How Variable Input Prices Change Short-Run Costs

Why Variable Input Prices Change Costs

A firm can face a different price for a variable input even when its production capacity and planned output have not changed. For example, a wage increase changes the cost of the labor hours used to make a given quantity of output. To isolate this price effect, hold output and the quantities of inputs required at that output fixed. This is a short-run comparison: it does not assume that the firm changes its production method or input mix in response to the new price.

The key distinction is between how much of an input is required and what each unit of that input costs. If the requirement is unchanged, a change in the input’s price changes variable cost in proportion to the quantity of that input used. Fixed cost does not change merely because a variable input becomes more or less expensive.

Definition: At a specified output, let $F$ be fixed cost, $x_j$ the quantity used of variable input $j$, and $p_j$ its price per unit. With input quantities held fixed, variable cost is $VC=\sum_j p_jx_j$, and total cost is $TC=F+VC$. Average variable cost and average total cost are $AVC=VC/q$ and $ATC=TC/q$, respectively, for positive output $q$. For a discrete output change, marginal cost is the change in total cost divided by the change in output.

These comparisons assume the same production technology, output level, input quantities, and fixed cost before and after the price change. If several variable inputs are used, the effect of changing one input’s price can be calculated separately, then added to the effects of any other changed input prices. The earlier material on fixed, variable, and total cost provides the basic cost categories; here the emphasis is on how input prices alter them.

How the Price Change Passes Through Cost Measures

For one variable input, write its quantity requirement at output $q$ as $x(q)$, and its price per unit as $p$. Then variable cost is $VC=px(q)$. If the price changes from $p_0$ to $p_1$, while $q$ and $x(q)$ remain unchanged, the cost difference is:

\Delta VC=(p_1-p_0)x(q)

Because fixed cost is unchanged, $\Delta TC=\Delta VC$. At the same positive output, dividing these changes by $q$ gives $\Delta AVC=\Delta ATC=\Delta VC/q$. This equality concerns the change in average cost: the levels of $AVC$ and $ATC$ still differ by average fixed cost, $F/q$.

For discrete marginal cost, compare two output levels $q_1$ and $q_2$, with corresponding input requirements $x_1$ and $x_2$. If the input price is $p$, the input’s contribution to the cost of moving between those outputs is $p(x_2-x_1)$. A change in that price therefore changes discrete marginal cost according to the input quantity needed for the additional output.

MC=\frac{TC(q_2)-TC(q_1)}{q_2-q_1} \qquad\text{and}\qquad \Delta MC=\frac{(p_1-p_0)(x_2-x_1)}{q_2-q_1}

The second expression applies when only this input’s price changes, fixed cost is unchanged, and all input requirements at both output levels stay the same. It is not a claim that marginal cost changes by the same percentage as the input price; the relevant quantity is the input used for the additional output.

Worked Cost Calculations

Worked Example: A Higher Wage and a Full Cost Comparison

A workshop has fixed cost of $300 per day and uses labor as its only variable input. At output of 100 units, it requires 20 labor-hours. The wage rises from $15 to $18 per labor-hour. At outputs of 80, 100, and 120 units, labor requirements are 12, 20, and 31 hours, respectively. Assume all these requirements and fixed cost remain unchanged. Calculate the before-and-after variable cost, total cost, average variable cost, average total cost at 100 units, and discrete marginal costs over the two adjacent output intervals.

Method. Multiply labor-hours by the wage to obtain variable cost; add the unchanged fixed cost for total cost; divide by output for the averages. For each marginal cost, use the change in labor cost divided by the change in output. This method avoids treating the wage increase as a change in fixed cost.

At 100 units, before the wage increase: $VC_0=20\times \$15=\$300$. Thus $TC_0=\$300+\$300=\$600$, $AVC_0=\$300/100=\$3.00$ per unit, and $ATC_0=\$600/100=\$6.00$ per unit.

