Chapter 4 The Theory of the Firm under Perfect Competition

Introduction to the Firm and Profit Maximisation

This chapter examines how a firm makes decisions about how much to produce and sell in a specific market environment.

A firm is an entity that undertakes production, which is the process of transforming inputs (like labour, capital, raw materials) into output (goods or services).

To acquire inputs, a firm incurs costs. By selling the output, the firm earns revenue.

Profit is the difference between the firm's total revenue and its total cost ($\Pi = TR - TC$).

A key assumption in this analysis is that the firm's primary objective is to maximise its profit.

The quantity of output a firm chooses to produce and supply to the market is the one that yields the highest possible profit.

Perfect Competition: Defining Features

A perfectly competitive market is a theoretical market structure characterized by several specific conditions:

• Large number of buyers and sellers: There are so many participants on both sides of the market that no single buyer or seller has any significant influence on the market price. Each individual unit is tiny relative to the overall market size.
• Homogenous product: All firms produce identical products. Buyers perceive the products from different firms as perfect substitutes. There is no product differentiation.
• Free entry and exit: Firms can easily enter or leave the market without facing significant barriers. This ensures that the number of firms can adjust in the long run.
• Perfect information: Both buyers and sellers have complete and accurate information about prices, product quality, and other relevant market conditions.

Price-Taking Behavior

The defining characteristics of perfect competition lead to a crucial outcome: price-taking behavior.

• From the firm's perspective: A perfectly competitive firm is a price-taker because it cannot influence the market price. If it tries to charge a price higher than the market price, buyers will simply purchase from other firms selling the identical product. The firm can sell as much output as it wants at the prevailing market price.
• From the buyer's perspective: Buyers are also price-takers. They can purchase as much as they want at the market price. If a buyer offers less than the market price, no firm will sell to them because they can sell their entire output at the market price.

Essentially, the market price is determined by the forces of overall market demand and supply, and individual firms and buyers must accept this price.

Revenue

Revenue is the income a firm earns from selling its output.

Total Revenue (TR)

Total Revenue (TR) is the total income a firm receives from selling a specific quantity of output. In a perfectly competitive market, where the firm is a price-taker, TR is calculated by multiplying the market price (p) by the quantity sold (q):

$TR = p \times q$

Since the price (p) is constant for a price-taking firm, Total Revenue increases proportionally with the quantity sold. The Total Revenue curve is an upward-sloping straight line that starts from the origin (because TR=0 when q=0).

The slope of the TR curve is equal to the market price (p).

Average Revenue (AR)

Average Revenue (AR) is the revenue earned per unit of output sold. It is calculated by dividing Total Revenue (TR) by the quantity sold (q):

$AR = \frac{TR}{q}$

Substituting the formula for TR ($p \times q$):

$AR = \frac{p \times q}{q} = p$

Thus, for a price-taking firm in perfect competition, Average Revenue is equal to the market price.

Marginal Revenue (MR)

Marginal Revenue (MR) is the additional revenue earned from selling one more unit of output. It is the change in Total Revenue resulting from a one-unit change in quantity sold:

$MR = \frac{\Delta TR}{\Delta q}$

In a perfectly competitive market, when a firm sells one more unit, the total revenue increases by exactly the price of that unit, because the firm can sell the additional unit at the market price without affecting it.

Therefore, for a price-taking firm, Marginal Revenue is equal to the market price.

Relationship between Price, AR, and MR in Perfect Competition

A key characteristic for a firm in perfect competition is that its Price (P), Average Revenue (AR), and Marginal Revenue (MR) are all equal:

$P = AR = MR$

This equality arises because the firm is a price-taker and faces a perfectly elastic demand curve at the market price. This means the firm can sell any quantity at the given market price (p), but nothing at a price higher than p.

Graphically, the demand curve facing a perfectly competitive firm is a horizontal straight line at the market price. This line is also the firm's Price Line, AR curve, and MR curve.

Profit Maximisation

A firm aims to produce the level of output ($q$) that maximises its profit ($\Pi$). Profit is the difference between Total Revenue (TR) and Total Cost (TC).

$\Pi(q) = TR(q) - TC(q)$

To find the output level that maximises profit, the firm considers how profit changes as output changes.

Conditions for Profit Maximisation

For a firm operating in a competitive market to maximise profit by producing a positive level of output ($q > 0$), three conditions must be met:

1. Condition 1: Price equals Marginal Cost ($p = MC$)

   As long as the additional revenue from selling one more unit (MR) is greater than the additional cost of producing that unit (MC), profit will increase by producing more. Conversely, if MR is less than MC, profit will decrease. Profit is maximised where the marginal benefit (MR) equals the marginal cost (MC). Since for a competitive firm, $MR=p$, the profit-maximising output level is where $p = MC$.

