The Theory of the Firm Under Perfect Competition — NCERT Solutions

Given/Concept: A perfectly competitive market is an idealised market structure with specific features.

Characteristics of a Perfectly Competitive Market:

1. Large number of buyers and sellers: There are so many buyers and sellers that no single buyer or seller can influence the market price. Each firm is a price-taker.

2. Homogeneous product: All firms sell an identical (homogeneous) product, so buyers have no preference for one seller over another.

3. Free entry and exit: Firms can freely enter or exit the industry in the long run without any barriers.

4. Perfect information: All buyers and sellers have complete and perfect knowledge about prices and market conditions.

5. Perfect mobility of factors: Factors of production can move freely between industries.

6. No transportation costs: There are no transportation costs, so the price is uniform everywhere.

Conclusion: Because of these features, a single market price prevails and every firm is a price-taker.

Given: A price-taking firm sells its output at the prevailing market price.

Concept/Formula:
$$\text{Total Revenue (TR)} = \text{Market Price (P)} \times \text{Quantity Sold (Q)}$$

Explanation:

• The market price $P$ is determined by the forces of demand and supply in the market. The firm accepts this price as given.
• If the firm sells $Q$ units at price $P$, its total revenue is simply $P \times Q$.
• Since $P$ is constant for a price-taking firm, TR increases proportionally with $Q$. For every additional unit sold, TR rises by exactly $P$.

Conclusion: Total revenue is directly and proportionally related to the quantity sold, with market price acting as the constant of proportionality: $TR = P \times Q$.

Concept: In a perfectly competitive market, a firm is a price-taker — it cannot influence the market price.

Definition: The price line (also called the demand curve faced by a competitive firm) is a horizontal straight line drawn at the level of the prevailing market price $P$.

Explanation:

• Since the firm can sell any quantity it wishes at the fixed market price $P$, the price remains constant regardless of the quantity sold by the firm.
• This horizontal line at $P$ represents both the Average Revenue (AR) and the Marginal Revenue (MR) of the firm, because:

$$AR = MR = P = \text{constant}$$

Conclusion: The price line is a horizontal line at the market price, indicating that the firm faces a perfectly elastic demand curve.

Given: For a price-taking firm, $TR = P \times Q$, where $P$ is constant.

Why it is an upward-sloping straight line:

• Since $P$ is constant, $TR = P \times Q$ is a linear function of $Q$.
• As $Q$ increases by 1 unit, $TR$ increases by a fixed amount equal to $P$.
• Therefore, the TR curve is a straight line with a positive (upward) slope equal to $P$.

Why it passes through the origin:

• When the quantity sold $Q = 0$, we get:

$$TR = P \times 0 = 0$$

• So when output is zero, total revenue is also zero.
• Hence the TR curve passes through the origin (the point where both $Q = 0$ and $TR = 0$).

Conclusion: The TR curve is an upward-sloping straight line through the origin with slope equal to the market price $P$.

Given: For a price-taking firm, market price $P$ is constant.

Formula:
$$AR = \frac{TR}{Q} = \frac{P \times Q}{Q} = P$$

Explanation:

• Average Revenue is defined as total revenue per unit of output sold.
• Since $TR = P \times Q$, dividing both sides by $Q$ gives $AR = P$.
• This means that for every level of output, the average revenue equals the market price.

Conclusion: For a price-taking firm, Average Revenue (AR) is always equal to the market price ($P$) at every level of output.

Given: For a price-taking firm, market price $P$ is constant.

Formula:
$$MR = \frac{\Delta TR}{\Delta Q}$$

Derivation:

• $TR$ at output $Q$ is $P \cdot Q$.
• $TR$ at output $Q+1$ is $P \cdot (Q+1)$.
• Therefore, $MR = P(Q+1) - PQ = P$.

Explanation:

• Since the firm sells each additional unit at the same market price $P$, the addition to total revenue from selling one more unit is always $P$.
• Hence $MR = P$ at every level of output.

Conclusion: For a price-taking firm, Marginal Revenue (MR) is always equal to the market price ($P$): $MR = P$.

Concept: A profit-maximising firm chooses output to maximise $\pi = TR - TC$.

Conditions for profit maximisation at a positive output level:

(i) Short Run:
Three conditions must hold simultaneously:

1. $P = SMC$ (Price equals Short-Run Marginal Cost) — the first-order condition.
2. $SMC$ is non-decreasing at that output — the second-order (stability) condition.
3. $P \geq AVC$ (Price must cover Average Variable Cost) — the shut-down condition.

(ii) Long Run:
Three conditions must hold simultaneously:

1. $P = LRMC$ (Price equals Long-Run Marginal Cost).
2. $LRMC$ is non-decreasing at that output.
3. $P \geq LRAC$ (Price must cover Long-Run Average Cost).

Explanation:

• Condition (1) ensures that profit cannot be increased by changing output.
• Condition (2) ensures it is a maximum and not a minimum.
• Condition (3) ensures the firm does not shut down (it at least covers variable/total costs).

Conclusion: All three conditions must hold together for a profit-maximising firm to produce positive output.

Answer: No, a profit-maximising firm in a competitive market cannot produce a positive level of output at which market price is not equal to marginal cost (given the other conditions are satisfied).