At 100 units, after the wage increase: $VC_1=20\times \$18=\$360$. Thus $TC_1=\$300+\$360=\$660$, $AVC_1=\$360/100=\$3.60$ per unit, and $ATC_1=\$660/100=\$6.60$ per unit. Variable cost, total cost, and each average cost rise by $60, $60, and $0.60 per unit for the respective measures.

Marginal cost from 80 to 100 units: output rises by $100-80=20$ units, and labor rises by $20-12=8$ hours. Before the wage change, the added labor cost is $8\times \$15=\$120$, so $MC_0=\$120/20=\$6.00$ per unit. Afterward, it is $8\times \$18=\$144$, so $MC_1=\$144/20=\$7.20$ per unit.

Marginal cost from 100 to 120 units: output rises by 20 units, and labor rises by $31-20=11$ hours. Before the wage change, $MC_0=(11\times \$15)/20=\$165/20=\$8.25$ per unit. Afterward, $MC_1=(11\times \$18)/20=\$198/20=\$9.90$ per unit.

Check: at 100 units, total cost equals fixed cost plus variable cost in both cases: $\$300+\$300=\$600$ and $\$300+\$360=\$660$. In each interval, the marginal-cost increase equals the extra labor hours times the $3 wage increase, divided by 20 output units.

Worked Example: A Price Change for One of Two Inputs

A small producer has fixed cost of $500. To make 250 units, it uses 30 labor-hours at $20 per hour and 50 units of packaging material. Packaging initially costs $4 per unit, then rises to $5 per unit. The producer’s input requirements are unchanged. At output of 200 units, it uses 24 labor-hours and 38 units of packaging. Calculate the cost measures at 250 units and the discrete marginal cost from 200 to 250 units before and after the packaging price change.

Method. Calculate each input’s cost separately and sum them. Then add fixed cost or divide by output as required. This avoids applying the new price to labor, whose price did not change.

At 250 units, before the packaging price change: labor cost is $30\times \$20=\$600$, and packaging cost is $50\times \$4=\$200$. Therefore $VC_0=\$600+\$200=\$800$, $TC_0=\$500+\$800=\$1{,}300$, $AVC_0=\$800/250=\$3.20$ per unit, and $ATC_0=\$1{,}300/250=\$5.20$ per unit.

At 250 units, after the packaging price change: labor cost remains $600, while packaging cost is $50\times \$5=\$250$. Thus $VC_1=\$600+\$250=\$850$, $TC_1=\$500+\$850=\$1{,}350$, $AVC_1=\$850/250=\$3.40$ per unit, and $ATC_1=\$1{,}350/250=\$5.40$ per unit.

Marginal cost from 200 to 250 units: output increases by $250-200=50$ units. Labor increases by $30-24=6$ hours, while packaging increases by $50-38=12$ units. Before the price change, the added variable cost is $6\times \$20+12\times \$4=\$120+\$48=\$168$, so $MC_0=\$168/50=\$3.36$ per unit. Afterward, the added variable cost is $6\times \$20+12\times \$5=\$120+\$60=\$180$, so $MC_1=\$180/50=\$3.60$ per unit.

Check: the $50 increase in variable cost at 250 units is exactly $50$ packaging units times the $1 increase in packaging price. For marginal cost, the increase is $12\times \$1/50=\$0.24$ per output unit, matching $\$3.60-\$3.36$.

Why the Cost Relationships Hold

The formulas follow directly from the definition of input cost. With one variable input, the cost of using $x(q)$ units at price $p$ is $px(q)$. At the same output, the input requirement is the same before and after the price change. Subtracting the two variable costs therefore gives:

$$ VC_1-VC_0=p_1x(q)-p_0x(q)=(p_1-p_0)x(q) $$

Adding the same fixed cost $F$ to each variable-cost total does not change their difference: $(F+VC_1)-(F+VC_0)=VC_1-VC_0$. Dividing each total by the same positive output $q$ gives the average-cost changes: $(VC_1-VC_0)/q$ for AVC, and $((F+VC_1)-(F+VC_0))/q$ for ATC. Both equal $(p_1-p_0)x(q)/q$.

