Input-Price Changes and Isocost Rotation

Why Input Prices Change the Production Choice

A firm choosing labor and capital must consider both the production target and the prices of those inputs. If labor becomes more expensive relative to capital, a firm may be able to maintain its output while using less labor and more capital. But a changed input mix and a changed total cost are distinct consequences: the first describes quantities, while the second describes expenditure.

This tutorial analyzes how input prices change isocost lines and input choices. The analysis assumes a firm treats input prices as given and, where specified, can adjust both inputs in the long run. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Let $L\geq 0$ denote labor, $K\geq 0$ capital, $w>0$ the price of labor per unit, and $r>0$ the price of capital per unit. The total cost of an input bundle is $C=wL+rK$. An isocost line is the set of bundles that have the same total cost $C$. Written with $K$ on the vertical axis, its equation is $K=C/r-(w/r)L$, so its slope is $-w/r$, its labor-axis intercept is $C/w$, and its capital-axis intercept is $C/r$.

If the firm holds $C$ and $r$ fixed while $w$ changes, the capital-axis intercept stays at $C/r$, while the slope and labor-axis intercept change. A higher labor price makes the line steeper and reduces its labor-axis intercept; a lower labor price makes it flatter and increases that intercept. This is the rotation of an isocost line. If instead the firm is minimizing expenditure for a fixed output target, the cost $C$ is not held fixed: the new minimum-cost isocost may have a different intercept as well as a different slope.

Price Changes and Cost-Minimizing Choices

For a fixed output target, the firm seeks the lowest-cost bundle on or above the target isoquant. In the standard smooth, convex case, an interior optimum occurs where the isoquant is tangent to the lowest attainable isocost line. Let $MP_L$ and $MP_K$ be the marginal products of labor and capital. The marginal rate of technical substitution, stated as a positive magnitude, is $MP_L/MP_K$: the amount of capital the firm can give up when it adds a unit of labor while holding output constant.

Condition for an interior tangency: If the production function is differentiable, both inputs are used in positive quantities, and the optimum is an interior tangency, then $MP_L/MP_K=w/r$. This condition equates the rate at which the technology permits substitution to the relative input price. It does not by itself describe corner solutions or technologies with fixed input proportions.

The condition explains why a relative price change can alter the mix. When labor becomes more expensive relative to capital, $w/r$ rises. For a smooth technology with a strictly convex isoquant, the firm can generally reduce labor relative to capital while maintaining output. The exact quantities depend on the production function and output target; the price ratio alone is not enough to calculate them.

Why the Tangency Condition Holds

At a smooth interior cost minimum, a small change along the isoquant must not reduce cost. For a change that preserves output, the isoquant’s slope is $dK/dL=-MP_L/MP_K$. Along an isocost line, keeping cost constant gives $dC=w\,dL+r\,dK=0$, and therefore $dK/dL=-w/r$. At tangency the slopes are equal, so $-MP_L/MP_K=-w/r$, which yields the stated condition.

Equivalently, a cost-minimizing firm compares marginal product per dollar. If $MP_L/w$ were greater than $MP_K/r$, shifting a small amount of expenditure from capital to labor would increase output at the same cost. The reverse shift would help if $MP_K/r$ were greater. At an interior optimum neither direction improves the choice, so the two ratios are equal. This reasoning relies on an interior choice; a boundary optimum need not satisfy equality.

Worked Changes in Input Prices

Worked Example: A Higher Labor Price at a Fixed Output Target

A firm must produce $Q=12$ units using $Q=\sqrt{LK}$. Initially $w=\$2$ and $r=\$2$. Then the labor price rises to $w=\$4$, while $r=\$2$. Find the cost-minimizing bundles and costs before and after the change. Assume both inputs are continuously adjustable and the optimum is interior.

Method. For this production function, $MP_L=\tfrac12\sqrt{K/L}$ and $MP_K=\tfrac12\sqrt{L/K}$, so $MP_L/MP_K=K/L$. At an interior cost minimum, set $K/L=w/r$, and combine it with the output requirement $LK=144$.

Before the price change. The relative price is $w/r=2/2=1$, so $K/L=1$, or $K=L$. The output condition gives $L^2=144$. Since input quantities are nonnegative, $L=12$ and $K=12$. The cost is $C=2(12)+2(12)=\$48$.

After the price change. Now $w/r=4/2=2$, so $K=2L$. Substitution into $LK=144$ gives $L(2L)=144$, hence $L^2=72$, $L=6\sqrt{2}\approx8.4853$, and $K=12\sqrt{2}\approx16.9706$. The new cost is $C=4(6\sqrt{2})+2(12\sqrt{2})=48\sqrt{2}\approx\$67.88$, rounded to the nearest cent.

Check and interpretation. The new bundle satisfies $LK=(6\sqrt{2})(12\sqrt{2})=144$ and $K/L=2$, as required. Labor use falls, capital use rises, and total cost rises by approximately $\$67.88-\$48=\$19.88$. A common mistake is to use $L=6,\ K=24$: although that bundle produces 12, its ratio $K/L=4$ does not equal $w/r=2$, so it is not the cost-minimizing bundle.

Worked Example: A Lower Labor Price with a Fixed Budget

A firm has a budget of $\$90$ and produces $Q=\sqrt{LK}$. Initially $w=\$3$ and $r=\$3$. The labor price falls to $\$1$, while the budget and capital price remain fixed. Find the output-maximizing bundles and outputs. Assume the firm can use fractional units of either input and the optimum is interior.

Method. With a fixed budget, the firm maximizes output subject to $wL+rK=90$. At an interior optimum, the marginal product per dollar is equal for both inputs. For $Q=\sqrt{LK}$, this implies $K/L=w/r$.

