Source: Microeconomic Theory, Texas A&M University
Tags: perfect competition, profit maximisation, marginal revenue, marginal cost, shutdown condition, supply curve, producer surplus, competitive equilibrium, price taking, short-run supply
In a perfectly competitive market, firms are price takers. The profit-maximising output is where price equals marginal cost, provided the firm covers its average variable cost. The firm's short-run supply curve is the portion of its MC curve above AVC. Market supply is the horizontal sum of individual supply curves, and equilibrium is where market supply meets market demand.
Perfectly competitive market
A market satisfying three assumptions: price taking, product homogeneity, and free entry and exit. Each firm faces a perfectly elastic (horizontal) demand curve at the market price.
Price taking
A firm has no influence over the market price and takes it as given.
Product homogeneity
Products of all firms are identical. No firm can raise its price without losing all its business.
Free entry and exit
No barriers prevent firms from entering or leaving the industry.
Revenue (R)
Total income from selling output. R(q) = p · q in a competitive market.
Profit (Π)
The difference between revenue and cost: Π(q) = R(q) − C(q).
Marginal revenue (MR)
The rate at which revenue changes when output changes by one unit. MR(q) = R'(q). In a competitive market, MR(q) = p.
Marginal profit
The rate at which profit changes when output changes: Π'(q) = R'(q) − C'(q) = MR − MC.
Producer surplus (PS)
The area below the market price and above the market supply curve, between 0 and Q*. Equivalently, PS = R(Q*) − VC(Q*). Note: in the short run, profit = PS − FC, so profit and producer surplus differ.
Competitive equilibrium
The price p* at which quantity demanded equals quantity supplied: Q(p*) = S(p*).
The individual firm faces a horizontal demand curve at the market price, even though the market demand curve slopes downward. Because the firm can sell as much as it likes at price p, every additional unit brings in exactly p, so MR = p.
In the short run, capital is fixed. The firm chooses how much to produce to maximise profit.
Key properties of the short-run competitive setting:
R(q) is linear (R = pq)
C(0) > 0 because of fixed costs
MC(q) is eventually increasing because of diminishing marginal returns
Case 1: Positive profit
The profit-maximising output q₀ satisfies Π'(q₀) = 0, which reduces to p = MC(q₀).
Case 2: Negative profit
There may be two output levels where p = MC (call them q₁ and q₂). The firm must compare Π(q) at those points and at q = 0 to determine the true optimum.
If Π(q₂) > Π(0), produce q₂
If Π(q₂) = Π(0), both q₂ and 0 are profit-maximising
If Π(q₂) < Π(0), shut down (produce 0)
A firm producing q* does at least as well as shutting down if Π(q*) ≥ Π(0). This simplifies to:
p · q − VC(q) − FC ≥ 0 − VC(0) − FC = −FC
Which reduces to:
p ≥ AVC(q*)
If price is below AVC at the candidate output, the firm is better off shutting down entirely.
A profit-maximising output level is q* > 0 if both conditions hold:
p = MC(q*), i.e. MR = MC and marginal profit is zero
p ≥ AVC(q*), i.e. the no-shutdown condition is met
If no such q* > 0 exists, the firm should shut down.
A firm may produce at a loss in the short run. The reason: by producing, it minimises its loss, because shutting down still means paying the entirety of its fixed cost.
Condition for profit maximisation (precise version):
Marginal revenue equals marginal cost at a point where the marginal cost curve is rising.
Profit at q* can be expressed as:
Π(q*) = p · q* − C(q*) = p · q* − ATC(q*) · q* = q* · (p − ATC(q*))
On a graph, profit is the rectangle with height (p − ATC(q*)) and width q*. If p < ATC(q*), the rectangle represents a loss.
Worked Example 1: Market price of coffee = $380.
Where p = MC(q), q = 22
Check: p = 380 ≥ AVC(22) = 258, so produce
Π(22) = (380 − 283) × 22 = 97 × 22 = 2,134
Worked Example 2 – Shutdown price:
The firm shuts down when p < min(AVC). From the graph, the minimum of AVC is 214, so the firm shuts down if and only if price is below 214.
Worked Example 3 – Negative profit threshold:
Profit is negative when p < min(ATC). From the graph, the minimum of ATC is 264, so the firm earns negative profit if and only if price is below 264.
