Tracing a Change in Monopoly Conditions
A change in demand or production cost can alter a monopolist’s best output, the price consumers pay, and the firm’s profit. These outcomes are connected but are not interchangeable: the firm first chooses output by comparing marginal revenue with marginal cost, then reads the price from the demand curve at that output. A change in one condition can therefore affect price and output in different directions.
The central method is to solve the firm’s choice under the original conditions, solve it again under the changed conditions, and compare the results. This tutorial focuses on a single-price monopolist and on demand and cost shifts that are specified clearly enough to calculate. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
A Linear-Demand Benchmark
A useful benchmark is inverse demand \(P=a-bQ\), where \(P\) is price, \(Q\) is output, \(a>0\) is the demand intercept, and \(b>0\) is the absolute slope. Suppose total cost is \(C(Q)=F+cQ\), where \(F\) is a fixed cost and \(c\geq 0\) is constant marginal cost. For the benchmark calculations below, assume there is no avoidable fixed cost that makes shutdown preferable; a fixed cost included in profit is treated as unavoidable over the decision period.
Total revenue is price times output. Differentiating it with respect to output gives marginal revenue:
The marginal-revenue curve has the same vertical intercept as demand and twice its absolute slope. This follows directly from differentiating total revenue, not from assuming the firm charges marginal revenue as its price. The price the firm charges at its chosen output is always read from demand.
The interior solution requires \(a>c\), so it gives positive output. If \(a\leq c\), marginal revenue at zero output is no greater than marginal cost, and under this linear-demand, constant-cost setup the best output is zero. If a fixed cost can be avoided by shutting down, the firm must also compare the profit from operating with the shutdown payoff; the interior condition by itself does not settle that decision.
Worked Example: A Parallel Increase in Demand
A fictional single-price producer faces inverse demand \(P=100-2Q\) and has total cost \(C(Q)=20Q\). Demand then shifts to \(P=130-2Q\), while cost is unchanged. Calculate and compare output, price, and profit before and after the shift. Use the marginal condition and identify a mistake the method prevents.
Method. Derive marginal revenue from each demand curve, set it equal to marginal cost, and use the relevant demand curve to find price. Here \(MC=20\), and profit is revenue less variable cost.
Before the shift: Demand is \(P=100-2Q\), so \(TR=100Q-2Q^2\) and \(MR=100-4Q\). Set \(MR=MC\):
At this output, price is \(P=100-2(20)=60\). Revenue is \(60(20)=1{,}200\), cost is \(20(20)=400\), and profit is \(1{,}200-400=800\).
After the shift: Demand is \(P=130-2Q\), so \(TR=130Q-2Q^2\) and \(MR=130-4Q\). Then:
The new price is \(P=130-2(27.5)=75\). Revenue is \(75(27.5)=2{,}062.5\), cost is \(20(27.5)=550\), and profit is \(2{,}062.5-550=1{,}512.5\). The shift raises output from 20 to 27.5, price from 60 to 75, and profit from 800 to 1,512.5. Checking the marginal condition gives \(MR=130-4(27.5)=20=MC\).
Common mistake avoided: Treating the demand intercept increase of 30 as an equal increase in the chosen price would predict a price of 90. The firm expands output, so its new price must be evaluated at \(Q=27.5\) on the shifted demand curve; it is 75, not 90.
What a Single Shift Tells You
For the linear benchmark, the formulas make the effects of certain isolated shifts explicit. If only the demand intercept \(a\) rises, with \(b\) and \(c\) fixed, then \(Q^*=(a-c)/(2b)\) rises and \(P^*=(a+c)/2\) rises. With no fixed cost, profit also rises because \((a-c)^2/(4b)\) increases when \(a>c\). These conclusions depend on the stated linear-demand and constant-marginal-cost assumptions; they should not be applied automatically to every demand shift or cost function.
If only constant marginal cost \(c\) rises, with \(a\) and \(b\) fixed and the firm still producing, the same formulas imply that output falls, price rises, and profit falls. In particular, the price increase is half the marginal-cost increase in this benchmark, not the full increase. With more general demand and cost curves, the exact size of pass-through and the directions of some effects need to be worked out from the new \(MR=MC\) condition.
