Why Long-Run Production Choices Differ
A firm may be able to produce a given output using different combinations of workers, equipment, space, or materials. Its preferred combination depends on the production technology and on what those inputs cost. The choice is different when some inputs cannot be changed quickly: a firm may be able to vary labor this month while remaining committed to a factory size chosen earlier.
In production theory, the long run is the planning horizon in which the firm can adjust every factor of production. It does not mean a particular number of months or years. The short run is a horizon in which at least one factor is fixed. A factor is fixed in this sense because the firm cannot adjust its quantity within the period, not because it has no economic cost.
A firm comparing plans must first check that each plan can meet the target. A plan that costs less but produces too little is not a valid alternative. Among plans that can produce the target, the firm compares the costs of all inputs, including inputs that are fixed in the short run but adjustable in the long run. The examples below use simple production benchmarks: each listed input combination is specified to produce at least the stated output.
Setting Up the Choice
Let \(L\) denote labor, \(K\) denote capital services such as equipment use, and \(f(L,K)\) denote the output produced with those inputs under the firm’s technology. Let \(w\) be the price per unit of labor and \(r\) the price per unit of capital services. For a target \(q\), an input plan \((L,K)\) is feasible if it can produce at least \(q\). With constant input prices and no adjustment costs, its input cost is \(wL+rK\).
This expression defines long-run cost for a specified output target: compare the costs of feasible plans in which both inputs can vary, and select the lowest. It does not say that the firm always produces \(q\), nor does it determine the firm’s profit-maximizing output. It answers the narrower question, “What is the least input cost of producing this target?” The cost comparison assumes that the technology and input prices are known, the firm can choose among the stated plans, and any costs included in the comparison are measured on the same basis and for the same period.
Worked Example: Comparing Feasible Plans for a Target
A small packaging firm needs to produce 100 crates in a shift. Its production manager identifies three available plans. Plan A uses 10 labor-hours and 4 machine-hours and produces 100 crates. Plan B uses 8 labor-hours and 5 machine-hours and produces 100 crates. Plan C uses 7 labor-hours and 4 machine-hours but produces only 92 crates. Labor costs $20 per hour and machine use costs $30 per hour. Which plan has the lowest cost among those that meet the target?
Method. First test each plan against the output requirement, then calculate the costs of the feasible plans. This avoids selecting a cheap plan that does not produce enough.
Plan A meets the target. Its cost is \(10\times \$20+4\times \$30=\$200+\$120=\$320\).
Plan B also meets the target. Its cost is \(8\times \$20+5\times \$30=\$160+\$150=\$310\).
Plan C does not meet the target because \(92<100\). Its cost would be \(7\times \$20+4\times \$30=\$140+\$120=\$260\), but it is not a feasible choice for this target.
Comparing the feasible costs, \(\$310<\$320\), so Plan B is the least-cost listed plan. The calculation is checked against the requirement: Plan B produces 100 crates and has the lower cost of the two plans that do so.
Short-Run Constraints and the Long-Run Choice Set
Suppose a firm has a fixed quantity of capital in the short run. It can still vary labor, but its available plans are restricted to those using that quantity of capital. In the long run it can change both labor and capital, so its choice set includes the short-run plans as well as additional plans. The two horizons therefore differ in the alternatives available, not necessarily in the production technology or input prices.
Here \(\bar K\) denotes the capital quantity fixed in the short run. If prices, technology, output target, and cost accounting are the same, the long-run cost-minimizing cost cannot be greater than the short-run cost-minimizing cost. The reason is that the long-run firm can choose the short-run plan if it wants to, while it may also have other plans available. More options cannot raise the minimum cost. This comparison does not promise that the difference will be large: the best long-run plan may use the same input quantities as the short-run plan.
Worked Example: Replacing a Costly Fixed Input
A workshop must make 60 components per day. At its current facility, capital use is fixed at \(K=4\) machine-units for the day. The production schedule shows that this facility needs \(L=12\) labor-hours to reach 60 components. Labor costs $10 per hour and machine use costs $60 per machine-unit per day. For a future operating period, the firm can change both inputs; one available alternative is \(K=2\), \(L=22\), which also produces 60 components. Compare the current short-run plan with that long-run alternative.
Method. Include the cost of both inputs in each plan, even though capital is fixed in the current period. A fixed input still has a cost; it is simply not adjustable in that short-run choice.
The current plan’s labor cost is \(12\times \$10=\$120\). Its capital cost is \(4\times \$60=\$240\). Thus, current total input cost is \(\$120+\$240=\$360\) per day.
The alternative plan’s labor cost is \(22\times \$10=\$220\). Its capital cost is \(2\times \$60=\$120\). Its total input cost is \(\$220+\$120=\$340\) per day.
Both plans produce 60 components, and \(\$340<\$360\). The stated long-run alternative saves \(\$360-\$340=\$20\) per day relative to the current plan. This establishes that the current plan is not least-cost among these two choices; it does not establish that the alternative is the least-cost plan among every possible long-run combination.
