Economic applications

Production function

The gradient of $Q$ records how output changes when either input changes. An optimal allocation compares those changes with input prices through the multiplier condition, which requires $\nabla Q$ to be parallel to the budget-line normal.

Budget constraint

The inequality includes every affordable input bundle. Its boundary $qE+pL=B$ is the budget line: the bundles that spend the entire budget.

Formulate a production problem

If output increases with either input, an allocation leaving budget unused cannot be optimal, because some additional productive input could be purchased. Without this monotonicity assumption (or if inputs are indivisible), keep the inequality and compare interior, axis and slack candidates instead.

Cobb-Douglas production function

The exponent $\varepsilon$ controls the relative contribution of capital and $1-\varepsilon$ controls the relative contribution of labour. The positive scale parameter $A$ changes output levels but, under the linear budget constraint, does not change the optimal input shares.

Optimal Cobb-Douglas allocation

Maximise

subject to

Assume $p,q,B>0$. A feasible bundle with $E,L>0$ produces positive output, whereas a bundle with $E=0$ or $L=0$ produces zero output. Hence an optimum has $E,L>0$, so the following first-order equations and division are valid.

The multiplier equations are

together with $qE+pL=B$.

Divide the first equation by the second. The common factors simplify to

Equivalently,

Combining this relation with the budget equation gives

Thus the optimal input quantities are

The firm spends the fraction $\varepsilon$ of its budget on capital and the fraction $1-\varepsilon$ on labour:

The scale parameter $A$ cancels from the allocation because it multiplies both multiplier equations in the same way.

Here $\lambda$ has units of output per currency unit and measures the marginal value of relaxing the budget. Then $\lambda q$ and $\lambda p$ have the units of the marginal products $Q_E$ and $Q_L$: output per unit of capital and output per unit of labour.

This interpretation can be checked directly. The first multiplier equation gives $\lambda=\varepsilon Q/(qE)=Q/B$ at the optimum. Since both optimal inputs are proportional to $B$, the optimal output is also proportional to $B$; thus $dQ_{\max}/dB=Q_{\max}/B=\lambda$.

The multiplier equations locate a candidate, not by themselves a global maximum. Along the budget line, $L=(B-qE)/p$ for $0<E<B/q$ and

This derivative is positive before $E=\varepsilon B/q$ and negative after it; output is zero at both endpoints. The candidate is therefore the global maximum on the feasible budget segment.