 for all t ≥ 1.

Using ( 4) and ( 5) we have that for all t ≥ 1:

Ktt = KS(wt-ι,τt-ι,st-ι,Le-ι,Nt-ι) =

c((1 - τ_t-1)w_t-1L_t^e_-1 + s_t-1 (N_t-1 - L_t^e_-1))   .              (6)

Equation ( 6) says that the supply of capital in period t, that is, the supply
of saving in period t-1, is equal to aggregate labour income times the saving’s
propensity.

We assume that the interest rate in period t, Rt, balances demand and
supply in at least one of the inputs markets, that is:

Definition 2.3.1 Given w_t-1, L_t^e_-1 and w_t; R_t^t is such that:

Kd = K S(wt-i ,Tt-i,St-i,Le-i,Nt-i )                   (7)

and

Lt^d ≤ Nt;                                (8)

or

Lt^d = Nt                               (9)

Kd ≤ Ks(wt-i, Tt-1, st-i,Le-i, Nt-1).                  (10)

Equations ( 7) and ( 8) say that the capital market is in equilibrium and
that there is an excess supply of labor. Equations ( 9) and ( 10) that the labor
market is in equilibrium and that there is an excess supply of capital.

The equilibrium interest rate, R_t^t, the competitive wage, w_t^c , the inputs’
demand functions and the effective quantity of labor employed are given by the
following proposition:

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