Input Prices Can Change the Preferred Plan
A plan that minimizes cost at one set of input prices need not do so at another. If labor becomes relatively expensive, a plan that uses less labor may become more attractive, provided its additional use of other inputs does not outweigh the saving. The production target and technology determine which plans are feasible; input prices determine how costly each feasible plan is.
Worked Example: Re-ranking Plans After a Price Change
A repair depot has three long-run plans that each complete at least 80 service jobs per week: Plan A uses 10 labor-hours and 2 equipment-units, Plan B uses 6 labor-hours and 4 equipment-units, and Plan C uses 4 labor-hours and 6 equipment-units. Initially, labor costs $18 per hour and equipment use costs $15 per unit. Later, the equipment-use price rises to $20 while the labor price stays at $18. Find the least-cost listed plan before and after the price change.
Method. Calculate each plan’s labor cost and equipment cost separately under each price schedule, then compare totals. This avoids carrying an old input price into the new comparison.
At the initial prices, Plan A costs \(10\times \$18+2\times \$15=\$180+\$30=\$210\). Plan B costs \(6\times \$18+4\times \$15=\$108+\$60=\$168\). Plan C costs \(4\times \$18+6\times \$15=\$72+\$90=\$162\). Since \(\$162<\$168<\$210\), Plan C is least costly at the initial prices.
After the equipment-use price rises to $20, Plan A costs \(10\times \$18+2\times \$20=\$180+\$40=\$220\). Plan B costs \(6\times \$18+4\times \$20=\$108+\$80=\$188\). Plan C costs \(4\times \$18+6\times \$20=\$72+\$120=\$192\). Now \(\$188<\$192<\$220\), so Plan B is least costly among the listed plans.
The price change makes the plan using fewer equipment-units more attractive, but Plan A’s labor use is high enough that it still is not least costly. The final ranking is checked by comparing all three recalculated totals, not just the old winner with one alternative.
Common Mistakes and Scope of the Choice
The phrase “long-run choice” can be misread as a claim that the firm changes every input immediately or that it should always replace its current equipment. It instead describes the set of quantities the firm can choose over the relevant planning horizon. A firm may have contracts, installation delays, or other constraints that limit actual adjustment; those constraints should be reflected in the feasible plans and, where relevant, in the costs.
- Ignoring a fixed input’s cost: “Fixed” means fixed in quantity during the short run, not free. Include its cost when comparing total costs on a consistent basis.
- Comparing plans with different output: Check feasibility before ranking costs. A lower-cost plan that misses the target cannot be selected to meet it.
- Treating long-run cost minimization as output choice: The firm’s least-cost plan for a specified target is not, by itself, a decision about which target to produce. That requires a separate comparison of benefits and costs.
- Assuming every input-price change changes the chosen plan: Recalculate the alternatives. A price change can alter costs without changing which plan is least costly.
The central idea is a choice-set comparison: in the long run all factors can vary, whereas in the short run at least one is fixed. For a given target, the firm compares the costs of plans that can actually produce it, using the relevant input prices and constraints.
Check Your Understanding
For each problem, check output feasibility before comparing costs. Treat each stated input cost as applying to the same production period.
- A print shop needs 50 posters. Plan A uses 5 labor-hours and 3 press-hours and produces 50 posters; Plan B uses 7 labor-hours and 1 press-hour and produces 50; Plan C uses 4 labor-hours and 2 press-hours and produces 46. Labor costs $14 per hour and press use costs $20 per hour. Which feasible plan is least costly? (Answer: Plan B, at $118.)
- A short-run plant has capital fixed at \(K=3\); producing 40 units requires 10 labor-hours. Labor costs $8 per hour and capital costs $15 per unit. In the long run, two alternatives are available: \(K=2,L=12\) and \(K=4,L=7\), each producing 40 units. Compare the short-run cost with the least-cost of these stated long-run alternatives. (Answer: Short-run cost $125; long-run alternative \(K=4,L=7\) costs $116.)
- Three plans each produce at least 30 batches: A uses 8 labor-hours and 2 machine-hours, B uses 5 labor-hours and 4 machine-hours, and C uses 3 labor-hours and 6 machine-hours. At labor $12 per hour and machine use $10 per hour, which plan is least costly, and what is its cost? (Answer: Plan C, at $96.)
- Using the plans in question 3, suppose machine use instead costs $16 per hour and labor remains $12 per hour. Recalculate all three costs and identify the least-cost plan. (Answer: A costs $128, B costs $124, and C costs $132; Plan B is least costly.)
- A firm’s short-run feasible plans for a target have costs of $240 and $265. In the long run, it can use either of those plans or a third feasible plan costing $225, with the same technology, prices, and cost accounting. What is the long-run least cost, and why can it not exceed the short-run least cost here? (Answer: $225; the long-run choice set includes both short-run plans and the additional feasible plan.)