Source: Varghese practice problems, textbook chapters
Tags: marginal revenue, marginal cost, marginal profit, profit maximisation, MR equals MC, total revenue, total cost, inverse demand, demand function, cost function, fixed cost, variable cost, ECON 323, Varghese
Marginal analysis is the core decision-making tool in managerial economics. A firm maximises profit by producing up to the point where marginal revenue equals marginal cost (MR = MC), or equivalently where marginal profit equals zero. This section covers how to derive demand and cost functions, calculate total revenue and profit, and apply the MR = MC rule in both table-based and equation-based problems.
Marginal revenue (MR)
The additional revenue a firm earns from selling one more unit of output. Calculated as the derivative of total revenue with respect to quantity, or as the change in total revenue when output increases by one unit.
Marginal cost (MC)
The additional cost incurred from producing one more unit of output. Calculated as the derivative of total cost with respect to quantity.
Marginal profit (Mπ)
The additional profit earned from producing one more unit. Marginal profit = MR − MC. Equivalently, it is the derivative of the profit function with respect to quantity.
Total revenue (TR)
Price multiplied by quantity sold. TR = P × Q. When demand is a function of Q, substitute the inverse demand equation for P.
Total cost (TC)
The sum of all costs of production at a given output level. May include a fixed component (which does not change with output) and a variable component (which does).
Profit (π)
Total revenue minus total cost. π = TR − TC.
Inverse demand function
Expresses price as a function of quantity: P = f(Q). This is the form you typically use for graphing demand, where P is on the y-axis.
Demand function
Expresses quantity as a function of price: Q = f(P). The algebraic rearrangement of the inverse demand function.
Fixed cost
A cost that does not vary with the level of output. Appears as a constant in the cost function (e.g. the "200" in C = 200 + 4Q).
Average cost (AC)
Total cost divided by quantity. AC = TC / Q.
A firm maximises profit at the output level where MR = MC, which is the same as saying marginal profit equals zero.
If marginal profit is positive (MR > MC), the firm should increase output. Each additional unit adds more to revenue than to cost.
If marginal profit is negative (MR < MC), the firm should decrease output.
The firm stops adjusting when marginal profit reaches zero.
Given a table of price, quantity, total revenue, and total cost:
Profit at any quantity = TR − TC
Marginal revenue = change in TR from one additional unit
Marginal cost = change in TC from one additional unit
Marginal profit = change in profit from one additional unit, or MR − MC
Worked example from Table 2-1:
At Q = 3: TR = 39, TC = 12, so profit = 39 − 12 = $27
At Q = 2: TR = 28, TC = 7, so profit = $21
Marginal profit of the 3rd unit = 27 − 21 = $6
Equivalently: MR of 3rd unit = 39 − 28 = $11, MC of 3rd unit = 12 − 7 = $5, so Mπ = 11 − 5 = $6
When given two price-quantity points, you can find the linear demand function by treating them as coordinates on a line.
Worked example (Lam's Garden):
Point 1: P = $3.85, Q = 125
Point 2: P = $4.29, Q = 100
Slope = (4.29 − 3.85) / (100 − 125) = 0.44 / (−25) = −0.0176
Using point-slope form for the inverse demand: P − 3.85 = −0.0176(Q − 125)
P = 3.85 − 0.0176Q + 2.20 = 6.05 − 0.0176Q
Inverse demand: P = 6.05 − 0.0176Q
Demand function: rearrange to Q = (6.05 − P) / 0.0176 ≈ 343.75 − 56.82P
Write the profit function π = TR − TC, take the derivative, set it to zero, and solve for Q.
Worked example (Problem 2.6): P = 100 − 2Q, C = 200 + 4Q
TR = P × Q = (100 − 2Q)Q = 100Q − 2Q²
π = TR − TC = 100Q − 2Q² − 200 − 4Q = −200 + 96Q − 2Q²
dπ/dQ = 96 − 4Q
Set to zero: 96 − 4Q = 0, so Q* = 24
P* = 100 − 2(24) = $52
π* = −200 + 96(24) − 2(24²) = −200 + 2304 − 1152 = $952
Find MR and MC separately, set them equal, and solve for Q. This should give the same answer as Method 1.
Using the same example:
TR = 100Q − 2Q², so MR = dTR/dQ = 100 − 4Q
TC = 200 + 4Q, so MC = dTC/dQ = 4
Set MR = MC: 100 − 4Q = 4, so 96 = 4Q, Q* = 24
Same result.
Demand: Q = 100 − P (so inverse demand: P = 100 − Q). MC = AC = $10.
TR = P × Q = (100 − Q)Q = 100Q − Q²
MR = 100 − 2Q
Set MR = MC: 100 − 2Q = 10, so Q* = 45
P* = 100 − 45 = $55
Profit = TR − TC = (55 × 45) − (10 × 45) = 2475 − 450 = $2,025
Inverse demand: P = 45 − 0.025Q. Cost: C = 30Q (so MC = 30).
