The Theory of the Firm Under Perfect Competition

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.

Concept: A unit tax is a fixed tax imposed per unit of output produced (e.g., Rs $t$ per unit).

Effect on Supply Curve:

- A unit tax of $t$ per unit increases the marginal cost of production by $t$ at every level of output.
- If the original MC was $MC_0$, after the tax it becomes $MC_0 + t$.
- Since the supply curve is the rising part of the MC curve, the MC curve shifts upward by $t$, which means the supply curve shifts to the left.
- At every price, the firm now supplies less output than before.

Conclusion: The imposition of a unit tax shifts the supply curve of a firm to the left (leftward shift), indicating a decrease in supply at every price level.

Concept: Inputs (like labour, raw materials, capital) are used in the production process. Their prices affect the cost of production.

Effect on Supply Curve:

- An increase in the price of an input raises the cost of production — both total cost and marginal cost increase at every level of output.
- Since the supply curve is the rising part of the MC curve, an upward shift in MC shifts the supply curve to the left.
- At every given price, the firm is now willing to supply less output than before.

Conversely: A decrease in input prices lowers MC and shifts the supply curve to the right.

Conclusion: An increase in the price of an input shifts the supply curve of a firm to the left, indicating a decrease in supply. A decrease in input price shifts the supply curve to the right.

Concept: The market supply curve is obtained by the horizontal summation of the supply curves of all individual firms in the market.

Effect of an Increase in the Number of Firms:

- When more firms enter the market, at every given price, the total quantity supplied in the market increases (since more firms are now supplying).
- Horizontally summing more supply curves results in a larger total quantity supplied at each price.
- This shifts the market supply curve to the right.

Example: If originally 10 firms each supply 5 units at price $P$, market supply = 50 units. If 5 more firms enter and each supplies 5 units, market supply = 75 units at the same price $P$.

Conclusion: An increase in the number of firms in a market shifts the market supply curve to the right, indicating an increase in market supply at every price level.

Meaning:
The price elasticity of supply measures the responsiveness of quantity supplied to a change in the market price of a good. It tells us by what percentage the quantity supplied changes when the price changes by one per cent.

Formula:
$$e_s = \frac{\text{Percentage change in quantity supplied}}{\text{Percentage change in price}}$$

$$e_s = \frac{\Delta Q / Q}{\Delta P / P} = \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}$$

where:
- $\Delta Q$ = change in quantity supplied
- $\Delta P$ = change in price
- $P$ = initial price
- $Q$ = initial quantity supplied

Interpretation:
- e_s > 1: Elastic supply (quantity supplied is highly responsive to price changes)
- e_s < 1: Inelastic supply
- $e_s = 1$: Unitary elastic supply
- $e_s = 0$: Perfectly inelastic supply
- $e_s = \infty$: Perfectly elastic supply

Conclusion: Price elasticity of supply is always non-negative (since supply curves are upward sloping) and is measured by the formula above.

Given: Market price $P = \text{Rs } 10$

Formulas:
- $TR = P \times Q = 10 \times Q$
- $MR = TR_Q - TR_{Q-1}$ (change in TR when one more unit is sold)
- $AR = TR / Q = P = 10$ (for all Q > 0)

Calculations:

| Quantity Sold ($Q$) | TR (Rs) | MR (Rs) | AR (Rs) |
|---|---|---|---|
| 0 | $10 \times 0 = 0$ | — | — |
| 1 | $10 \times 1 = 10$ | $10 - 0 = 10$ | $10/1 = 10$ |
| 2 | $10 \times 2 = 20$ | $20 - 10 = 10$ | $20/2 = 10$ |
| 3 | $10 \times 3 = 30$ | $30 - 20 = 10$ | $30/3 = 10$ |
| 4 | $10 \times 4 = 40$ | $40 - 30 = 10$ | $40/4 = 10$ |
| 5 | $10 \times 5 = 50$ | $50 - 40 = 10$ | $50/5 = 10$ |
| 6 | $10 \times 6 = 60$ | $60 - 50 = 10$ | $60/6 = 10$ |

Observation: Since price is constant at Rs 10, $MR = AR = P = \text{Rs } 10$ at every level of output. TR increases by Rs 10 for each additional unit sold.

Given: TR and TC schedules as in the table.

