Source: Velez & Guo lecture notes, Pindyck & Rubinfeld Ch. 6-9 | Texas A&M University
Tags: production function, marginal product, diminishing marginal returns, isoquant, MRTS, returns to scale, cost function, marginal cost, average cost, profit maximisation, competitive market, supply, aggregate supply, sales tax, competitive equilibrium, consumer surplus, producer surplus, deadweight loss, welfare analysis, intermediate microeconomics
Producer theory models firms as "black boxes" that turn inputs (labour, capital) into output, then asks how profit-maximising firms behave in competitive markets. The key results: firms produce where price equals marginal cost, competitive equilibrium maximises total welfare, and any deviation (taxes, price controls, market power) creates deadweight loss.
Production function, f(L, K)
Assigns to each combination of labour (L) and capital (K) the maximum output the firm can produce. Analogous to the utility function in consumer theory, but the output number has cardinal meaning (actual units produced).
Marginal product of labour (MPL)
MPL(L, K) = df/dL. The additional output from one more unit of labour, holding capital fixed. Graphically, it is the slope of the production function when K is constant.
Marginal product of capital (MPK)
MPK(L, K) = df/dK. The additional output from one more unit of capital, holding labour fixed.
Law of diminishing marginal returns
As one input increases while others stay fixed, its marginal product eventually decreases (or hits zero and never rises again). This is about adding more of one input with everything else constant, not about scaling all inputs together.
Isoquant
The set of input combinations (L, K) that produce the same output level. Analogous to indifference curves. Shape depends on the production function: smooth curves, straight lines (perfect substitutes), or L-shapes (perfect complements).
Marginal rate of technical substitution (MRTS)
MRTS_LK(L, K) = MPL / MPK. The absolute value of the slope of the isoquant. Represents the maximum capital that can be replaced by one additional unit of labour while keeping output constant.
Returns to scale
Compares f(2L, 2K) with 2f(L, K) when all inputs are doubled.
Increasing returns to scale: f(2L, 2K) > 2f(L, K)
Constant returns to scale: f(2L, 2K) = 2f(L, K)
Decreasing returns to scale: f(2L, 2K) < 2f(L, K)
Economic cost
The value of a resource in its best alternative use (opportunity cost). Differs from accounting cost, which records only explicit payments.
Fixed cost (F)
Cost that does not vary with output. When only labour is variable, F = rK (capital cost is sunk).
Variable cost, VC(q)
Cost that varies with output: VC(q) = w * L(q, K), where L(q, K) is the labour needed to produce q units.
Total cost, C(q)
C(q) = F + VC(q).
Average cost, AC(q)
AC(q) = C(q) / q.
Average variable cost, AVC(q)
AVC(q) = VC(q) / q.
Average fixed cost, AFC(q)
AFC(q) = F / q. Always decreasing.
Marginal cost, MC(q)
MC(q) = C'(q) = dC/dq. The cost of producing one additional unit.
Iso-cost line
The set of input combinations costing the same total: rK + wL = c. Slope is -w/r.
Cost minimisation tangency rule
When both inputs are variable, the cost-minimising input combination satisfies MRTS_LK = w/r, together with f(L*, K*) = q.
Competitive market
A market where no agent (consumer or firm) can affect prices, products are identical, and agents are small relative to the market.
Supply function, S(p)
The quantity a firm produces at each price p, derived from profit maximisation.
Consumer surplus, CS(q, p)
The net area under the demand curve and above the price line, up to quantity q. Measures the welfare gain consumers enjoy from purchasing at price p rather than not purchasing at all.
Producer surplus, PS(q, p)
The net area above the supply curve and below the price line, up to quantity q. Measures the firm's gain from selling at price p rather than not producing.
Total welfare (aggregate surplus), W
W = CS + PS (+ government net revenue if taxes exist).
Deadweight loss (DWL)
The reduction in total welfare relative to the competitive equilibrium. DWL = W(competitive) - W(actual). Always non-positive for deviations from the competitive outcome.
