Module M18 Quiz: Technology, Production, and Profit Maximisation

25 questions · Intermediate · Multiple choice, calculation, and short answer

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Attempt each question before clicking Show Answer.

Question 1

Lesson L01 · Production

A production function f(L,K) exhibits constant returns to scale if:

Type: Multiple Choice

A) f(L,K)=L+K
B) f(tL,tK) = t×f(L,K) for all t>0
C) Output is constant regardless of inputs
D) Marginal products are constant

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Answer: B) f(tL,tK) = t×f(L,K) for all t>0

CRS: doubling all inputs exactly doubles output. Formally: f is homogeneous of degree 1.

Question 2

Lesson L01 · Production

For f(L,K) = L^0.3 × K^0.7, what are the returns to scale?

Type: Calculation

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Answer: CRS (constant returns to scale — exponents sum to 1)

f(tL,tK)=(tL)^0.3×(tK)^0.7 = t^(0.3+0.7)×L^0.3×K^0.7 = t×f(L,K). Exponents 0.3+0.7=1 → CRS.

Question 3

Lesson L01 · Production

The MRTS (marginal rate of technical substitution) equals:

Type: Multiple Choice

A) w/r
B) MP_L/MP_K — the rate at which capital can be substituted for labour while holding output constant
C) The slope of the isocost line
D) Output per worker

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Answer: B) MP_L/MP_K — the rate at which capital can be substituted for labour while holding output constant

MRTS = |dK/dL| along an isoquant = MP_L/MP_K. It equals the ratio of marginal products.

Question 4

Lesson L01 · Production

Explain the difference between technical efficiency and allocative efficiency in production.

Type: Short Answer

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Answer: Technical efficiency: producing on the production frontier (no wasted inputs for given output). Allocative efficiency: choosing the cost-minimising input combination given input prices (MRTS = w/r). A firm can be technically efficient but allocatively inefficient if it uses too much of one input.

Technical: on isoquant. Allocative: at the right point on the isoquant given prices.

Question 5

Lesson L01 · Production

Which production function exhibits diminishing MRTS (convex isoquants)?

Type: Multiple Choice

A) f=aL+bK (perfect substitutes)
B) f=min(aL,bK) (Leontief)
C) f=L^a × K^b with a,b>0 (Cobb-Douglas)
D) f=L+K

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Answer: C) f=L^a × K^b with a,b>0 (Cobb-Douglas)

Cobb-Douglas has strictly convex isoquants — MRTS diminishes as L/K rises, reflecting diminishing marginal products.

Question 6

Lesson L02 · Production

The MRTS for a Leontief (fixed-proportions) production function f=min(L,2K) at optimal production is:

Type: Multiple Choice

A) Equal to w/r
B) Undefined (isoquant has a kink) — the firm always uses inputs in fixed ratio L/K=2 regardless of prices
C) Equal to 1
D) Zero

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Answer: B) Undefined (isoquant has a kink) — the firm always uses inputs in fixed ratio L/K=2 regardless of prices

Leontief isoquants are L-shaped — the MRTS is not defined at the kink (corner of the L). Cost minimisation always chooses the corner: L=2K.

Question 7

Lesson L02 · Production

For f=L^0.4 × K^0.6 with w=4 r=6, find the cost-minimising K/L ratio.

Type: Calculation

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Answer: K/L = (0.6/0.4)×(w/r) = 1.5×(4/6) = 1

MRTS = MP_L/MP_K = (0.4K)/(0.6L) = w/r = 4/6 → K/L = (0.4/0.6)×(6/4) = 1.

Question 8

Lesson L02 · Production

Euler's theorem for a homogeneous degree-r function states f(x) = r × Σ(xᵢ × ∂f/∂xᵢ). For CRS (r=1) this means:

Type: Multiple Choice

A) Profits are always zero in competitive equilibrium
B) Output exhausts factor payments: f = L×MP_L + K×MP_K — paying each factor its marginal product exactly exhausts revenue
C) Firms earn normal profit
D) All production functions are Cobb-Douglas

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Answer: B) Output exhausts factor payments: f = L×MP_L + K×MP_K — paying each factor its marginal product exactly exhausts revenue

With CRS, paying each input its marginal product exactly uses up all output — no surplus for economic profit in long-run competitive equilibrium.

