Source: Intermediate Microeconomics, Problem Set 3 (Questions 1 & 3)

Tags: short-run costs, marginal product of labour, diminishing marginal returns, MC, AVC, AC, AFC, profit maximisation, shutdown rule, competitive firm, break-even, lump-sum tax, fixed cost, variable cost

TL;DR

A competitive firm in the short run has at least one fixed input (typically capital). You can build its entire cost structure from a production table and input prices, then use the MC = MR rule to find the profit-maximising output. If price drops below the minimum AVC, the firm shuts down to limit its losses to fixed costs alone.

Key Terms

Marginal product of labour (MPL)

The additional output produced by one more unit of labour, holding all other inputs constant. Calculated as ΔQ / ΔL.

Diminishing marginal returns

The point at which each additional unit of a variable input (labour) adds less output than the previous unit. MPL begins to fall.

Fixed cost (FC)

Costs that do not change with output. In the short run, FC = r × K, where r is the rental rate and K is the fixed capital.

Variable cost (VC)

Costs that change with output. VC = w × L, where w is the wage rate and L is the quantity of labour.

Total cost (TC)

TC = FC + VC.

Average fixed cost (AFC)

AFC = FC / q. Falls continuously as output rises (spreading overhead).

Average variable cost (AVC)

AVC = VC / q. Typically U-shaped.

Average cost (AC)

AC = TC / q, or equivalently AFC + AVC.

Marginal cost (MC)

The cost of producing one more unit of output. MC = ΔTC / Δq = w / MPL. MC is inversely related to MPL.

Marginal revenue (MR)

For a perfectly competitive firm, MR = P (the market price), because the firm is a price taker.

Shutdown rule

A firm should shut down in the short run if P < min AVC. At that point, revenue cannot even cover variable costs, so the firm loses more by operating than by producing nothing.

Profit-maximisation rule

Produce where MR = MC (with MC rising). For a competitive firm, this means P = MC on the upward-sloping portion of the MC curve.

Core Content

Building a Cost Table from Production Data (Question 1)

The problem gives a production schedule (output q for each level of labour L) and fixed parameters:

• Capital: K = 10 (fixed)

• Wage: w = 20

• Rental rate: r = 15

• Output price: p = 2

Step 1: Compute FC, VC, TC

• FC = r × K = 15 × 10 = 150 (constant for every row)

• VC = w × L (increases by 20 for each additional worker)

• TC = FC + VC

Step 2: Compute MPL

MPL = Δq / ΔL. Since ΔL = 1 for each row, MPL is simply the change in output.

Step 3: Compute MC

MC = w / MPL = 20 / MPL. This is equivalent to ΔVC / Δq, since each extra worker costs 20 and produces MPL units.

Step 4: Revenue columns

• R = p × q = 2q

• MR = p = 2 (constant for a competitive firm)

Step 5: Average cost columns

• AFC = 150 / q

• AVC = VC / q

• AC = TC / q

Completed table:

Identifying the Production Period and Diminishing Returns (1a)

This is a short-run scenario. Capital is fixed at 10 units, and only labour varies. The short run is defined by having at least one input that cannot be adjusted.

Diminishing marginal returns begin after L = 5 (q = 75). MPL rises from 5 to 25 over the first five workers, then falls from 25 to 20 at L = 6. The peak MPL of 25 occurs at L = 5.

A common mistake: diminishing returns do not mean negative returns. MPL is still positive at L = 10 (MPL = 1), output is still growing, just more slowly.

Profit Maximisation at p = 2 (1b)

The rule: produce where MR = MC, on the rising portion of MC.

• MR = 2 throughout

• MC = 2 occurs at two output levels: q = 15 (MC falling) and q = 120 (MC rising)

• Only q = 120 satisfies MR = MC with MC rising

At q = 120:

• Revenue = 2 × 120 = 240

• TC = 310

• Profit = 240 − 310 = −70

The firm is making a loss of 70. But it should still operate, because the alternative (shutting down) means losing the entire FC of 150. By operating, the firm covers all its variable costs and chips away at fixed costs.

Check: at q = 120, AVC = 1.33. Since p = 2 > 1.33 = AVC, operating is better than shutting down.

The Shutdown Decision at p = 1 (1c)

When price falls to p = 1, find the minimum AVC from the table.

Looking at the AVC column, the lowest value is 1.26 at q = 95.

Since p = 1 < 1.26 = min AVC, the firm shuts down.

If it tried to operate (producing at q = 95, where MC = 1 on the rising portion):

• Revenue = 1 × 95 = 95

• TC = 270

• Loss from operating = 95 − 270 = −175

Loss from shutting down = −FC = −150.

Shutting down is less painful (−150 vs. −175), which confirms the AVC rule. The firm cannot even cover its variable costs at this price.