2. Condition 2: Marginal Cost must be non-decreasing at the profit-maximising output level.

   The MC curve must be upward-sloping or flat at the output level where $p=MC$. If MC were falling at this point, increasing output slightly would mean MC falls below the constant price, making additional units profitable. So, the firm would increase output further, meaning the initial point could not have been the maximum.

3. Condition 3: Price must be greater than or equal to Average Variable Cost (AVC) in the short run, and greater than or equal to Average Cost (AC) in the long run.
   • Short Run ($p \ge AVC_{min}$): In the short run, the firm should produce only if the market price is high enough to cover its average variable costs. If $p < AVC$ at the output level where $p=MC$, the revenue per unit is not even covering the variable cost per unit. In this case, the firm would incur a loss greater than its Total Fixed Cost (TFC) by producing. By shutting down (producing zero output), the firm only loses its TFC. Therefore, if $p < AVC_{min}$, the profit-maximising output in the short run is zero. The condition $p \ge AVC$ ensures the firm covers its variable costs and contributes towards covering fixed costs, making production potentially more profitable (or less lossy) than shutting down.
   • Long Run ($p \ge LRAC_{min}$): In the long run, all costs are variable. The firm will only produce if the market price is high enough to cover its long run average cost ($p \ge LRAC$). If $p < LRAC$ at the output level where $p=LRMC$, the firm is making a loss ($TR < TC$). In the long run, a firm can exit the market and avoid all costs, resulting in zero profit. Therefore, if $p < LRAC_{min}$, the profit-maximising output in the long run is zero. The condition $p \ge LRAC$ ensures the firm makes at least zero economic profit (normal profit).

Graphical Representation of Profit Maximisation

Profit maximisation can be shown graphically using the firm's cost curves and the price line (which is also AR and MR). The firm produces at the intersection of the MR (price line) and the MC curve, provided MC is rising and the price is above the relevant average cost curve (AVC in SR, LRAC in LR).

In the figure, at output $q_0$, the price line intersects the SMC curve from below (SMC is rising). If $p \ge AVC$ at $q_0$, the firm produces $q_0$. Profit is represented by the area between the price line and the SAC curve up to output $q_0$ (if price is above SAC), or loss if price is below SAC.

Specifically, Total Revenue (TR) is the area of the rectangle $p \times q_0$. Total Cost (TC) is $SAC \times q_0$. Profit is $TR - TC = (p - SAC) \times q_0$. Graphically, if $p > SAC$ at $q_0$, profit is the area of the rectangle formed by the price level, the SAC level at $q_0$, and the quantity $q_0$.

Supply Curve of a Firm

A firm's supply curve shows the quantities of output that a firm is willing and able to produce and sell at different market prices, holding other factors constant (technology, input prices).

A supply schedule is a table listing these quantities at various prices. A supply curve is the graphical representation of the supply schedule.

Due to the different conditions regarding fixed inputs, we distinguish between the short run and long run supply curves.

Short Run Supply Curve

The short run supply curve of a perfectly competitive firm is derived directly from its profit maximisation conditions in the short run ($p=SMC$, SMC non-decreasing, $p \ge AVC$).

If the market price ($p$) is below the minimum Average Variable Cost ($AVC_{min}$), the firm will produce zero output. This is because the price is not even covering the variable cost of production, and producing would lead to losses greater than the fixed costs.

If the market price ($p$) is at or above the minimum Average Variable Cost ($AVC_{min}$), the firm will produce the output level where $p = SMC$ and the SMC curve is upward-sloping.

Therefore, the firm's Short Run Supply Curve is the portion of its Short Run Marginal Cost (SMC) curve that lies above the minimum point of the Average Variable Cost (AVC) curve. For prices below $AVC_{min}$, the supply quantity is zero.

Long Run Supply Curve

The long run supply curve of a perfectly competitive firm is derived from its profit maximisation conditions in the long run ($p=LRMC$, LRMC non-decreasing, $p \ge LRAC$).

In the long run, all costs are variable. If the market price ($p$) is below the minimum Long Run Average Cost ($LRAC_{min}$), the firm will produce zero output and exit the industry. This is because the firm cannot cover its total costs in the long run at that price.