For discrete marginal cost, the variable input used to produce the additional output is $x(q_2)-x(q_1)$. Its cost at price $p$ is $p[x(q_2)-x(q_1)]$. The fixed-cost term cancels when total costs at the two output levels are subtracted. Dividing by $q_2-q_1$ gives the input’s contribution to marginal cost. Replacing $p_0$ with $p_1$ changes that contribution by $(p_1-p_0)[x(q_2)-x(q_1)]/(q_2-q_1)$, as stated above.

Interpreting the Change and Avoiding Traps

A price change does not generally produce the same percentage change in total cost. If variable cost is only part of total cost, fixed cost cushions the percentage movement in total cost. For instance, a 20% rise in a variable input price raises variable cost by 20% only when the relevant input quantity is unchanged and that is the only variable cost affected. With positive fixed cost, total cost rises by a smaller percentage than 20% in that comparison.

Common mistake: Suppose variable cost is $400 and fixed cost is $600, and the only variable input price rises by 20% while input use stays fixed. A wrong calculation applies 20% to total cost and reports a $200 increase: $0.20\times(\$400+\$600)=\$200$. The correction applies the price increase to variable cost: $0.20\times \$400=\$80$. Total cost moves from $1,000 to $1,080, an 8% increase, not 20%.

Another trap is to add the dollar change in an input’s unit price directly to AVC, ATC, or MC. The price increase must first be multiplied by the appropriate input quantity: total input use for a cost level, or the additional input use over an output interval for discrete marginal cost. Also check that the comparison holds output fixed when calculating average costs. If output or required input quantities change, the price effect is no longer the only source of the cost difference.

Key takeaway: With output and input requirements fixed, a variable input’s price change affects variable cost by the price change times the quantity used. Total cost changes by the same dollar amount when fixed cost is unchanged; average-cost changes divide that amount by output. For discrete marginal cost, use the input required for the additional output, not the input used for all output.

Check Your Understanding

For each problem, hold output, input requirements, and fixed cost constant across the stated price comparison unless the question gives two output levels.

A workshop has fixed cost of $240. At 80 units of output it uses 12 labor-hours. The wage rises from $10 to $12 per hour. Find variable cost, total cost, AVC, and ATC before and after the change. (Answer: $VC=\$120$ to $\$144$; $TC=\$360$ to $\$384$; $AVC=\$1.50$ to $\$1.80$; $ATC=\$4.50$ to $\$4.80$ per unit.)
At 100 units, a producer uses 10 labor-hours at $12 per hour and 25 material units. Fixed cost is $300. Material price rises from $3 to $4 per unit. Calculate total variable cost, total cost, AVC, and ATC before and after. (Answer: $VC=\$195$ to $\$220$; $TC=\$495$ to $\$520$; $AVC=\$1.95$ to $\$2.20$; $ATC=\$4.95$ to $\$5.20$ per unit.)
A firm’s fixed cost is $200. It uses 7 labor-hours at 40 units of output and 10 labor-hours at 50 units. The wage rises from $16 to $20 per hour. Find AVC and ATC at 50 units and discrete MC from 40 to 50 units, before and after. (Answer: at 50 units, $AVC=\$3.20$ to $\$4.00$ and $ATC=\$7.20$ to $\$8.00$ per unit; discrete $MC=\$4.80$ to $\$6.00$ per unit.)
At output of 200 units, fixed cost is $600 and variable cost is $200. The only variable input price rises by 25%, with input use unchanged. Find the new total cost and its percentage increase from the original total cost. (Answer: new total cost is $\$850$; the increase is $\$50/\$800=6.25\%$.)
From 90 to 120 units of output, labor use rises from 6 to 9 hours at $15 per hour, and material use rises from 20 to 26 units. Material price rises from $4 to $5 per unit. At 120 units, fixed cost is $180. Calculate discrete marginal cost before and after the change, and total cost at 120 units before and after. (Answer: discrete $MC=\$2.30$ to $\$2.50$ per unit; total cost at 120 units is $\$419$ before and $\$445$ after.)

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