TR = (45 − 0.025Q)Q = 45Q − 0.025Q²
MR = 45 − 0.05Q
Set MR = MC: 45 − 0.05Q = 30, so 0.05Q = 15, Q* = 300
P* = 45 − 0.025(300) = 45 − 7.50 = $37.50
Profit = TR − TC = (37.50 × 300) − (30 × 300) = 11,250 − 9,000 = $2,250
When a firm sells in a competitive market at a fixed price (say $12 per unit), MR equals that fixed price for every unit. The firm's demand curve is perfectly horizontal.
MR = $12 (constant)
If C = 200 + 4Q, then MC = $4
Since MR > MC at every output level, the firm would want to keep expanding output. The constraint becomes capacity or the assumption that cost eventually rises with scale.
In this simplified problem, MR always exceeds MC, so the firm produces as much as it can.
If the cost of raw materials increases, the MC curve shifts upward. The MR = MC intersection moves to a lower quantity and a higher price. The firm produces less and charges more.
A change in fixed cost does not affect MC (because MC is the derivative of TC, and the derivative of a constant is zero). It also does not affect MR. Therefore, a change in fixed cost has no effect on the profit-maximising output or price. It only affects the level of total profit.
Total revenue: TR = P × Q
Profit: π = TR − TC
Marginal revenue: MR = dTR/dQ
Marginal cost: MC = dTC/dQ
Profit maximisation condition: MR = MC (equivalently, dπ/dQ = 0)
Marginal profit: Mπ = MR − MC = dπ/dQ
For a linear inverse demand P = a − bQ: TR = aQ − bQ², so MR = a − 2bQ. Notice that MR has the same intercept as the inverse demand curve but twice the slope.
Profit function (general form with linear demand and linear cost): π = (a − bQ)Q − (F + cQ) = aQ − bQ² − F − cQ = −F + (a − c)Q − bQ²
⚠️ Marginal revenue is additional revenue from one more unit of output, not from a change in price or profit. Watch the wording carefully.
⚠️ When marginal profit is positive, increase output. When it is negative, decrease output. Stop when marginal profit is zero. Do not confuse this with "increase until MR is negative" or "decrease until marginal profit is negative."
⚠️ For a linear inverse demand P = a − bQ, the MR curve has the same y-intercept but double the slope: MR = a − 2bQ. This comes up repeatedly.
⚠️ Fixed costs do not affect optimal price or quantity. They affect total profit only. If the exam asks what happens when fixed costs change, marginal revenue and marginal cost are both unchanged.
⚠️ Know both methods (profit function and MR = MC) and be ready to use either. The exam may specify which method to apply.
⚠️ When deriving a demand function from two points, be careful about which variable is the "y" and which is the "x." The inverse demand function has P on the left; the demand function has Q on the left.
⚠️ At the profit-maximising output, profit from the table is found by computing TR − TC at that quantity, not by looking at marginal values alone.
Q: What is marginal revenue?
A: The amount of additional revenue from a one-unit increase in output.
Q: Using Table 2-1, what is the firm's profit from selling 3 units?
A: $27. Profit = TR − TC = 39 − 12 = 27.
Q: Using Table 2-1, what is the marginal profit of the 3rd unit?
A: $6. Profit at Q=3 is $27, profit at Q=2 is $21, so marginal profit = 27 − 21 = 6.
Q: If marginal profit is positive at the current output level, what should the firm do?
A: Increase output until marginal profit is zero. Each additional unit still adds to profit, so there is room to produce more.
Q: The profit function is π = −200 + 80Q − 0.2Q². What is the profit-maximising output?
A: Q = 200. Take the derivative: dπ/dQ = 80 − 0.4Q. Set to zero: 80 = 0.4Q, so Q = 200.
Q: If a firm's inverse demand is P = 100 − 2Q and its cost function is C = 200 + 4Q, what are the optimal price and quantity?
A: Q* = 24, P* = $52. MR = 100 − 4Q, MC = 4. Set MR = MC: 100 − 4Q = 4, Q = 24. P = 100 − 2(24) = 52.
Q: What happens to the profit-maximising price and quantity if raw material costs rise?
A: Price increases and quantity decreases. The MC curve shifts up, so the MR = MC intersection moves to a lower Q and a higher P.
Q: A decrease in fixed costs will have what effect on marginal revenue and marginal cost?
A: Neither changes. Fixed costs do not enter the MR or MC calculations.
Q: For the firm with demand Q = 100 − P and constant MC = AC = $10, what is the profit-maximising quantity and total profit?
A: Q* = 45, profit = $2,025. MR = 100 − 2Q = 10 gives Q = 45. P = 55. Profit = (55)(45) − (10)(45) = 2,025.
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