Section 7 - Firm

The Firm Problem & Market Clearing

Firms are competitive and own no capital - they rent it from households in spot markets. Each instant they solve a static problem: choose $K$ and $L$ to maximise profit given prices $(r + \delta, w)$ and the technology $Y = F(K, L)$ .

Static profit maximisation

\max_{K, L}\; F(K, L) - (r + \delta) K - w L

The firm pays households the gross rental rate

r + \delta

for capital - the household's net return

r

after covering depreciation.

With constant returns to scale, profit is zero in equilibrium. The first-order conditions are:

Step 1
$\frac{\partial F}{\partial K} = r + \delta \;\Longleftrightarrow\; r = f'(k) - \delta$
The marginal product of capital equals the gross user cost; the net return equals the marginal product minus depreciation.
Step 2
$\frac{\partial F}{\partial L} = w \;\Longleftrightarrow\; w = f(k) - k f'(k)$
Euler's theorem for homogeneous functions: with CRS, wages equal output net of capital's marginal product.

Cobb-Douglas case

For $Y = A K^\alpha L^{1-\alpha}$ , the per-worker production function is $f(k) = A k^\alpha$ , and the factor prices specialise nicely:

r + \delta = \alpha A k^{\alpha - 1}, \qquad w = (1 - \alpha) A k^\alpha.

Cobb-Douglas factor prices. Each factor earns its share of output.

Market clearing - assets equal capital

Households own all assets in the economy. Capital is the only asset in the baseline RCK model - there are no government bonds, no foreign claims, no non-reproducible factors with positive net supply. Therefore:

a(t) = k(t) \quad \text{for all } t \geq 0.

market-clearing

The asset market clearing condition.

Substituting $a = k$ and the firm's first-order conditions into the per-worker budget constraint:

Step 1
$\dot k = (r - n) k + w - c$
Budget constraint with $a = k$ .
Step 2
$= \bigl(f'(k) - \delta - n\bigr) k + \bigl(f(k) - k f'(k)\bigr) - c$
Substitute factor prices.
Step 3
$\boxed{\;\dot k = f(k) - c - (n + \delta) k\;}$
The capital accumulation equation - identical to Solow's, except $c$ is endogenous.