2.4 Cost minimization and cost curves

Cost Minimization in Production

Objective and Principles

Cost minimization is about finding the cheapest way to produce a given quantity of output. A firm chooses the combination of inputs (typically labor and capital) that minimizes total cost while still hitting its output target. This is the flip side of profit maximization: you can't maximize profit if you're wasting money on an inefficient input mix.

The solution depends on two things: the production function (how inputs translate into output) and input prices (the wage $w$ for labor and the rental rate $r$ for capital). The firm's total cost is:

$TC = wL + rK$

The goal is to minimize this subject to the constraint that output equals some target level $\bar{q} = f(L, K)$.

This framework applies in both the short run and the long run, but the constraints differ. In the short run, at least one input (usually capital) is fixed, so the firm can only adjust variable inputs. In the long run, all inputs are adjustable, giving the firm full flexibility to find the true cost-minimizing combination.

Applications in Different Time Frames

• Short run: The firm optimizes only its variable inputs (e.g., labor) since fixed inputs like capital can't be changed. With capital locked in, "cost minimization" reduces to choosing the level of labor that produces $\bar{q}$ given $\bar{K}$, so there's no real optimization over input mix.
• Long run: All inputs are variable. The firm can adjust both labor and capital, switch technologies, or change plant size. This is where the tangency condition (below) fully applies.
• Short-run decisions affect immediate profitability, while long-run choices shape the firm's sustainable cost structure and competitiveness.

Cost Minimization Condition

Derivation and Interpretation

The cost minimization problem is solved using constrained optimization, typically with the Lagrangian method. You minimize $wL + rK$ subject to $f(L, K) = \bar{q}$.

Here are the steps:

1. Set up the Lagrangian: $\mathcal{L} = wL + rK - \lambda[f(L,K) - \bar{q}]$

2. Take first-order conditions (FOCs) with respect to $L$, $K$, and $\lambda$:

   • $\frac{\partial \mathcal{L}}{\partial L} = w - \lambda \cdot MP_L = 0$
   • $\frac{\partial \mathcal{L}}{\partial K} = r - \lambda \cdot MP_K = 0$
   • $\frac{\partial \mathcal{L}}{\partial \lambda} = \bar{q} - f(L,K) = 0$

3. From the first two FOCs, you get $\lambda = w / MP_L = r / MP_K$, which rearranges to the tangency condition:

$\frac{MP_L}{MP_K} = \frac{w}{r}$

1. Combine with the constraint $f(L,K) = \bar{q}$ to solve for the optimal $L^*$ and $K^*$.

Note that the Lagrange multiplier $\lambda$ has an economic interpretation: it equals the marginal cost of output ($\lambda = MC$), since it measures how much minimum cost rises when the output target $\bar{q}$ increases by one unit.

The tangency condition can be rearranged to an equivalent and often more intuitive form:

$\frac{MP_L}{w} = \frac{MP_K}{r}$

This says: the last dollar spent on labor should yield the same additional output as the last dollar spent on capital. If it doesn't, the firm can reduce costs by shifting spending toward the input that's more productive per dollar, while keeping output constant.

Graphically, this condition means the isoquant (constant output curve) is tangent to the isocost line (constant cost line). The slope of the isoquant is the marginal rate of technical substitution ($MRTS_{LK} = MP_L / MP_K$), and the slope of the isocost line is $w/r$. At the optimum, these slopes are equal, so no reallocation of spending can reduce cost without reducing output.

As you vary the target output $\bar{q}$, the set of optimal input bundles traces out the expansion path, which shows how the firm scales its input mix as production grows.

Practical Implications

• If wages rise, the tangency condition tells you the firm should substitute toward capital (and vice versa). How much it substitutes depends on the elasticity of substitution between inputs.
• The condition provides a benchmark for comparing efficiency across firms. A firm violating the condition is spending more than necessary for its output level.
• Technological change that raises $MP_L$ relative to $MP_K$ shifts the optimal mix toward more labor (holding input prices constant), because labor has become more productive per dollar.

Cost Curves: Types and Interpretation

Total and Average Cost Curves

The cost function $C(q)$ gives the minimum total cost of producing each output level $q$. You derive it by plugging the optimal $L^*(q)$ and $K^*(q)$ from the cost minimization problem back into $wL + rK$.