Worked Example: A Rise in Marginal Cost
A hypothetical service firm faces inverse demand \(P=100-Q\). Its total cost initially is \(C(Q)=100+10Q\), and then marginal cost rises so that total cost becomes \(C_{\mathrm{new}}(Q)=100+22Q\). The fixed cost of 100 is unavoidable in the decision period. Find output, price, and profit in both cases, and check the marginal conditions.
Method. The inverse demand has \(TR=100Q-Q^2\), so \(MR=100-2Q\). The marginal costs are 10 before the shift and 22 afterward. The unavoidable fixed cost affects profit but not the output choice.
Before the shift:
Revenue is \(55(45)=2{,}475\). Total cost is \(100+10(45)=550\), so profit is \(2{,}475-550=1{,}925\). Check: \(MR=100-2(45)=10=MC\).
After the shift:
Revenue is \(61(39)=2{,}379\). Total cost is \(100+22(39)=958\), so profit is \(2{,}379-958=1{,}421\). Check: \(MR=100-2(39)=22=MC\). Output falls by 6 units, price rises by 6, and profit falls by 504.
Common mistake avoided: Adding the full marginal-cost increase of 12 to the old price would give 67. That is not the new price. The new price is 61, calculated from demand at the new output. In this linear benchmark, price rises by half the cost increase because \(P^*=(a+c)/2\).
Simultaneous Shifts and the Need to Recalculate
When demand and cost change together, it is unsafe to add up separate verbal predictions without checking the new optimum. For example, in the linear benchmark, if the demand intercept rises by the same amount that constant marginal cost rises, output is unchanged: the numerator \(a-c\) in \(Q^*=(a-c)/(2b)\) stays the same. Price rises by the full common amount because both \(a\) and \(c\) enter \(P^*=(a+c)/2\). With no fixed cost, profit is unchanged because \(a-c\) is unchanged.
This is a result for the specified parallel demand shift and constant marginal cost, not a general rule that simultaneous shifts leave output or profit unaffected. A change in the demand slope, a nonlinear cost change, or a different size of the shifts can produce other outcomes. The reliable procedure is to compute the new marginal revenue, compute the new marginal cost, solve for output, and then evaluate price and profit.
Worked Example: Equal Intercept and Marginal-Cost Increases
A fictional firm initially faces \(P=100-Q\) and has cost \(C(Q)=20Q\). A change in its market raises the demand curve to \(P=120-Q\), while production costs change to \(C_{\mathrm{new}}(Q)=40Q\). Calculate the two outcomes. Explain why output and profit do not change even though price does.
Method. Solve \(MR=MC\) separately under each set of conditions, then calculate price from demand and profit from revenue less cost. There is no fixed cost.
Initially: \(TR=100Q-Q^2\), so \(MR=100-2Q\). Equating this to \(MC=20\) gives \(100-2Q=20\), hence \(Q^*=40\). Price is \(100-40=60\), revenue is \(60(40)=2{,}400\), cost is \(20(40)=800\), and profit is \(1{,}600\). The check is \(MR=100-2(40)=20=MC\).
After both shifts: \(TR=120Q-Q^2\), so \(MR=120-2Q\). Equating this to \(MC=40\) gives \(120-2Q=40\), hence \(Q^*=40\). The new price is \(120-40=80\), revenue is \(80(40)=3{,}200\), cost is \(40(40)=1{,}600\), and profit is \(3{,}200-1{,}600=1{,}600\). The check is \(MR=120-2(40)=40=MC\).
Output is unchanged because both the demand intercept and marginal cost rise by 20, leaving \(a-c\) unchanged. Price rises by 20, while the per-unit margin \(P-MC\) remains \(80-40=60-20=40\); with output fixed at 40, total profit is unchanged.
Common mistake avoided: Concluding that the demand increase alone must raise output overlooks the simultaneous cost change. The correct comparison uses the new marginal revenue and new marginal cost together.
Profit, Fixed Costs, and a Zero-Output Check
A change in fixed cost is different from a change in marginal cost. If \(F\) rises while the firm remains in operation, marginal cost is unchanged, so the \(MR=MC\) condition gives the same output and price. Profit falls by the increase in \(F\). If the fixed cost is avoidable, however, the firm must consider whether operating is better than shutting down; a fixed cost can affect that operating decision even though it does not alter the conditional best output when the firm operates.