Formula: $\text{Profit} = TR - TC$

Market Price: Since the firm is competitive, $P = AR = TR/Q$ (for Q > 0).
At $Q = 1$: $P = TR/Q = 5/1 = \text{Rs } 5$

Profit Calculations:

| Quantity Sold | TR (Rs) | TC (Rs) | Profit = TR − TC (Rs) |
|---|---|---|---|
| 0 | 0 | 5 | $0 - 5 = -5$ |
| 1 | 5 | 7 | $5 - 7 = -2$ |
| 2 | 10 | 10 | $10 - 10 = 0$ |
| 3 | 15 | 12 | $15 - 12 = +3$ |
| 4 | 20 | 15 | $20 - 15 = +5$ |
| 5 | 25 | 23 | $25 - 23 = +2$ |
| 6 | 30 | 33 | $30 - 33 = -3$ |
| 7 | 35 | 40 | $35 - 40 = -5$ |

Market Price: $P = \text{Rs } 5$ per unit (since TR increases by Rs 5 for each unit sold).

Profit-maximising output: At $Q = 4$, profit is maximum at Rs 5.

Given: $P = \text{Rs } 10$, TC schedule as given.

Formula:
- $TR = P \times Q = 10Q$
- $\text{Profit} = TR - TC$

Calculations:

| Output ($Q$) | TC (Rs) | TR = 10Q (Rs) | Profit = TR − TC (Rs) |
|---|---|---|---|
| 0 | 5 | 0 | $0 - 5 = -5$ |
| 1 | 15 | 10 | $10 - 15 = -5$ |
| 2 | 22 | 20 | $20 - 22 = -2$ |
| 3 | 27 | 30 | $30 - 27 = +3$ |
| 4 | 31 | 40 | $40 - 31 = +9$ |
| 5 | 38 | 50 | $50 - 38 = +12$ |
| 6 | 49 | 60 | $60 - 49 = +11$ |
| 7 | 63 | 70 | $70 - 63 = +7$ |
| 8 | 81 | 80 | $80 - 81 = -1$ |
| 9 | 101 | 90 | $90 - 101 = -11$ |
| 10 | 123 | 100 | $100 - 123 = -23$ |

Verification using MC = P:

| Output | MC = $\Delta TC$ |
|---|---|
| 1 | 10 |
| 2 | 7 |
| 3 | 5 |
| 4 | 4 |
| 5 | 7 |
| 6 | 11 |
| 7 | 14 |
| 8 | 18 |

At $Q = 5$: MC = 7 < 10 = P; at $Q = 6$: MC = 11 > 10 = P. So $P = MC$ is approximately satisfied between $Q = 5$ and $Q = 6$. Since we deal in whole units, profit is highest at $Q = 5$ (Rs 12).

Conclusion: The profit-maximising level of output is $Q = 5$, where profit = Rs 12.

Concept: Market supply = Horizontal summation of individual supply curves.
$$SS_{\text{market}} = SS_1 + SS_2$$

Calculations:

| Price (Rs) | SS1 (units) | SS2 (units) | Market Supply = SS1 + SS2 (units) |
|---|---|---|---|
| 0 | 0 | 0 | $0 + 0 = 0$ |
| 1 | 0 | 0 | $0 + 0 = 0$ |
| 2 | 0 | 0 | $0 + 0 = 0$ |
| 3 | 1 | 1 | $1 + 1 = 2$ |
| 4 | 2 | 2 | $2 + 2 = 4$ |
| 5 | 3 | 3 | $3 + 3 = 6$ |
| 6 | 4 | 4 | $4 + 4 = 8$ |

Conclusion: The market supply schedule is obtained by adding the quantities supplied by both firms at each price level.

Concept: Market supply = $SS_1 + SS_2$ at each price.

Calculations:

| Price (Rs) | SS1 (kg) | SS2 (kg) | Market Supply = SS1 + SS2 (kg) |
|---|---|---|---|
| 0 | 0 | 0 | $0 + 0 = 0$ |
| 1 | 0 | 0 | $0 + 0 = 0$ |
| 2 | 0 | 0 | $0 + 0 = 0$ |
| 3 | 1 | 0 | $1 + 0 = 1$ |
| 4 | 2 | 0.5 | $2 + 0.5 = 2.5$ |
| 5 | 3 | 1 | $3 + 1 = 4$ |
| 6 | 4 | 1.5 | $4 + 1.5 = 5.5$ |
| 7 | 5 | 2 | $5 + 2 = 7$ |
| 8 | 6 | 2.5 | $6 + 2.5 = 8.5$ |

Conclusion: The market supply schedule is obtained by adding the quantities supplied by both firms at each price. Note that Firm 2 starts supplying only from price Rs 4 onwards.