If the market price ($p$) is at or above the minimum Long Run Average Cost ($LRAC_{min}$), the firm will produce the output level where $p = LRMC$ and the LRMC curve is upward-sloping.

Therefore, the firm's Long Run Supply Curve is the portion of its Long Run Marginal Cost (LRMC) curve that lies above the minimum point of the Long Run Average Cost (LRAC) curve. For prices below $LRAC_{min}$, the supply quantity is zero.

The Shut Down Point

The shut down point is the price and output level at which a firm is indifferent between producing and shutting down temporarily (in the short run) or exiting the market permanently (in the long run).

• Short Run Shut Down Point: This occurs at the minimum point of the Average Variable Cost (AVC) curve. If the price falls below this point, the firm cannot even cover its variable costs and is better off shutting down, producing zero output, and only incurring fixed costs. The supply curve starts from this point (or above).
• Long Run Shut Down Point: This occurs at the minimum point of the Long Run Average Cost (LRAC) curve. If the price falls below this point, the firm cannot cover all its costs in the long run and will exit the market, producing zero output. The long run supply curve starts from this point (or above).

Normal Profit and Break-even Point

Normal Profit is the minimum level of profit required to keep a firm in business in the long run. It is considered a part of the firm's total costs, specifically as the opportunity cost of the entrepreneur's time and capital.

Super-normal Profit (or Economic Profit) is any profit earned above the normal profit.

The Break-even Point is the output level at which a firm earns exactly zero economic profit, meaning it earns just enough revenue to cover all its costs, including normal profit.

In perfect competition, the break-even point occurs at the minimum point of the Average Cost (AC) curve (SAC in the short run, LRAC in the long run), where Price = AC. At this point, $TR = TC$, and profit is zero.

Opportunity Cost (Concept Note)

Opportunity cost is the value of the next best alternative that is forgone when a particular choice is made.

For example, if you invest $\textsf{₹}1,000$ in your business, the opportunity cost is the amount of money you *could* have earned by investing that $\textsf{₹}1,000$ elsewhere (e.g., the interest you would have received from the best available bank deposit). Normal profit for a firm includes the opportunity cost of the owner's capital and time.

Determinants Of A Firm’s Supply Curve

A firm's supply curve is based on its marginal cost curve. Therefore, anything that causes the firm's cost structure to change will affect its supply curve.

Technological Progress

Improvements in technology (like more efficient production methods or better machinery) generally lead to a decrease in the cost of production per unit of output.

With technological progress, a firm can produce the same amount of output with fewer inputs, or more output with the same inputs. This results in the firm's Marginal Cost (MC) curve shifting downwards (or rightwards).

Since the supply curve is based on the MC curve, technological progress causes the firm's supply curve to shift to the right. At any given price, the firm is willing and able to supply a larger quantity.

Input Prices

Changes in the prices of inputs used in production also affect costs.

An increase in the price of an input (e.g., higher wages for labour, more expensive raw materials) raises the cost of producing each unit of output. This causes the firm's Marginal Cost (MC) curve to shift upwards (or leftwards).

Consequently, the firm's supply curve shifts to the left. At any given price, the firm will supply a smaller quantity.

Conversely, a decrease in input prices lowers costs, shifts the MC curve downwards (rightwards), and shifts the supply curve to the right.

Impact of a Unit Tax

A unit tax is a tax levied by the government on each unit of output sold by a firm. For example, a tax of $\textsf{₹}t$ per unit means the firm must pay $\textsf{₹}t$ to the government for every unit it sells.

A unit tax directly increases the firm's cost of production by the amount of the tax per unit. This causes both the Average Cost (AC) and Marginal Cost (MC) curves to shift upwards by the amount of the tax ($t$).

Since the firm's supply curve is determined by its MC curve, the imposition of a unit tax causes the supply curve to shift upwards (or leftwards) by the amount of the tax. At any given price, the firm will supply a lower quantity because its marginal cost is effectively higher.

Market Supply Curve

The market supply curve represents the total quantity of a good that all firms in the market are willing and able to supply at different market prices, at a specific point in time.

Horizontal Summation

The market supply curve is derived by summing up the quantities supplied by all individual firms in the market at each possible price level.

This is known as horizontal summation because we sum the quantities (measured on the horizontal axis) supplied by each firm at each given price (measured on the vertical axis).

If there are $n$ firms in the market, say Firm 1, Firm 2, ..., Firm $n$, and at a given price $p$, Firm $i$ supplies $q_i(p)$, then the total market supply $Q(p)$ at that price is:

$Q(p) = q_1(p) + q_2(p) + \dots + q_n(p)$

Numerical Example with Two Firms:

Suppose Firm 1's supply is $S_1(p)$ and Firm 2's supply is $S_2(p)$.