In the short run, total cost splits into two components:

$TC = FC + VC(q)$

• Fixed costs (FC) don't change with output (e.g., rent on a factory, lease payments on equipment). On a graph of TC vs. $q$, FC is the vertical intercept.
• Variable costs (VC) increase with output (e.g., labor, raw materials). The shape of VC reflects the underlying production function and diminishing returns.

From total cost, you get the per-unit cost curves:

• Average total cost: $ATC = \frac{TC}{q}$
• Average variable cost: $AVC = \frac{VC}{q}$
• Average fixed cost: $AFC = \frac{FC}{q}$, which declines continuously as output rises (spreading fixed costs over more units)

The typical U-shape of the ATC curve comes from two opposing forces: declining AFC pulls ATC down at low output, while rising AVC (due to diminishing returns) pushes ATC up at high output. The minimum of ATC occurs where these two effects exactly offset.

Marginal Cost and Relationships

Marginal cost measures the additional cost of producing one more unit:

$MC = \frac{dTC}{dq} = \frac{dVC}{dq}$

MC depends only on variable costs since fixed costs don't change with output.

The relationship between MC and the average cost curves follows a mathematical rule worth internalizing:

• When $MC < ATC$, producing another unit costs less than the current average, so ATC is falling.
• When $MC > ATC$, producing another unit costs more than the current average, so ATC is rising.
• Therefore, MC crosses ATC at its minimum point. The same logic applies to AVC.

This is just like grade point averages: if your next semester GPA (the "marginal") is below your cumulative GPA (the "average"), your cumulative drops.

You can prove this formally by differentiating $ATC = TC/q$ with respect to $q$:

$\frac{d(ATC)}{dq} = \frac{MC - ATC}{q}$

So ATC is falling when $MC < ATC$, rising when $MC > ATC$, and flat (at its minimum) when $MC = ATC$.

In the short run, the MC curve typically falls initially (reflecting increasing marginal returns to the variable input) and then rises (reflecting diminishing marginal returns). This U-shaped MC is what drives the U-shape of AVC and, combined with declining AFC, the U-shape of ATC.

Production and Costs: Relationship

Scale Economies and Returns

The production function directly determines the shape of cost curves. The connection between returns to scale and long-run costs is precise:

• Increasing returns to scale → doubling inputs more than doubles output → cost per unit falls as output grows → economies of scale (declining LRAC). Example: automobile manufacturing, where large-scale assembly lines and specialization reduce per-unit costs.
• Constant returns to scale → doubling inputs exactly doubles output → cost per unit stays flat → constant LRAC.
• Decreasing returns to scale → doubling inputs less than doubles output → cost per unit rises → diseconomies of scale (rising LRAC). Example: large organizations where coordination and management complexity grow faster than output.

A quick note: economies of scale and returns to scale are related but not identical concepts. Returns to scale is a property of the production function (a purely technical relationship). Economies of scale refer to how costs behave as output changes, which can also be affected by input prices changing with scale (e.g., bulk discounts). For most of this course, the distinction won't matter much, but it's worth knowing.

The long-run average cost (LRAC) curve is the envelope of all short-run ATC curves. Each short-run ATC corresponds to a different level of the fixed input (e.g., a different plant size). The LRAC shows the lowest achievable average cost at each output level when the firm can freely choose its plant size. At any given output, the LRAC is tangent to the relevant short-run ATC, and the LRAC is always at or below every short-run ATC.

Technological and Input Factors

• The distinction between short-run and long-run cost curves exists because some inputs are fixed in the short run. Once all inputs become variable, the firm operates on its long-run cost curves. Long-run costs are always weakly below short-run costs at every output level, because more flexibility can never increase minimum cost.
• Technological progress shifts cost curves downward by increasing the marginal products of inputs, meaning less input is needed per unit of output.
• The elasticity of substitution between inputs determines how sensitive costs are to input price changes. With a high elasticity of substitution (inputs are easily swapped), a wage increase won't raise costs much because the firm can shift toward capital. With a low elasticity (think fixed-proportions or Leontief technology), the firm has little room to adjust, and cost increases are harder to avoid.