Source: Exam 1 Objectives (Section 2) | Varghese, Texas A&M
Tags: marginal analysis, MR equals MC, profit maximisation, revenue function, cost function, marginal revenue, marginal cost, sensitivity analysis, pure selling, Panera Bread demand estimation, ECON 323, microeconomics
This section is about how firms find the price and quantity that maximise profit. The central insight is the MR = MC condition: keep producing as long as the revenue from one more unit exceeds the cost of producing it. You need to find this optimum using algebra, tables, and graphs, and understand what happens when costs shift.
Marginal revenue (MR)
The additional revenue earned from selling one more unit of output. Found by taking the derivative of the total revenue function with respect to quantity.
Marginal cost (MC)
The additional cost incurred from producing one more unit of output. Found by taking the derivative of the total cost function with respect to quantity.
Marginal profit
The additional profit from one more unit of output. Equal to MR − MC. At the profit-maximising quantity, marginal profit equals zero.
Profit maximisation condition (MR = MC)
A firm maximises profit at the quantity where marginal revenue equals marginal cost. If MR > MC, the firm should produce more. If MR < MC, it should produce less.
Revenue function
Total revenue as a function of quantity: R(Q) = P(Q) × Q. If the inverse demand is P = a − bQ, then R(Q) = aQ − bQ².
Cost function
Total cost as a function of quantity: C(Q). Typically includes a fixed component (overhead) and a variable component that depends on output.
Sensitivity analysis ("what if" analysis)
Examining how the optimal price and quantity change when an underlying parameter shifts, such as an increase in raw material costs or overhead.
Pure selling
A situation where the firm has no production costs (or they are sunk/fixed), so the only decision is what price to charge. MC = 0, so the optimum is where MR = 0.
Margin (price-cost margin)
Defined in this course as (P − MC) / P. This is the share of the price that is not eaten up by marginal cost. The textbook calls this "markup," but the course uses "margin."
Markup (mark-up over cost)
Defined in this course as (P − MC) / MC. This is how much the price exceeds marginal cost, expressed as a fraction of marginal cost. Used in "rule of thumb" pricing.
If you are given two (P, Q) data points, you can find the linear demand function passing through them.
Find the slope: m = (Q₂ − Q₁) / (P₂ − P₁).
Plug one point into Q = mP + b and solve for b.
This is the Panera Bread approach from class.
Models simplify reality to focus on the forces that matter most.
Assumptions (like profit maximisation, rational behaviour, ceteris paribus) are deliberate simplifications, not claims about literal truth.
You should be able to explain why a given assumption is useful, not just state it.
Start from the inverse demand function P = a − bQ.
Revenue: R(Q) = P × Q = aQ − bQ².
Cost: typically given directly, e.g. C(Q) = F + cQ, where F is fixed cost and c is constant marginal cost.
Profit: π(Q) = R(Q) − C(Q).
Take the derivative: π′(Q) = R′(Q) − C′(Q) = MR − MC.
Set π′(Q) = 0, solve for Q*.
Plug Q* back into the inverse demand function to get P*.
Find MR by differentiating R(Q).
Find MC by differentiating C(Q).
Set MR = MC, solve for Q*.
Plug Q* back into inverse demand to get P*.
Both methods give the same answer. The MR = MC framing is more intuitive for explaining why the answer is optimal.
If MR > MC, producing one more unit adds more to revenue than to cost, so profit increases.
If MR < MC, the last unit costs more than it brings in, so profit would rise if you produced less.
Only at MR = MC is there no way to improve profit by adjusting output.
Table (schedule) method: list Q, R, C, and π for each quantity; find the row where π is highest. This is the approach from introductory economics.
Math method: use derivatives as described above.
Graphical method: draw MR and MC curves; the intersection gives Q*.
Word problem method: not required for this exam.
The revenue function is an inverted parabola (rises then falls) when demand is linear.
The cost function typically rises, often linearly if MC is constant.
The profit function is the vertical distance between R and C; it peaks where MR = MC.
You do not need to draw these from scratch, but you should recognise their general shapes and how they relate.
MR is a downward-sloping line (for linear demand, MR has the same intercept as inverse demand but twice the slope).
MC may be horizontal (constant) or upward-sloping.
The intersection determines Q*. From Q*, go up to the demand curve to find P*.
If raw material costs rise, MC shifts up.
The new MR = MC intersection occurs at a lower Q* and (via the demand curve) a higher P*.
If overhead (fixed costs) rises, MC is unchanged, so Q* and P* do not change. Only total profit falls.
When MC = 0, set MR = 0 to find Q*.
This gives the revenue-maximising quantity, which is also the profit-maximising quantity when there are no variable costs.
When a firm sells more than one product, the pricing of one can affect the demand for another (complements or substitutes within the firm's portfolio).
You need to understand the economics of this interaction, not solve it mathematically.
Revenue from linear inverse demand P = a − bQ: R(Q) = aQ − bQ²
Marginal revenue from linear inverse demand: MR = a − 2bQ
Profit function: π(Q) = R(Q) − C(Q)
Profit-maximising condition: MR = MC, or equivalently, dπ/dQ = 0
Price-cost margin (course definition): Margin = (P − MC) / P
Markup over cost (course definition): Markup = (P − MC) / MC
⚠️ The course defines margin and markup differently from the textbook. Margin is (P − MC) / P. Markup is (P − MC) / MC. Use the course definitions on the exam.
⚠️ For linear inverse demand P = a − bQ, the MR line has the same vertical intercept (a) but twice the slope (−2b). This is a quick shortcut worth memorising.
⚠️ An increase in fixed costs (overhead) does not change the optimal price or quantity. Only variable cost changes shift MC and therefore change the optimum.
⚠️ An increase in raw materials raises MC, which leads to higher P* and lower Q*.
⚠️ In a pure selling problem, set MR = 0 (since MC = 0). Do not set price equal to zero.
Q: A firm's inverse demand is P = 100 − 2Q and its cost function is C = 200 + 4Q. Find the optimal price and quantity using the profit function method.
A: π(Q) = (100Q − 2Q²) − (200 + 4Q) = −2Q² + 96Q − 200. Set π′(Q) = −4Q + 96 = 0, so Q* = 24. Then P* = 100 − 2(24) = 52.
Q: Find the same optimum using MR = MC.
A: MR = 100 − 4Q. MC = 4. Set 100 − 4Q = 4, so 4Q = 96, Q* = 24. P* = 100 − 48 = 52. Same answer.
Q: Suppose raw material costs increase. What happens to the firm's price and output?
A: MC shifts up, so the MR = MC intersection moves left (lower Q*) and, via the demand curve, P* rises. Answer: price increases, quantity decreases. Choice (b).
Q: A firm faces inverse demand P = 50 − Q and has zero variable costs. What quantity maximises profit?
A: This is a pure selling problem. R(Q) = 50Q − Q². MR = 50 − 2Q. Set MR = 0: Q* = 25. P* = 50 − 25 = 25.
Q: Why is MR = MC the profit-maximising condition?
A: At any quantity where MR > MC, the firm gains more revenue than cost from the next unit, so it should expand. Where MR < MC, the last unit costs more than it earns, so the firm should contract. Only at MR = MC can profit not be improved by changing output.
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