Firm 1 Supply:

$S_1(p) = \begin{cases} 0 & \text{if } p < 10 \\ p - 10 & \text{if } p \ge 10 \end{cases}$

Firm 2 Supply:

$S_2(p) = \begin{cases} 0 & \text{if } p < 15 \\ p - 15 & \text{if } p \ge 15 \end{cases}$

The market supply $S_m(p)$ is the sum of individual supplies:

$S_m(p) = S_1(p) + S_2(p)$

We need to consider different price ranges:

• If $p < 10$: Both firms supply 0. $S_m(p) = 0 + 0 = 0$.
• If $10 \le p < 15$: Firm 1 supplies $p-10$, Firm 2 supplies 0. $S_m(p) = (p - 10) + 0 = p - 10$.
• If $p \ge 15$: Firm 1 supplies $p-10$, Firm 2 supplies $p-15$. $S_m(p) = (p - 10) + (p - 15) = 2p - 25$.

Combining these, the market supply schedule is:

$S_m(p) = \begin{cases} 0 & \text{if } p < 10 \\ p - 10 & \text{if } 10 \le p < 15 \\ 2p - 25 & \text{if } p \ge 15 \end{cases}$

This shows how the market supply curve is constructed segment by segment by adding the quantities supplied by firms that are active at each price level.

The market supply curve also shifts if the number of firms in the market changes. An increase in the number of firms shifts the market supply curve to the right (more output supplied at each price), and a decrease shifts it to the left.

Price Elasticity Of Supply

The price elasticity of supply ($e_S$) is a measure of the responsiveness of the quantity supplied of a good to a change in its market price, holding other factors constant.

It tells us the percentage change in quantity supplied for a one percent change in price.

The formula for price elasticity of supply is:

$e_S = \frac{\text{Percentage change in Quantity Supplied}}{\text{Percentage change in Price}}$

Which can also be written as:

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

Where $\Delta Q$ is the change in quantity supplied, $Q$ is the initial quantity supplied, $\Delta P$ is the change in price, and $P$ is the initial price.

Since supply curves are typically upward-sloping (firms supply more at higher prices), the price elasticity of supply is usually positive.

• If $e_S > 1$, supply is elastic (Quantity supplied is relatively responsive to price changes).
• If $e_S < 1$, supply is inelastic (Quantity supplied is relatively unresponsive to price changes).
• If $e_S = 1$, supply is unit elastic (Quantity supplied changes proportionally to price changes).
• If $e_S = 0$, supply is perfectly inelastic (Quantity supplied does not change with price - a vertical supply curve).
• If $e_S = \infty$, supply is perfectly elastic (Firms supply any quantity at a specific price, and zero at any other price - a horizontal supply curve).

Calculation Example

Answer:

The Geometric Method

For a straight-line supply curve, the price elasticity of supply at any point can be determined geometrically by examining where the extended line intersects the price or quantity axis.

For a straight-line supply curve passing through point $S$ at $(Q_0, P_0)$, the elasticity at point S is given by the ratio of the distance from the quantity intercept to $Q_0$ ($MQ_0$) divided by the quantity $Q_0$ ($OQ_0$). The formula is $e_S = \frac{MQ_0}{OQ_0}$.

• Panel (a): Supply curve intersects the Price axis (positive intercept on price axis). The extended supply line intersects the quantity axis at a negative value (M is to the left of O). In this case, $MQ_0$ (distance from M to $Q_0$) is greater than $OQ_0$ (distance from O to $Q_0$). So, $e_S = \frac{MQ_0}{OQ_0} > 1$. Supply is elastic at all points on such a curve.
• Panel (b): Supply curve passes through the Origin (0,0). The supply line intersects the quantity axis at the origin (M coincides with O). In this case, $MQ_0 = OQ_0$. So, $e_S = \frac{OQ_0}{OQ_0} = 1$. Supply is unit elastic at all points on such a curve.
• Panel (c): Supply curve intersects the Quantity axis (positive intercept on quantity axis). The extended supply line intersects the quantity axis at a positive value (M is to the right of O). In this case, $MQ_0$ (distance from M to $Q_0$) is less than $OQ_0$ (distance from O to $Q_0$). So, $e_S = \frac{MQ_0}{OQ_0} < 1$. Supply is inelastic at all points on such a curve.