[DEFINITIONS] Economic applications

Production function

A production function describes output $Q$ as a function of input quantities:

Q=f(E,L),

where $E,L\ge0$ , $E$ denotes capital equipment or investment and $L$ denotes labour. The formula is understood on the economically feasible part of this non-negative quadrant.

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

If $q>0$ is the price per unit of capital, $p>0$ is the price per unit of labour and $B>0$ is the available budget, the feasible input choices satisfy

qE+pL\le B, \qquad E\ge0,\quad L\ge0.

The products $qE$ and $pL$ are both measured in currency units, as is $B$ ; this keeps the budget equation dimensionally meaningful.

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

[1]:Formulate a production problem

Cobb-Douglas production function

A Cobb-Douglas production function has the form

Q(E,L)=AE^\varepsilon L^{1-\varepsilon}, \qquad A>0, \qquad 0<\varepsilon<1.

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.

[1]:Production function
[2]:Optimal Cobb-Douglas allocation

Definitions from Economic applications

Production function

Budget constraint

Cobb-Douglas production function

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