§ 1Two Horizons
A competitive firm produces output $Y = f(L,K)$, sells it at price $p$, and pays $w$ per unit of labour and $r$ per unit of capital. Profit is
What the firm can choose depends on the time horizon:
Short run. Capital is sunk at level $\bar K$. The firm picks only $L$, taking $\bar K$ as given. The first-order condition is
Long run. Both factors are flexible. The firm picks $L$ and $K$ jointly, giving two FOCs:
Because the firm has more margins of adjustment in the long run, the long-run labour demand curve is flatter (more wage-elastic) than the short-run curve — a corollary of the Le Chatelier principle. The two curves cross at the wage where the existing $\bar K$ happens to coincide with the long-run optimal $K$.
§ 2Interactive Model
Toggle the horizon to switch between the two problems. In short run, $\bar K$ is a slider you control. In long run, $K$ is solved for and the slider is locked. The right panel always shows both demand curves so the Le Chatelier comparison is visible at a glance — use Set $\bar K$ = $K^{LR}$ to make them cross at the current wage.
Horizon
Technology
Prices
Capital stock
Technology parameters
Production & Cost
Profit Function
Labour Demand
§ 3Reading the Three Panels
Tangency at $L^*$
Black curve: revenue $pf(L,K)$ at the relevant $K$ ($\bar K$ in short run, $K^{LR}$ in long run). Red dashed line: total wage cost $wL$. The optimum is where the slope of revenue equals $w$. The vertical gap is variable profit, before subtracting $rK$.
The profit hill
Profit $\pi(L)$ peaks at $L^*$. Raising $w$ flattens it and slides it left; raising $r$ shifts it down in parallel without moving the peak — capital cost is independent of $L$, so $r$ never appears in the FOC for labour. Switching to long run: the firm picks the $K$ that maximises the height of this hill jointly with $L$.
SR vs. LR (Le Chatelier)
Solid red: short-run demand at the current $\bar K$. Dashed green: long-run demand, with $K$ optimally chosen at every wage. The two curves cross at the wage where $\bar K$ matches $K^{LR}$. Away from the crossing, the LR curve is flatter: more adjustment margins ⇒ more wage-elastic.
§ 4Closed Forms & Returns to Scale
To keep the long-run problem well-posed, the production function exhibits decreasing returns to scale with parameter $\nu \in (0,1]$. Without DRS, long-run profit is unbounded (or zero) and labour demand is degenerate.
Cobb–Douglas (DRS): $\;Y = A\,(L^{\alpha}K^{1-\alpha})^{\nu}.$ Short-run labour demand is
In the long run both FOCs hold and yield $K^{LR}/L^{LR} = \tfrac{1-\alpha}{\alpha}\cdot\tfrac{w}{r}$, with
The wage exponent $\tfrac{(1-\alpha)\nu-1}{1-\nu}$ in the unconditional curve is more negative than the short-run exponent $\tfrac{-1}{1-\alpha\nu}$: the long-run curve is strictly more wage-elastic.
CES (DRS): $\;Y = A\,\big[\alpha L^{\rho} + (1-\alpha)K^{\rho}\big]^{\nu/\rho}, \;\rho = (\sigma-1)/\sigma.$ Here both horizons are solved numerically. With $\sigma > 1$ (substitutes) the long-run labour demand is markedly more wage-elastic than the short-run one, and a wage rise tilts the firm's input mix sharply toward capital ($K/L$ rises steeply). With $\sigma < 1$ (complements) the LR–SR gap is small and the $K/L$ ratio responds little to wages — labour and capital move together. Note that, under DRS, both unconditional factor demands $L^{LR}(w,r)$ and $K^{LR}(w,r)$ are downward-sloping in $w$ regardless of $\sigma$: the scale effect of a higher wage (lower output, hence less of both factors) outweighs the substitution effect on capital. What $\sigma$ governs is the relative response, i.e. how fast $K/L$ changes.
Limit case $\nu \to 1$ (constant returns): long-run profit becomes scale-free and the long-run labour demand becomes perfectly elastic at the unit-cost wage. Slide $\nu$ to 0.98 and watch the LR curve flatten dramatically — that is the textbook "long-run zero-profit" envelope emerging.