Constructing A Profit Function

Introduction

A firm’s revenue is not the same as its profit. Revenue records the value of sales; profit accounts for the cost of producing those sales as well. To calculate profit at a particular output, or express profit as a function of output, first put total revenue and total cost on the same basis and then subtract cost from revenue.

This tutorial develops that calculation for a stated production period. It covers both a firm selling at a fixed price and a firm facing a price that depends on output. The aim is to construct and evaluate the profit function, not to determine which output a firm should choose.

Definition: For a specified period and output level $q$, total revenue $TR(q)$ is the money received from sales, total cost $TC(q)$ is the cost assigned to producing that output, and profit is $$ \pi(q)=TR(q)-TC(q). $$ A positive value is a profit, zero is break-even in the chosen cost measure, and a negative value is a loss.

Set the Basis Before Calculating

Before writing a profit function, identify the period, the output unit, and the cost concept. For example, if $q$ is the number of units produced in one month, then revenue and cost must both be monthly amounts. A monthly revenue amount cannot be compared directly with an annual cost amount unless one is converted to the other’s time basis.

The same consistency is needed for units. If $q$ counts hundreds of units, then a formula written in terms of $q$ uses hundreds, not individual units. Each output value must be substituted using the definition of $q$. A price stated per individual item must also be multiplied by the number of individual items sold.

Total cost may contain fixed costs as well as costs that vary with output. If a stated cost function includes a fixed cost, it is already part of $TC(q)$ and must be subtracted once. If the question asks for economic profit, total cost includes opportunity costs as well as explicit costs. If only explicit costs are included, the resulting measure is accounting profit, not economic profit.

Model assumption: The revenue and cost functions in a calculation refer to the same firm, period, output measure, and currency. Output $q$ is nonnegative. Use the cost definition supplied by the problem; do not add or omit a cost without a stated reason.

Constructing the Function

If the firm sells every unit at a fixed price $p$, then total revenue is price times quantity. If the price per unit varies with output, write that price as $p(q)$; total revenue is then the price at output $q$ multiplied by $q$. In either case, the profit function is formed by subtracting total cost from the resulting total revenue.

TR(q)=p(q)q \qquad\text{and}\qquad \pi(q)=p(q)q-TC(q).

The fixed-price case is $p(q)=p$, so $TR(q)=pq$. In the output-dependent case, $p(q)$ is a price per unit, not total revenue. Multiplying by $q$ is essential to obtain total revenue. These expressions describe sales revenue; they do not by themselves specify the firm’s costs.

The subtraction has a direct accounting justification. For each period, profit is the amount of revenue remaining after the costs assigned to that period’s production have been paid or accounted for. If sales bring in $TR(q)$ dollars and production uses $TC(q)$ dollars of resources, the net amount is $TR(q)-TC(q)$. For economic profit, the cost total also counts the value of resources in their next-best alternative use.

Identify the output variable and period.
Write down what one unit of $q$ means and whether the amounts are, for example, weekly or annual.

Write total revenue.
For a fixed price, use price times quantity. For an output-dependent price, multiply $p(q)$ by $q$.

Subtract total cost.
Use the cost function or cost schedule on the same time and output basis as revenue.

Evaluate and check.
Substitute the specified output into revenue and cost, then calculate revenue minus cost. Check the sign and the units of the result.

Worked Calculations

Worked Example: Fixed Price With a Fixed Cost

A hypothetical workshop sells its entire output at $18 per item during one month. Its monthly total cost, in dollars, is $TC(q)=120+6q+0.5q^2$, where $q$ is the number of items produced and sold. Construct its monthly profit function and calculate profit at $q=8$ and $q=12$.

Method. Because the price is fixed, total revenue is price times quantity: $TR(q)=18q$. Subtract the full stated total cost:

\pi(q)=18q-(120+6q+0.5q^2) =-0.5q^2+12q-120.

At $q=8$, revenue is $TR(8)=18(8)=144$ dollars. Cost is $TC(8)=120+6(8)+0.5(8^2)=120+48+32=200$ dollars. Therefore, $\pi(8)=144-200=-56$ dollars.

At $q=12$, revenue is $TR(12)=18(12)=216$ dollars. Cost is $TC(12)=120+6(12)+0.5(12^2)=120+72+72=264$ dollars. Therefore, $\pi(12)=216-264=-48$ dollars.

Check. Substituting into the simplified function gives $\pi(8)=-0.5(64)+12(8)-120=-32+96-120=-56$, and $\pi(12)=-0.5(144)+144-120=-72+144-120=-48$. Both calculations agree. The common mistake this method avoids is omitting the $120 fixed cost because it does not change with output. It remains part of total cost and therefore part of profit at both output levels.

Worked Example: Revenue When Price Depends on Output

A hypothetical producer faces the inverse demand function $p(q)=50-2q$, with price in dollars per unit and output $q$ in units per week. Weekly total cost is $TC(q)=100+10q+q^2$ dollars. Construct the weekly profit function and evaluate it at $q=5$ and $q=10$. Use only output levels for which the stated price is nonnegative.

