# Thinking about Inter-temporal production and the costs of hiring

Suppose that production requires time. The worker hired to produce today will not generate revenue today, but next period and the firm will hire the worker if the present value of his marginal revenue product is larger than his present wage. The rationale is that even though investors are lending the firm capital (or bonds) during the period of production workers are not lending to the firm, but require immediate payment. This distinction becomes relevant when production requires time. But if production was instantaneous, there would be no reason for the firm to borrow from creditors. The fundamental model period should be the time required for production to occur.

If 1 period is the time lag between payments to variable inputs and revenues from the finished output, and if $b(n)$ is the (nominal) single period bond rate in period n, the first order condition for the firm becomes:

(*) $E_n(p_{n+1})F_L(K_n,L_n) = (1+b_n)w_n$

If capital (a stand-in for fixed costs more generally)  is rented out in the current period, but payment to capital is only supplied in the subsequent period (when proceeds of sales are realized), then the bond discount factor does not apply directly to capital rental payments, but only to the variable payments. In a simple model, labor is the one to take the hit.

However, in equilibrium, the firm can stagger production so that sales of the previous period’s production fund the current period’s labor expenses, which are in turn based on expectations of next period’s revenues, etc.

The black boxes correspond to outlays, and the grey boxes receipts.

In equilibrium the firm can pay all current labor expenses out of the proceeds of past production with no need to maintain working capital.

But if the economy is growing, so that firms are increasing output in every period, then there is a gap between present outlays and present receipts, forcing the firm to incur bond debts in the form of working capital. In that case, the firm needs to be more profitable each period in order to retire the debts. It is as if the firm needs to borrow more capital from shareholders than what is needed for physical production. It needs to borrow capital in order to grow, to bridge the difference in time between production and revenue.

The red bar represents the growth in working capital required of the growing firm. Payment of interest on working capital is part of the firm’s expansion costs.

The green bar represents the reduction in working capital needs, corresponding to a reduction in interest payments and a reduction in the firm’s overall costs.

It is enough to push the firm, for some reason, off the optimal quantity of labor. In that case, there will be a wedge of the form $1 + b_n$ between the wage rate and the marginal revenue product of labor, and the firm will not return to the optimal quantity of labor unless the time value of money is zero.

Note: Updated for clarity.