Before the price change. The ratio is $K/L=3/3=1$, so $K=L$. The budget gives $3L+3L=90$, or $L=15$, and therefore $K=15$. Output is $Q=\sqrt{15\cdot15}=15$. The expenditure check is $3(15)+3(15)=\$90$.

After the price change. Now $K/L=1/3$, so $K=L/3$. The budget equation becomes $L+3(L/3)=90$, or $2L=90$. Thus $L=45$ and $K=15$. Output is $Q=\sqrt{45\cdot15}=\sqrt{675}=15\sqrt{3}\approx25.9808$. The expenditure check is $1(45)+3(15)=\$90$.

Check and interpretation. The new bundle has the required ratio $15/45=1/3$ and uses the entire budget. The fall in labor’s price permits more labor to be purchased, while capital use remains 15 in this example; output rises from 15 to about 25.9808. Do not treat this as a fixed-output cost comparison: here the budget is fixed, and output changes.

Worked Example: A Price Change with Fixed-Proportions Technology

A firm produces according to $Q=\min(L,K/2)$ and must produce $Q=8$. Initially, $w=\$2$ and $r=\$1$. The labor price then rises to $\$5$, with the capital price unchanged. Find the required cost-minimizing bundle and total cost in each case.

Method. This is a fixed-proportions technology: one unit of output requires one unit of labor and two units of capital. To produce at least 8, the firm must have $L\geq8$ and $K\geq16$. Because both input prices are positive, using more than these required amounts cannot lower cost. The efficient bundle is therefore $L=8,\ K=16$ at either set of prices.

Before the price change. Cost is $C=2(8)+1(16)=\$32$.

After the price change. The bundle remains $L=8,\ K=16$, but cost is $C=5(8)+1(16)=\$56$.

Check and interpretation. At either bundle, $Q=\min(8,16/2)=8$. The price rise increases total cost by $\$24$, with no substitution toward capital: the production recipe does not allow the firm to replace labor with capital. The isocost slope changes from $-2$ to $-5$, but the fixed-proportions isoquant’s kink remains the relevant input choice.

Input Substitution Is Not the Same as Total Cost

For a fixed output target, minimum cost is weakly increasing when an input price rises, holding all other prices and the production technology fixed. To see why, let the labor price rise from $w$ to $w'\geq w$, and consider any feasible bundle $(L,K)$. Since $L\geq0$, its new cost satisfies $w'L+rK\geq wL+rK$. Every feasible bundle therefore costs at least as much at the new prices as at the old prices, so the minimum over feasible bundles cannot be lower. A price decrease gives the corresponding weak inequality in the other direction.

This cost result does not say how much of each input the firm will use. Under a smooth, strictly convex technology with an interior solution, a higher relative price of labor typically leads the firm to substitute away from labor at a fixed output target. With fixed-proportions technology, the quantities may not change at all. At a corner, or with a nonstandard technology, an interior tangency calculation may not apply. The correct procedure is to identify the technology and check whether the proposed optimum is interior before using the tangency condition.

Common Mistakes to Avoid

Holding the wrong quantity fixed. A fixed-output problem holds the required output constant; a fixed-budget problem holds expenditure constant. Do not use the answer to one as if it solved the other.
Reversing the input ratio. With the convention $MP_L/MP_K=w/r$, the square-root technology implies $K/L=w/r$, not $L/K=w/r$. Check the resulting bundle against the condition.
Assuming every technology allows substitution. A steeper isocost does not guarantee a changed input mix. Fixed proportions can make the required mix unchanged even when relative prices change.
Confusing the rotated line with the new minimum-cost line. The isocost through a fixed budget or fixed cost rotates when one price changes. For a fixed-output cost-minimization problem, the minimum cost itself may change, so its isocost need not share the old intercept.

Key takeaway: A change in one input price changes the slope of the isocost line and can change the cost-minimizing input mix. Under standard interior conditions, the firm equates marginal product per dollar across inputs. Always keep the output target or budget clear, verify the resulting bundle, and compare total cost separately from input quantities.

Check Your Understanding

For each problem, identify what is held fixed before calculating. Assume nonnegative inputs and the stated technology applies.

A firm must produce $Q=10$ using $Q=\sqrt{LK}$, with $w=\$1$ and $r=\$4$. Using an interior cost-minimizing choice, find $L$, $K$, and total cost. (Answer: $L=20,\ K=5,\ C=\$40$.)
A firm must produce $Q=6$ using $Q=\sqrt{LK}$. Initially $w=\$2,\ r=\$1$; then $w$ rises to $\$8$, with $r$ unchanged. Find the new cost-minimizing bundle and its cost. (Answer: $L=3/\sqrt{2}\approx2.1213,\ K=12\sqrt{2}\approx16.9706,\ C=24\sqrt{2}\approx\$33.94$.)
A firm has a $\$120$ budget and produces $Q=\sqrt{LK}$. Its input prices are $w=\$2$ and $r=\$6$. Find the output-maximizing bundle and output, assuming an interior optimum. (Answer: $L=30,\ K=10,\ Q=10\sqrt{3}\approx17.3205$.)
A firm must produce $Q=5$ using $Q=\min(L,K/3)$. Prices change from $w=\$1,\ r=\$2$ to $w=\$4,\ r=\$2$. Find the required bundle and both costs. (Answer: $L=5,\ K=15$ in both cases; costs are $\$35$ initially and $\$50$ afterward.)
For a fixed output target and positive labor use, explain why raising $w$ while holding $r$ fixed cannot reduce the minimum attainable cost, even if the optimal input mix changes. (Answer: Every feasible bundle’s cost weakly rises because $w'L+rK\geq wL+rK$ when $w'\geq w$ and $L\geq0$; minimizing over the same feasible set cannot produce a lower minimum.)

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