Method. The price is not constant, so first multiply the price at output $q$ by the quantity: $TR(q)=(50-2q)q=50q-2q^2$. The price is nonnegative when $50-2q\geq 0$, or $0\leq q\leq25$. Both requested output levels satisfy this condition. Now subtract total cost:

\pi(q)=(50q-2q^2)-(100+10q+q^2) =-100+40q-3q^2.

At $q=5$, the price is $p(5)=50-2(5)=40$ dollars per unit. Thus $TR(5)=40(5)=200$ dollars per week. Cost is $TC(5)=100+10(5)+5^2=100+50+25=175$ dollars per week. Profit is $200-175=25$ dollars per week.

At $q=10$, the price is $p(10)=50-2(10)=30$ dollars per unit. Thus $TR(10)=30(10)=300$ dollars per week. Cost is $TC(10)=100+10(10)+10^2=100+100+100=300$ dollars per week. Profit is $300-300=0$ dollars per week.

Check. The simplified function gives $\pi(5)=-100+40(5)-3(25)=-100+200-75=25$, and $\pi(10)=-100+400-300=0$. A common error is to use $50-2q$ as total revenue. That expression is price per unit; multiplying it by $q$ is necessary before subtracting total cost.

Worked Example: Profit From a Cost and Revenue Schedule

A hypothetical design studio sells licenses over a quarter. The following schedule gives its total revenue and explicit costs at three output levels. The studio’s owner also gives up an alternative use of time worth $100 per quarter, regardless of output. Calculate economic profit at each listed level.

Licenses, $q$	Total revenue	Explicit total cost
0	$0	$300
5	$450	$420
10	$800	$620

Method. Add the $100 opportunity cost to explicit cost to obtain economic total cost, then subtract that amount from total revenue. The total economic costs are $300+$100=$400 at $q=0$, $420+$100=$520 at $q=5$, and $620+$100=$720 at $q=10$.

At $q=0$, economic profit is $0-400=-400$ dollars. At $q=5$, economic profit is $450-520=-70$ dollars. At $q=10$, economic profit is $800-720=80$ dollars.

Check. In every row, economic profit plus economic total cost equals total revenue: $-400+400=0$, $-70+520=450$, and $80+720=800$. The common mistake avoided here is to subtract only explicit costs when the question asks for economic profit. If only those explicit costs were subtracted, the figures would instead be accounting profit: $-300$, $30$, and $180$ dollars, respectively.

Interpreting the Result and Avoiding Errors

The sign of $\pi(q)$ describes the result at the specified output and under the specified cost measure. A positive value means revenue exceeds cost; zero means revenue equals cost; and a negative value means cost exceeds revenue. A zero economic profit does not imply that the firm receives no return at all: opportunity costs are included in economic cost, so zero means revenue covers those costs as well as explicit costs.

When a profit function is simplified algebraically, preserve parentheses around total cost until the subtraction has been applied. For example, if $TC(q)=a+bq+cq^2$, then $TR(q)-TC(q)=TR(q)-a-bq-cq^2$. The minus sign applies to every term in the cost function. Dropping it from one term changes the calculated profit.

Do not confuse price with revenue. If price depends on output, multiply the per-unit price by output to get total revenue.
Do not omit a fixed cost. A fixed cost does not vary with output, but it is still included in total cost and subtracted from revenue.
Do not mix cost concepts. Explicit costs alone give accounting profit; economic profit also deducts opportunity costs.
Do not mix periods or units. Confirm that revenue and cost use the same period, currency, and output unit before subtracting.

A final check is to calculate profit in two ways when possible: subtract total cost from total revenue at the specified output, and substitute that output into the constructed profit function. Agreement helps catch algebra and substitution errors.

Key takeaway: Construct profit on a consistent basis by writing total revenue first, subtracting the relevant total cost, and then evaluating the resulting function at the stated output. Interpret the sign using the cost concept the problem specifies.

Check Your Understanding

For each problem, keep the stated output units and cost measure unchanged.

A hypothetical seller charges $24 per item. Its total cost is $TC(q)=90+8q$ dollars per week. Construct the weekly profit function and find profit at $q=7$. (Answer: $\pi(q)=16q-90$; $\pi(7)=\$22$ per week.)
A producer’s inverse demand function is $p(q)=40-q$ dollars per unit, and its total cost is $TC(q)=80+6q+q^2$ dollars. Construct $\pi(q)$ and evaluate it at $q=5$. (Answer: $\pi(q)=-80+34q-2q^2$; $\pi(5)=\$40$.)
A firm’s total revenue and total cost, both in dollars per month, are $TR(q)=30q$ and $TC(q)=200+12q+2q^2$. Find profit at $q=6$, and state whether it is positive or negative. (Answer: $\pi(6)=180-(200+72+72)=-\$164$ per month, a loss.)
A hypothetical seller has $p(q)=64-2q$ dollars per unit and $TC(q)=150+8q+q^2$ dollars per week. Construct the profit function and evaluate it at $q=12$. (Answer: $\pi(q)=-150+56q-3q^2$; $\pi(12)=\$90$ per week.)
In one quarter, a hypothetical business earns $900 in revenue and incurs $520 in explicit costs. The owner’s opportunity cost is $250 for the quarter. Find both accounting profit and economic profit. (Answer: Accounting profit is $900-520=\$380$; economic profit is $900-(520+250)=\$130$ for the quarter.)

What You'll Learn