Reading Cost Curves from a Graph (Question 3)

Question 3 provides a graph showing MC, AC, and AVC curves. All readings are approximate.

Key features visible in the graph:

• MC is U-shaped, with its minimum around q = 35

• AVC reaches its minimum of roughly 20 at about q = 55

• AC reaches its minimum of roughly 45 at about q = 85

• MC crosses AVC from below at the AVC minimum, and crosses AC from below at the AC minimum

Profit Maximisation at p = 45 (3a, 3b)

MR = p = 45 for a competitive firm. Find where MC = 45 on the rising portion of MC.

From the graph, MC = 45 at approximately q = 85.

At q = 85, AC ≈ 45 as well (MC crosses AC at the AC minimum). This puts the firm roughly at break-even.

Profit ≈ (p − AC) × q ≈ (45 − 45) × 85 ≈ 0

The firm is at (or very close to) the break-even point, earning zero economic profit.

If the graph reading places AC slightly below 45 at q = 85 (say AC ≈ 35), then profit = (45 − 35) × 85 = 850. The precise answer depends on careful graph reading. Check your own reading against the curve.

Effect of a Lump-Sum Tax of 1,320 (3c)

A lump-sum tax is a fixed amount, independent of output. It increases FC but does not change VC or MC.

• MC is unchanged, so the profit-maximising output stays at q ≈ 85

• AC shifts up by Tax / q = 1,320 / 85 ≈ 15.53 at q = 85

• New profit = Old profit − 1,320

If the firm was earning roughly zero profit before the tax, new profit ≈ 0 − 1,320 = −1,320.

If the firm was earning 850 before the tax, new profit ≈ 850 − 1,320 = −470.

The critical point: a lump-sum tax never changes the firm's short-run output decision or shutdown decision (it does not affect MC or AVC).

Shutdown Analysis at p = 20 (3d, 3e, 3f)

The minimum AVC from the graph is approximately 20, at roughly q = 55.

Since p = 20 ≈ min AVC, the firm is right at the shutdown point. It is indifferent between operating and shutting down, because losses are roughly the same either way.

If operating (3e):

• Produce at q ≈ 55 (where MC = 20 on the rising portion)

• At q = 55, AC ≈ 40 (read from graph)

• Loss = (p − AC) × q = (20 − 40) × 55 = −1,100

If shutting down (3f):

• Loss = −FC

• FC = (AC − AVC) × q = (40 − 20) × 55 = 1,100

• Loss = −1,100

The losses are approximately equal, confirming this is the shutdown point.

Formulas / Key Relationships

Why It Matters / Exam Flags

⚠️ When finding profit-maximising output, always pick the point where MC = MR on the rising portion of MC. If MC = MR at two outputs, the one with falling MC is a loss-maximising point.

⚠️ MC = w / MPL. This is the link between production theory and cost theory. When MPL rises, MC falls, and vice versa. Diminishing marginal returns cause the MC curve to slope upward.

⚠️ A firm can be making a loss and still operate. The question is whether revenue covers variable costs (P > AVC). Shutting down avoids VC but still incurs FC.

⚠️ A lump-sum tax adds to FC. It raises AC but leaves MC and AVC untouched. The short-run output choice and shutdown decision are unaffected.

⚠️ At the shutdown point (P = min AVC), the loss from operating equals the loss from shutting down, and both equal FC.

⚠️ MC passes through the minimum of both AVC and AC from below. This is a mathematical property, not a coincidence.

Practice Q&A

Q: Why does the firm in Question 1 still operate at p = 2 despite making a loss?

A: Because p = 2 exceeds the minimum AVC of 1.26 (at q = 95). By operating, the firm's loss is 70, which is less than the FC of 150 it would lose by shutting down. Revenue covers all variable costs plus 80 of fixed costs.

Q: If MC = MR at two different output levels, how do you choose?

A: Pick the one where MC is rising. On the falling portion of MC, the firm could increase profit by producing more (since MR would exceed the next unit's MC). Only the rising-MC intersection is a true maximum.

Q: A government imposes a lump-sum tax of 500 on a competitive firm. What happens to its output in the short run?

A: Nothing. A lump-sum tax raises fixed costs and therefore AC, but it does not change MC or AVC. The firm produces the same quantity. Its profit falls by exactly 500.

Q: A firm has FC = 200, and at its best output level, revenue is 300 and VC is 250. Should it operate?

A: Yes. Revenue (300) exceeds VC (250), so the firm covers all variable costs and contributes 50 toward fixed costs. Operating yields a loss of 150. Shutting down yields a loss of 200.

Q: Where do diminishing marginal returns start, and why does it matter for costs?

A: They start where MPL begins to decline (in Question 1, after L = 5). Once MPL falls, each additional unit of output requires proportionally more labour, so MC begins to rise. This is why the MC curve eventually slopes upward.

Related Terms / Search Tags

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