Question 9

Lesson L02 · Production

A solar panel manufacturer doubles all inputs and output more than doubles. This represents:

Type: Multiple Choice

A) Constant returns to scale
B) Decreasing returns to scale
C) Increasing returns to scale — possibly due to specialisation and more efficient use of plant
D) Diminishing marginal returns

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Answer: C) Increasing returns to scale — possibly due to specialisation and more efficient use of plant

IRS: f(tL,tK) > t×f(L,K). Common in industries with high fixed costs or strong specialisation benefits at scale.

Question 10

Lesson L02 · Production

Why do isoquants typically have diminishing MRTS?

Type: Short Answer

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Answer: As we substitute more labour for capital (moving down the isoquant), each additional unit of labour has less capital to work with — its marginal product falls. Simultaneously more capital is released making it more productive. Hence more labour is needed to replace each unit of capital — MRTS falls.

Diminishing marginal products of both inputs cause the MRTS to diminish along an isoquant.

Question 11

Lesson L03 · Production

The cost-minimisation condition MRTS = w/r means:

Type: Multiple Choice

A) Labour and capital have the same marginal product
B) The last dollar spent on labour yields the same extra output as the last dollar spent on capital (MP_L/w = MP_K/r)
C) Wages equal the rental rate
D) The firm is at a corner solution

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Answer: B) The last dollar spent on labour yields the same extra output as the last dollar spent on capital (MP_L/w = MP_K/r)

Optimal input mix: bang per buck is equalised. MP_L/w = MP_K/r — or equivalently MRTS = w/r.

Question 12

Lesson L03 · Production

For f=L^0.5 × K^0.5 with w=4 r=1 and output y=10: find cost-minimising L* and K* and minimum cost.

Type: Calculation

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Answer: L*=5, K*=20, C*=40

MRTS=K/L=w/r=4 → K=4L. Constraint: L^0.5×(4L)^0.5=2L=10 → L*=5, K*=20. C=4×5+1×20=$40.

Question 13

Lesson L03 · Production

Shephard's Lemma for the cost function states ∂c(w,y)/∂w_i = x_i*(w,y). This means:

Type: Multiple Choice

A) Wages determine all costs
B) The derivative of the minimum cost function with respect to input price equals the conditional factor demand — useful for recovering factor demands from the cost function
C) Factor demands are always linear
D) Cost always rises with wages

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Answer: B) The derivative of the minimum cost function with respect to input price equals the conditional factor demand — useful for recovering factor demands from the cost function

Shephard's Lemma: differentiating the cost function w.r.t. input prices gives factor demands directly — avoids re-solving the optimisation.

Question 14

Lesson L03 · Production

A wheat farm uses labour (w=$20/hr) and machinery (r=$100/hr). Optimal: L*=500hrs, K*=200hrs for 10,000 tonnes. If wage rises to $25, use Shephard's Lemma to predict the direction of change in labour demand.

Type: Short Answer

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Answer: ∂L*/∂w = ∂²c/∂w² ≤ 0 (cost function is concave in w). Labour demand falls when wages rise — the farm substitutes toward machinery.

Shephard's Lemma gives L* = ∂c/∂w; differentiating again: ∂L*/∂w = ∂²c/∂w² ≤ 0.

Question 15

Lesson L03 · Production

The cost function c(w1,w2,y) is homogeneous of degree 1 in (w1,w2). This means:

Type: Multiple Choice

A) Doubling output doubles cost
B) Doubling all input prices doubles the minimum cost — the firm cannot do better by substituting
C) Cost is always linear in y
D) Firms always have CRS

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Answer: B) Doubling all input prices doubles the minimum cost — the firm cannot do better by substituting

If all input prices double, the cheapest way to produce y costs exactly twice as much — no input substitution can help when all prices double proportionally.

Question 16

Lesson L04 · Production

A competitive firm's profit-maximising condition p×MP_L = w means:

Type: Multiple Choice

A) The firm hires labour until it runs out of budget
B) The firm hires labour until the value of the marginal product of labour equals the wage — the firm's labour demand condition
C) Wages equal output prices
D) MP_L is maximised

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Answer: B) The firm hires labour until the value of the marginal product of labour equals the wage — the firm's labour demand condition

Value of marginal product (VMP_L = p×MP_L) equals the wage at the optimum — standard factor demand condition.

Question 17

Lesson L04 · Production

Hotelling's Lemma states ∂π(p,w)/∂p = y*(p,w). For π = p²/(4w) with w=4: find y* and verify Hotelling's Lemma.

Type: Calculation

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Answer: y* = p/2w = p/8; ∂π/∂p = 2p/(4w) = p/2w = y* ✓

π = p²/(4w). ∂π/∂p = 2p/4w = p/2w = y*. Hotelling's Lemma verified.

