Section 7

The Firm Problem

Profit maximisation

A representative competitive firm rents $K_t$ units of capital at gross rental price $\widetilde r_t$ and hires $L_t$ units of labour at wage $w_t$ , both each period. Its profit is

\Pi_t = F(K_t, L_t) - \widetilde r_t K_t - w_t L_t.

eq:firm-profit

Standard constant-returns-to-scale arguments give the two FOCs:

Step 1
$\frac{\partial \Pi_t}{\partial K_t} = F_K(K_t, L_t) - \widetilde r_t = 0 \;\Rightarrow\; \widetilde r_t = F_K(K_t, L_t) = f'(k_t)$
Marginal product of capital equals rental rate.
Step 2
$\frac{\partial \Pi_t}{\partial L_t} = F_L(K_t, L_t) - w_t = 0 \;\Rightarrow\; w_t = F_L(K_t, L_t) = f(k_t) - k_t f'(k_t)$
MPL = $f(k) - kf'(k)$ (Euler's theorem for homogeneous functions).

From rental rate to interest rate

The household receives the *net* return on its savings — the rental rate plus what is left of the capital after depreciation. If $\delta$ is the per-period depreciation rate:

1 + r_{t+1} = f'(k_{t+1}) + (1 - \delta).

eq:gross-return

Gross real return on one unit of savings: rental income

f'(k)

plus the undepreciated portion of the capital good itself.

With **full depreciation** $\delta = 1$ (our canonical assumption for closed forms), this simplifies to

1 + r_{t+1} = f'(k_{t+1}),

eq:gross-return-full-dep

and the household's claim on savings is essentially a claim on tomorrow's gross output, mediated by the marginal product.

Closed form: Cobb–Douglas

We will use the Cobb–Douglas production function $F(K, L) = A K^{\alpha} L^{1-\alpha}$ throughout. In intensive form $f(k) = A k^{\alpha}$ , and the factor prices become:

Quantity	Formula	Notes
$f'(k)$	$\alpha A k^{\alpha-1}$	Marginal product of capital.
$1 + r$	$\alpha A k^{\alpha-1}$ (with $\delta = 1$ )	Gross return on savings.
$f(k) - k f'(k)$	$(1 - \alpha) A k^{\alpha}$	Marginal product of labour = wage.
$w$	$(1 - \alpha) A k^{\alpha}$	Wage per unit of labour.
$w/y$	$1 - \alpha$	Labour share — constant under Cobb–Douglas.

Cobb–Douglas factor prices.

Two facts will matter in what follows:

w_t = (1 - \alpha) A k_t^{\alpha}, \qquad 1 + r_{t+1} = \alpha A k_{t+1}^{\alpha - 1}.

eq:cd-prices

Note how each is anchored to a *different* capital stock: wages depend on the capital the young find when they start work; the interest rate depends on the capital that exists when they retire. This timing is essential to the dynamics in Section 9.