It is also important to describe profit changes precisely when costs rise. If costs increase at every positive output but the cost at zero output is unchanged, profit is lower at each positive output, but it need not be lower at zero output. The optimized profit cannot increase: at every positive output the new profit is no greater than before, and at zero it is unchanged under this assumption. It may remain exactly the same if the firm’s best choice after the change is zero output and zero-output profit is unchanged.
This distinction prevents an overly strong conclusion such as “profit must strictly fall after any cost increase.” A cost increase makes each positive-output option less attractive, but strict decline in optimized profit depends on the firm’s available choices and whether it continues to select a positive output.
Identify the new demand curve, total cost, marginal revenue, and marginal cost. Keep fixed costs separate from marginal costs.
For an interior optimum, solve \(MR(Q)=MC(Q)\) and check that the solution maximizes profit and is feasible. Consider zero output or shutdown when applicable.
Substitute the chosen output into demand to get price. Then calculate profit as \(P(Q)Q-C(Q)\), including fixed costs that apply.
Report the change in output, price, and profit separately. Check the marginal condition at an interior solution and verify the arithmetic in total revenue and cost.
Common Exam Traps
- Using demand price in the marginal condition. A monopolist chooses output using \(MR=MC\), not \(P=MC\). Price is found afterward from demand at the selected quantity.
- Assuming a price change equals the cost change. Pass-through depends on the demand and cost conditions. In the linear benchmark, a change in constant marginal cost changes price by half as much when demand is fixed.
- Ignoring a simultaneous shift. An increase in demand does not determine the outcome by itself if marginal cost changes at the same time. Solve the new problem with both changes in place.
- Forgetting fixed cost in profit. Fixed cost does not change marginal cost, but it must be included when calculating total profit. If it is avoidable, shutdown may also need to be considered.
- Claiming profit must fall strictly. A cost increase at positive outputs lowers profit there, but zero-output profit may be unchanged. The maximum profit cannot rise, yet it need not fall strictly if zero output is optimal.
Check Your Understanding
For each problem, calculate the requested outcomes using the stated conditions. Assume an interior optimum unless the problem indicates otherwise.
- A firm faces inverse demand \(P=90-2Q\) and has cost \(C(Q)=10Q\). Demand shifts to \(P=110-2Q\), with cost unchanged. Find output, price, and profit before and after the shift. (Answer: Before: \(Q=20\), \(P=50\), profit \(=800\); after: \(Q=25\), \(P=60\), profit \(=1{,}250\).)
- A firm faces \(P=140-Q\) and initially has \(C(Q)=20Q\). Marginal cost rises so that cost becomes \(C_{\mathrm{new}}(Q)=32Q\). Find output, price, and profit in both cases. (Answer: Before: \(Q=60\), \(P=80\), profit \(=3{,}600\); after: \(Q=54\), \(P=86\), profit \(=2{,}916\).)
- A firm initially faces \(P=96-2Q\) with marginal cost 16. The demand intercept and marginal cost each rise by 12, giving new demand \(P=108-2Q\) and new marginal cost 28. Find output, price, and profit before and after, assuming no fixed costs. (Answer: Before: \(Q=20\), \(P=56\), profit \(=800\); after: \(Q=20\), \(P=68\), profit \(=800\).)
- A firm faces \(P=70-Q\) and has total cost \(C(Q)=10Q+0.5Q^2\). Demand then shifts to \(P=82-Q\), with cost unchanged. Find output, price, and profit in both cases. (Answer: Before: \(Q=20\), \(P=50\), profit \(=600\); after: \(Q=24\), \(P=58\), profit \(=864\).)
- A firm initially faces \(P=120-Q\) and has constant marginal cost 20. It later faces \(P=150-2Q\) and has constant marginal cost 30. With no fixed costs, calculate each outcome and state whether output, price, and profit rose or fell. (Answer: Before: \(Q=50\), \(P=70\), profit \(=2{,}500\); after: \(Q=30\), \(P=90\), profit \(=1{,}800\); output and profit fell, while price rose.)