Given: Three identical firms, so $SS_2 = SS_3 = SS_1$ at every price.

Concept: Market supply = $SS_1 + SS_2 + SS_3 = 3 \times SS_1$

Calculations:

| Price (Rs) | SS1 (units) | Market Supply = 3 × SS1 (units) |
|---|---|---|
| 0 | 0 | $3 \times 0 = 0$ |
| 1 | 0 | $3 \times 0 = 0$ |
| 2 | 2 | $3 \times 2 = 6$ |
| 3 | 4 | $3 \times 4 = 12$ |
| 4 | 6 | $3 \times 6 = 18$ |
| 5 | 8 | $3 \times 8 = 24$ |
| 6 | 10 | $3 \times 10 = 30$ |
| 7 | 12 | $3 \times 12 = 36$ |
| 8 | 14 | $3 \times 14 = 42$ |

Conclusion: Since all three firms are identical, the market supply at each price is simply three times the supply of firm 1.

Given:
- Initial price: $P_1 = \text{Rs } 10$, Initial TR: $TR_1 = \text{Rs } 50$
- New price: $P_2 = \text{Rs } 15$, New TR: $TR_2 = \text{Rs } 150$

Step 1: Find initial and final quantities.
$$Q_1 = \frac{TR_1}{P_1} = \frac{50}{10} = 5 \text{ units}$$
$$Q_2 = \frac{TR_2}{P_2} = \frac{150}{15} = 10 \text{ units}$$

Step 2: Calculate changes.
$$\Delta Q = Q_2 - Q_1 = 10 - 5 = 5 \text{ units}$$
$$\Delta P = P_2 - P_1 = 15 - 10 = \text{Rs } 5$$

Step 3: Apply the price elasticity of supply formula.
$$e_s = \frac{\Delta Q}{\Delta P} \times \frac{P_1}{Q_1} = \frac{5}{5} \times \frac{10}{5} = 1 \times 2 = 2$$

Conclusion: The price elasticity of the firm's supply curve is $\boxed{e_s = 2}$.

Given:
- $P_1 = \text{Rs } 5$, $P_2 = \text{Rs } 20$
- $\Delta Q = 15$ units
- $e_s = 0.5$

Step 1: Calculate the change in price.
$$\Delta P = P_2 - P_1 = 20 - 5 = \text{Rs } 15$$

Step 2: Use the elasticity formula to find $Q_1$.
$$e_s = \frac{\Delta Q}{\Delta P} \times \frac{P_1}{Q_1}$$
$$0.5 = \frac{15}{15} \times \frac{5}{Q_1}$$
$$0.5 = 1 \times \frac{5}{Q_1}$$
$$Q_1 = \frac{5}{0.5} = 10 \text{ units}$$

Step 3: Find $Q_2$.
$$Q_2 = Q_1 + \Delta Q = 10 + 15 = 25 \text{ units}$$

Conclusion:
- Initial output level: $Q_1 = \mathbf{10}$ units
- Final output level: $Q_2 = \mathbf{25}$ units

Given:
- $P_1 = \text{Rs } 10$, $Q_1 = 4$ units
- $P_2 = \text{Rs } 30$
- $e_s = 1.25$

Step 1: Calculate the change in price.
$$\Delta P = P_2 - P_1 = 30 - 10 = \text{Rs } 20$$

Step 2: Use the elasticity formula to find $\Delta Q$.
$$e_s = \frac{\Delta Q}{\Delta P} \times \frac{P_1}{Q_1}$$
$$1.25 = \frac{\Delta Q}{20} \times \frac{10}{4}$$
$$1.25 = \frac{\Delta Q}{20} \times 2.5$$
$$1.25 = \frac{2.5 \cdot \Delta Q}{20}$$
$$\Delta Q = \frac{1.25 \times 20}{2.5} = \frac{25}{2.5} = 10 \text{ units}$$

Step 3: Find the new quantity supplied.
$$Q_2 = Q_1 + \Delta Q = 4 + 10 = 14 \text{ units}$$

Conclusion: At the new price of Rs 30, the firm will supply $\boxed{14}$ units of output.