Question 18

Lesson L04 · Production

If p=MC at the profit-maximising output, why does a competitive firm not just set p>MC to earn more profit?

Type: Short Answer

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Answer: A competitive firm is a price-taker — it faces a horizontal demand at the market price and cannot raise its price above the market price without losing all customers. Given it must accept p, producing where p=MC maximises profit.

Price-taking means the firm has no power to set price; it only chooses quantity.

Question 19

Lesson L04 · Production

The supply function y*(p,w) is non-decreasing in p because:

Type: Multiple Choice

A) Higher prices reduce costs
B) By the envelope theorem ∂y*/∂p = ∂²π/∂p² ≥ 0 — the profit function is convex in p
C) Firms always expand when prices rise
D) Supply elasticity is always 1

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Answer: B) By the envelope theorem ∂y/∂p = ∂²π/∂p² ≥ 0 — the profit function is convex in p*

Convexity of the profit function in p (from the envelope theorem) implies ∂y*/∂p ≥ 0 — output supply is non-decreasing in price.

Question 20

Lesson L04 · Production

An Australian winery's profit function is π = 50y − y² − 200. Find profit-maximising output and profit.

Type: Calculation

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Answer: y*=25, π=$425

MR=MC: 50-2y=0 → y*=25. π=50×25-25²-200=1250-625-200=$425.

Question 21

Lesson L05 · Production

In the short run a firm has fixed capital K̄. Its short-run cost function c^S(w,r,y,K̄) satisfies:

Type: Multiple Choice

A) c^S < c^L always
B) c^S ≥ c^L for all y — the long-run cost is always at least as low since the firm has more flexibility
C) c^S = c^L always
D) Short-run costs are irrelevant for pricing

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Answer: B) c^S ≥ c^L for all y — the long-run cost is always at least as low since the firm has more flexibility

Long-run is always weakly cheaper: the firm can always do at least as well in the LR by choosing optimal K. SR restricts K to K̄.

Question 22

Lesson L05 · Production

The Le Chatelier principle states that:

Type: Multiple Choice

A) Long-run demand is more elastic than short-run demand
B) Long-run supply is more elastic than short-run supply — firms can adjust all inputs in the LR so their output response to price changes is larger
C) Short-run and long-run elasticities are equal
D) Firms always exit in the long run

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Answer: B) Long-run supply is more elastic than short-run supply — firms can adjust all inputs in the LR so their output response to price changes is larger

Le Chatelier: LR supply is more price elastic than SR supply because more inputs can be adjusted. Same principle applies to factor demands.

Question 23

Lesson L05 · Production

A dairy farm has fixed milking equipment worth $500,000 and variable costs VC(y)=0.5y². Find MC, AVC and ATC at y=100.

Type: Calculation

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Answer: MC=$100; AVC=$50; ATC=$55

MC=VC'=y=$100. AVC=VC/y=0.5y=$50. ATC=(VC+FC)/y=0.5y+500000/y=$50+$5000=$5050 (note: if y=100, AFC=500000/100=$5000; ATC=$5050). Check y scale — if y in tonnes and VC=$0.5y² then at y=100: VC=$5000, AVC=$50, AFC=$5000, ATC=$5050.

Question 24

Lesson L05 · Production

When should a firm shut down in the short run vs exit in the long run?

Type: Short Answer

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Answer: Short-run shutdown: when P < AVC (revenue doesn't cover variable costs; shutting down loses only FC). Long-run exit: when P < ATC (revenue doesn't cover all costs including the opportunity cost of capital — the firm leaves the industry permanently).

SR: lose only FC if shutdown (FC are sunk). LR: all costs are variable so exit when P < min LRATC.

Question 25

Lesson L05 · Production

Why does the long-run average cost curve act as an 'envelope' of all short-run average cost curves?

Type: Multiple Choice

A) LR costs are always higher
B) For each output level the firm chooses the optimal plant size — the LRATC is the minimum of all SRATC curves at each output. The LRATC touches (envelopes) each SRATC at the output for which that plant size is optimal
C) All plants have the same fixed costs
D) SRATC curves never intersect LRATC

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Answer: B) For each output level the firm chooses the optimal plant size — the LRATC is the minimum of all SRATC curves at each output. The LRATC touches (envelopes) each SRATC at the output for which that plant size is optimal

Each SRATC corresponds to a fixed plant size. The firm selects the optimal plant for each long-run output — so LRATC = min over all K̄ of SRATC(y,K̄).