Representing a firm
A firm is a "black box" that transforms inputs into output
For tractability, we use two inputs: labour (L) and capital (K)
The production function f(L, K) gives the maximum output from any input combination
Three canonical production function forms
Linear (perfect substitute inputs): f(L, K) = aL + bK
Example: a call centre where workers and machines independently answer calls, f = 30L + 240K
Isoquants are straight lines; MRTS is constant
Leontief (perfect complement inputs): f(L, K) = c * min{L, K}
Example: a newspaper where each journalist needs exactly one workstation, f = 3min{L, K}
Isoquants are L-shaped; inputs must be used in fixed proportions
Cobb-Douglas: f(L, K) = A L^a K^b
Example: an orange juice company, f = 5L^(1/9) * K^(8/9)
Isoquants are smooth curves
When only one input is variable
Fix K and study f(L, K-bar) as a function of L alone
The graph of this function shows total output against labour
MPL is the slope of this graph at each point
Under diminishing marginal returns, MPL eventually falls
When both inputs are variable
The firm chooses the cheapest input mix to produce a target output
Graphically: find the lowest iso-cost line that touches the target isoquant
For smooth production functions: MRTS_LK = w/r (tangency rule for cost minimisation)
For linear production functions: compare MRTS_LK with w/r to determine whether the firm uses only labour, only capital, or any mix
Returns to scale for Cobb-Douglas
For f(L, K) = A L^a K^b:
a + b > 1: increasing returns to scale
a + b = 1: constant returns to scale
a + b < 1: decreasing returns to scale
Free entry in an industry can be modelled by assuming constant returns to scale, since any entrant can replicate existing production.
Economic vs. accounting cost
A firm owns a truck it could rent for $1,000/month. Even if fully depreciated on the books, its economic cost is $1,000/month.
A manager who earns $20,000 running his own company but could earn $80,000 elsewhere has an economic labour cost of $80,000.
Firms should maximise economic profit, not accounting profit. A firm reporting positive accounting profit may still be making negative economic profit.
Cost when only labour is variable
Find L(q, K-bar) by inverting the production function: solve q = f(L, K-bar) for L
C(q) = rK-bar + w * L(q, K-bar)
All cost statistics (AC, AVC, AFC, MC) follow from this expression
Graphical construction of cost curves
Starting from the production function graph:
Rotate and flip the production function to get L(q, K-bar)
Multiply by w to get VC(q); add F = rK-bar to get C(q)
Three critical output levels determine the shape of MC, AVC, and AC:
q1: where the slope of VC starts decreasing (MC reaches its minimum)
q2: where the line from the origin is tangent to VC (AVC reaches its minimum; MC = AVC here)
q3: where the line from the origin is tangent to C (AC reaches its minimum; MC = AC here)
Key relationships between cost curves
MC intersects AVC at AVC's minimum
MC intersects AC at AC's minimum
AC is always above AVC (because AC = AVC + AFC)
AFC is always declining (F/q as q grows)
MC starts at the same value as AVC (both approach the same limit as q approaches zero)
Cost when both inputs are variable
No fixed costs: C(0) = 0
Cost minimisation uses the tangency rule MRTS_LK = w/r
With decreasing returns to scale, MC and AC are both increasing
With constant returns to scale, MC = AC = constant (flat)
Competitive market assumptions
Price-taking: no agent influences market price
Identical products
Agents are small relative to the market
Profit maximisation conditions
The firm produces q* satisfying:
Output rule (slope condition): p = MC(q*)
No shut-down rule:
If only labour is variable: p >= AVC(q*)
If both inputs are variable: p >= AC(q*)
If no q* satisfies both rules, the firm produces zero.
Intuition for the output rule
If p > MC(q*), the firm gains by producing one more unit
If p < MC(q*), the firm gains by producing one less unit
At p = MC(q*), the last unit produced just breaks even
Intuition for the shut-down rule
When only labour is variable, fixed costs are sunk regardless. The firm operates as long as revenue covers variable costs.
When both inputs are variable, all costs are avoidable. The firm operates as long as revenue covers total costs.
Constructing the supply curve
The supply curve is the portion of the MC curve at or above AVC (when K is fixed) or at or above AC (when both inputs are variable)
Below the relevant average cost, the firm produces zero
Supply curve shapes by returns to scale
Decreasing returns: upward-sloping MC, upward-sloping supply
Constant returns (free entry): horizontal MC = AC, horizontal supply at c
Aggregate supply
S(p) = S1(p) + S2(p) + ... + Sl(p). Graphically, horizontal summation of individual supply curves.
Sales taxes
Ad-valorem tax (proportion alpha of revenue): firm receives p(1 - alpha) per unit. Supply shifts left: S_alpha(p) corresponds to the old S evaluated at p(1 - alpha).
Specific tax (amount tau per unit): firm receives p - tau per unit. Supply shifts up by tau.
Profit identification on a graph
Profit = q* (p - AC(q)). This is the area of the rectangle with height (p - AC) and width q*. If p < AC, the firm makes a loss but may still operate if p >= AVC (covering variable costs and part of fixed costs is better than covering none).
Competitive equilibrium
A price p* and quantity q* such that:
Consumers maximise preferences at p*
Firms maximise profits at p*
Market clears: demand equals supply, Q(p*) = S(p*)
Found by solving Q(p) = S(p) for p.