When the wage (w) increases, the MC curve shifts upward (from MC₁ to MC₂). If the market price is still above AVC, the firm still produces, but at a lower output level.
The short-run supply curve S(p) has two parts:
For p ≥ min(AVC): the supply curve is the portion of MC that lies at or above AVC
For p < min(AVC): the firm shuts down, so S(p) = 0
Analytical example: C(q) = 12 + q²/8.
MC(q) = q/4 = 0.25q
Set p = MC: q* = 4p
AVC(q*) = (q*/8) = 4p/8 = 0.5p, which is less than p for all p > 0, so no shutdown
Supply function: S(p) = 4p
Supply curve shifts
An increase in the price of a variable input shifts the supply curve upward (leftward)
A decrease in input price or a technology improvement that reduces MC shifts the supply curve downward (rightward)
The market supply curve is the horizontal summation of all individual firms' supply curves.
If 10 identical firms each have Sᵢ(p) = 5p, market supply is S(p) = 50p
If Firm 1 has S₁(p) = 2p and Firm 2 has S₂(p) = 5p, market supply is S(p) = 7p
When prices are low enough that some firms shut down, only the remaining firms contribute to market supply. As price rises and more firms produce, the market supply curve becomes flatter.
Producer surplus:
PS = p · Q* − VC(Q*) = R(Q*) − VC(Q*)
In the short run, profit = PS − FC. They are not the same.
Finding competitive equilibrium:
Set demand equal to supply: Q(p*) = S(p*). Solve for p* (equilibrium price) and then compute q* = Q(p*) or S(p*) (equilibrium quantity).
Worked example: Q(p) = 10 − p, S(p) = 2p − 1.
10 − p = 2p − 1
11 = 3p, so p* = 11/3
q* = 10 − 11/3 = 19/3
Revenue: R(q) = pq (competitive market)
Profit: Π(q) = R(q) − C(q) = pq − C(q)
MR = p (competitive market)
Profit-max condition: p = MC(q*) and p ≥ AVC(q*)
Profit visualised: Π = q* × (p − ATC(q*))
Producer surplus: PS = R(Q*) − VC(Q*)
Profit = PS − FC (short run)
Competitive equilibrium: Q(p*) = S(p*)
⚠️ The two-part rule (p = MC and p ≥ AVC) is the single most important result for exam problems on competitive firms. Always check both conditions.
⚠️ A firm can produce at a loss in the short run. It does so to minimise the loss from its unavoidable fixed costs.
⚠️ The shutdown price is the minimum of AVC, not the minimum of ATC. This is a very common exam trap.
⚠️ Negative profit begins when p falls below the minimum of ATC. Shutdown happens below the minimum of AVC. The range in between is where the firm produces at a loss.
⚠️ Producer surplus and profit are not the same in the short run; they differ by the fixed cost.
⚠️ The short-run supply curve is the MC curve above the AVC curve. Below min(AVC) the firm produces zero.
Q: A competitive firm has MC(q) = 2q and AVC(q) = q. If the market price is 10, what is the profit-maximising output?
A: Set p = MC: 10 = 2q, so q* = 5. Check: p = 10 ≥ AVC(5) = 5. Produce 5 units.
Q: Why might a firm continue to produce even when profit is negative?
A: If price is below ATC but above AVC, the firm's revenue covers all variable costs and contributes something towards fixed costs. Shutting down would mean losing the entire fixed cost with no offset.
Q: What is the difference between producer surplus and profit in the short run?
A: Producer surplus = revenue minus variable cost. Profit = revenue minus total cost. The difference is the fixed cost: profit = PS − FC.
Q: If there are 5 identical firms each with supply Sᵢ(p) = 3p, what is the market supply curve?
A: S(p) = 5 × 3p = 15p.
Q: If Q(p) = 20 − 2p and S(p) = p − 1, what are the equilibrium price and quantity?
A: 20 − 2p = p − 1, so 21 = 3p, giving p* = 7. q* = 20 − 14 = 6.
perfectly competitive market, price taker, product homogeneity, free entry and exit, profit maximisation, marginal revenue equals marginal cost, MR = MC, shutdown condition, shutdown price, average variable cost, AVC, supply curve, short-run supply, market supply, horizontal summation, producer surplus, competitive equilibrium, equilibrium price, equilibrium quantity, factor price change, profit at a loss