Explanation:

• Case 1: $P > MC$ — If price exceeds marginal cost, the revenue from selling one more unit ($P$) is greater than the cost of producing it ($MC$). The firm can increase profit by producing more. So this cannot be a profit-maximising position.

• Case 2: $P < MC$ — If price is less than marginal cost, the revenue from the last unit is less than its cost. The firm can increase profit by reducing output. So this also cannot be a profit-maximising position.

• Case 3: $P = MC$ — Only when price equals marginal cost is there no incentive to change output. Profit cannot be increased by either expanding or contracting output.

Conclusion: Therefore, at the profit-maximising positive output level, it is necessary that $P = MC$. Any deviation from this condition means the firm can do better by adjusting output.

Answer: No, a profit-maximising firm will not produce at a level of output where marginal cost is falling.

Explanation:

• One of the necessary conditions for profit maximisation is that $P = MC$ and $MC$ must be non-decreasing (i.e., the $MC$ curve must be rising or flat) at the chosen output level.

• Suppose the firm is at an output where $P = MC$ but $MC$ is falling. This means that for the next unit of output, $MC$ will be even lower than $P$. So the firm can earn more profit by producing that additional unit. This contradicts the assumption that the current output is profit-maximising.

• More formally, when $MC$ is falling, the point $P = MC$ corresponds to a profit minimum, not a profit maximum.

Conclusion: A profit-maximising firm will only produce where $MC$ is non-decreasing (rising part of the $MC$ curve). It will never choose to produce in the range where $MC$ is falling.

Answer: No, a profit-maximising firm will not produce a positive output in the short run if $P < \min(AVC)$.

Explanation:

• In the short run, total cost = Total Fixed Cost (TFC) + Total Variable Cost (TVC).
• If the firm shuts down (produces zero output), it still incurs TFC. Its loss = TFC.
• If the firm produces a positive output $Q$ at price $P < AVC$:

$$TR = P \times Q < AVC \times Q = TVC$$

• So $TR < TVC$, which means the firm cannot even cover its variable costs.
• Loss from producing = $TC - TR = TFC + TVC - TR > TFC$.
• This loss is greater than the loss from shutting down (which is only TFC).

Conclusion: When $P < \min(AVC)$, the firm minimises its loss by shutting down and producing zero output. It will not produce any positive output in the short run.

Answer: No, a profit-maximising firm will not produce a positive output in the long run if $P < \min(AC)$.

Explanation:

• In the long run, there are no fixed costs — all costs are variable. The firm can exit the industry without any loss.
• If the firm shuts down in the long run, its profit = 0 (no revenue, no cost).
• If $P < AC$ at every level of output, then $TR < TC$ for every positive output level, meaning the firm makes a loss (negative profit) at every positive output.
• Since producing zero gives profit = 0, which is better than a negative profit, the firm will choose to exit the industry.

Conclusion: In the long run, a firm will produce positive output only if $P \geq \min(LRAC)$. If $P < \min(AC)$, the firm will shut down and exit the market, producing zero output.

Concept: The short-run supply curve shows the quantity a profit-maximising firm is willing to supply at each market price in the short run.

Short-Run Supply Curve:

The short-run supply curve of a firm consists of two parts:

1. When $P \geq \min(AVC)$: The firm produces at the output where $P = SMC$ and $SMC$ is rising. So the supply curve is the rising (upward-sloping) portion of the SMC curve at and above the minimum AVC point.

2. When $P < \min(AVC)$: The firm shuts down and supplies zero output.

Formally:
$$q^s = \begin{cases} \text{output where } P = SMC \text{ (rising part)} & \text{if } P \geq \min(AVC) \\ 0 & \text{if } P < \min(AVC) \end{cases}$$

Conclusion: The short-run supply curve is the rising part of the SMC curve from and above the minimum AVC, combined with zero output for all prices below the minimum AVC.

Concept: The long-run supply curve shows the quantity a profit-maximising firm is willing to supply at each market price in the long run, when all inputs are variable.

Long-Run Supply Curve:

The long-run supply curve of a firm consists of two parts:

1. When $P \geq \min(LRAC)$: The firm produces at the output where $P = LRMC$ and $LRMC$ is rising. So the supply curve is the rising (upward-sloping) portion of the LRMC curve at and above the minimum LRAC point.

2. When $P < \min(LRAC)$: The firm exits the industry and supplies zero output.

Formally:
$$q^s = \begin{cases} \text{output where } P = LRMC \text{ (rising part)} & \text{if } P \geq \min(LRAC) \\ 0 & \text{if } P < \min(LRAC) \end{cases}$$

Conclusion: The long-run supply curve is the rising part of the LRMC curve from and above the minimum LRAC, combined with zero output for all prices below the minimum LRAC.

Concept: Technological progress means the firm can produce the same output using fewer inputs, or more output using the same inputs — i.e., it reduces the cost of production.

Effect on Supply Curve:

• Technological progress reduces the marginal cost (MC) of production at every level of output.
• Since the supply curve of a firm is the rising part of its MC curve (above minimum AVC), a fall in MC shifts the MC curve downward and to the right.
• This means the firm is now willing to supply more output at every given price than before.

Conclusion: Technological progress shifts the supply curve of a firm to the right (rightward shift), indicating an increase in supply at every price level.