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In ordinary equilibrium models, we can pin down equilibrium quantity by supply-demand functions.

Is this the same in RBC models? Can we pin down equilibrium output and amount of labor and capital without initial conditions with stochastic variations? Or do we need initial conditions (time $0$ conditions) for labor and capital?

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The stock of capital in the RBC model is a predetermined state variable, meaning that the capital $K_t$ available at time $t$ was determined at time $t-1$. Usually we have an accumulation relation of the form $$K_t = I_{t-1}+(1-\delta)K_{t-1}$$ When solving an RBC model starting at time $t=0$, we must exogenously assume some initial capital stock $K_0>0$ to get the model started, since we don't explicitly consider what happened at time $t=-1$.

Labor, on the other hand, generally is not specified ahead of time. Instead, in many RBC models the household has an exogenous certain time endowment, perhaps normalized to 1, and then "leisure" $L_t=1-N_t$ is the amount of time remaining after "labor" $N_t$. Leisure then enters into the utility function just like consumption. (An equivalent and common alternative is to directly put labor $N_t$ as a "bad" into the utility function.)

The basic contrast is that capital is a stock while labor is a flow. We can choose the level of investment flow $I_t$ in each period, and the stock $K_t$ (which we can no longer control once we've reached time $t$) reflects the accumulation of past investment flows $$K_t = \sum_{s=0}^\infty (1-\delta)^s I_{t-s-1}$$ Meanwhile, labor $N_t$ is a flow that the household can choose optimally at time $t$.

Things becomes more complicated in a model with either (A) human capital or (B) search frictions. With (A) human capital, the effective supply of labor is partly a stock, since it reflects past flow investments in human capital; whereas with (B), the number of ongoing employer-employee relationship is a stock that reflects accumulated matches. Neither of these are present in the baseline RBC model, however.

Instead, the baseline RBC model is essentially the Ramsey growth model augmented with a stochastic process for TFP and endogenous labor supply. In this model, capital $K_t$ is the only endogenous linkage between periods.

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