Optimal Factor Choice: Isoquants and Isocosts

Optimal Factor Choice: Isoquants and Isocosts
In the long-run period, a firm chooses not only the volume of production, but also the combination of factors. How to produce a given volume at minimal cost? Or: with a given budget—maximize output? This is an optimization problem, analogous to consumer choice.

Isocosts
Isocost line — a line showing all combinations of L and K that the firm can purchase given a certain cost.
Equation of the isocost:
$C = wL + rK$
Where $C$ is total cost, $w$ is the price of labor (wage), $r$ is the price of capital (rent, interest).

Slope of the isocost: $-w/r$ — relative price of factors in the market.

Intersection with axes:
On the L axis: $C/w$ (all money spent on labor)
On the K axis: $C/r$ (all money spent on capital)

Shifts of the isocost:
Increase in budget → parallel shift outward
Change in relative prices → change in slope

Cost Minimization
Problem: produce volume $Q^$ at minimal cost.
Graphically: find the isocost touching the isoquant $Q^$ as close as possible to the origin.

Optimum condition:
$MRTS = w / r$
Or equivalently:
$\frac{MPL}{w} = \frac{MPK}{r}$

Interpretation: the last ruble spent on labor must provide the same marginal increase to output as the last ruble spent on capital.

If $MPL/w > MPK/r$ — hire more labor, less capital.

Output Maximization with Budget
Alternative formulation: with fixed budget $C^$ achieve maximum output.
Mathematically, it is the same point—touching of isocost $C^$ and the highest attainable isoquant.

Condition is the same: $MRTS = w/r$.

Expansion Path
When scale of production changes, the optimal combinations of L and K change.
Expansion path: a line connecting optimal points at different outputs (given unchanging factor prices).

Form of the expansion path:

• Ray from the origin — fixed factor proportions (homothetic function)
• Curved — proportions change with scale

Response to Changes in Factor Prices
Rise in wages ($w$): Isocosts become steeper
Optimum shifts: less labor, more capital
Substitution effect: capital substitutes labor
Long-term trend: rising wages → automation, robotization.
Cheap labor → labor-intensive technologies.

Country differences:
In countries with cheap labor — labor-intensive production.
In countries with expensive labor — capital-intensive.

This explains offshoring and differences in technologies.

Types of Technologies
By shape of isoquants:

• Perfect substitutes:
  Isoquants are straight lines
  One factor can be entirely replaced with another
  Optimum — corner solution (only one factor)
  Rare in reality
• Perfect complements (Leontief function):
  Isoquants are right angles
  Factors used in fixed proportions
  No substitution
  Example: one driver per one truck
• Imperfect substitutes (Cobb-Douglas, others):
  Isoquants are smooth convex curves
  Partial substitution is possible
  Most realistic case

Cobb-Douglas Production Function
$Q = A \times L^\alpha \times K^\beta$
Where $A$ is technological parameter, $\alpha$ and $\beta$ are the output elasticities for labor and capital.

Properties:

• $\alpha + \beta = 1$: constant returns to scale
• $\alpha + \beta > 1$: increasing returns
• $\alpha + \beta < 1$: decreasing returns

Factor shares in cost = $\alpha$ and $\beta$ (under perfect competition)

Popularity: simplicity, good approximation for many industries, convenient mathematical properties.

For the Investor
Technological flexibility:

• Companies able to substitute factors — more adaptive to price changes
• Rigid proportions — vulnerable to factor price shocks

Impact of rising wages:

• Capital-intensive companies — suffer less from rising wages
• Labor-intensive — benefit from cheap labor, suffer from labor becoming more expensive

Automation:
Long-term trend — labor replaced by capital
Companies at the forefront of automation — potential winners
Industries with high share of manual labor — at risk or candidates for transformation