Free-entry equilibrium
Constant returns to scale implies MC = AC = c (a constant)
In equilibrium, p* = c and economic profit is zero
Zero economic profit does not mean firms earn nothing; they earn exactly the opportunity cost of their inputs
Consumer surplus
For a downward-sloping demand curve, CS is the triangle (or area) between the demand curve and the price line
Measures the total "bonus" consumers get from paying one price for units they valued at different (higher) amounts
Producer surplus
Area between the price line and the supply curve, up to the quantity sold
When only labour is variable and the supply curve starts above zero, PS may exceed profit by the amount of fixed cost
Total welfare and the first welfare theorem
W = CS + PS at the competitive equilibrium
No other (q, p) combination produces higher total welfare
DWL(q, p) = W(q, p) - W(q*, p*) <= 0 for any deviation
Deadweight loss from a specific tax
Tax shifts supply up by tau
New equilibrium has higher price, lower quantity
DWL = the triangle between supply and demand, from the new quantity to the old quantity
Government revenue (b + c in the standard diagram) may or may not compensate consumer welfare loss
Rationale for taxes despite DWL
Markets need institutions (property rights, competition enforcement) that cost money
Markets do not distribute welfare gains fairly; redistribution has social value
The analysis implies taxes should be kept as low as feasible to minimise distortion
Marginal product of labour
MPL(L, K) = df(L, K)/dL
Marginal product of capital
MPK(L, K) = df(L, K)/dK
MRTS
MRTS_LK(L, K) = MPL(L, K) / MPK(L, K)
Returns to scale test (Cobb-Douglas)
f = A L^a K^b: increasing if a + b > 1, constant if a + b = 1, decreasing if a + b < 1
Cost function (K fixed)
C(q) = rK + w * L(q, K)
Profit
pi(q) = pq - C(q)
Profit maximisation
p = MC(q*), and p >= AVC(q*) [K fixed] or p >= AC(q*) [both variable]
Competitive equilibrium
Q(p*) = S(p*)
Total welfare
W = CS + PS + GNR (government net revenue, if applicable)
⚠️ Diminishing marginal returns is about increasing one input with others fixed. Do not confuse it with decreasing returns to scale (which scales all inputs).
⚠️ Economic cost includes opportunity cost. A firm reporting accounting profit may be making zero or negative economic profit.
⚠️ The supply curve is only the portion of MC above AVC (K fixed) or above AC (both variable). Below that, the firm shuts down.
⚠️ When both inputs are variable, C(0) = 0 and there is no sunk cost. The shut-down condition becomes p >= AC(q*), not p >= AVC(q*).
⚠️ Zero economic profit in free-entry equilibrium does not mean firms "work for nothing." They earn exactly the opportunity cost of their resources.
⚠️ The competitive equilibrium maximises total welfare. Any deviation creates DWL. This is the core result of welfare economics in this course.
⚠️ When constructing cost curves from a production function, remember to "rotate and flip" the production function to get L(q, K). Students frequently skip this step and get the cost shape wrong.
⚠️ For linear production functions, cost minimisation is a corner solution. Compare MRTS with w/r; the firm uses only the cheaper-per-unit-of-output input.
Q: A firm has production function f(L, K) = 2L^(1/2) * K^(1/2), w = 2, r = 3, and K is fixed at 4. What is the cost of producing 12 units?
A: Solve 12 = 2L^(1/2) 4^(1/2) = 2L^(1/2) 2 = 4L^(1/2). So L^(1/2) = 3, L = 9. Cost = rK + wL = 3(4) + 2(9) = 12 + 18 = $30.
Q: For f(L, K) = A L^(0.4) K^(0.7), does this production function exhibit increasing, constant, or decreasing returns to scale?
A: a + b = 0.4 + 0.7 = 1.1 > 1, so increasing returns to scale.
Q: A firm has MC(q) = q/4, AVC(q) = q/8, and fixed cost F = 12. The market price is p = 10. What is the firm's optimal output and profit?
A: From p = MC(q*): 10 = q*/4, so q* = 40. Check p >= AVC(q*): 10 >= 40/8 = 5. Yes. Profit = 10(40) - (12 + 40^2/8) = 400 - 12 - 200 = $188.
Q: Why is the competitive equilibrium efficient?
A: At the competitive equilibrium, the marginal cost of the last unit produced equals the price consumers are willing to pay for it. Producing one more unit would cost more than consumers value it; producing one fewer unit would forgo a unit consumers value more than it costs. Total welfare (CS + PS) is maximised.
Q: A specific tax of $5 per unit is imposed on producers. Before the tax, equilibrium was p = 20, q = 100. After the tax, the new price is p = 23 and the new quantity is q = 85. Who bears more of the tax burden, consumers or producers?**
A: Consumers pay $3 more per unit (from 20 to 23). Producers receive $2 less per unit (23 - 5 = 18, down from 20). Consumers bear more of the burden. The incidence depends on relative elasticities, not on who physically remits the tax.
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