Microeconomics: Perfect Competition and the Supply Curve

Production Decision

• Firms in a competitive market make production decisions to determine their supply curve.
• Firms aim to maximize profits by choosing output levels.
• Profit depends on cost structure:
  • Short Run:
    • Variable costs determine profit-maximizing output.
    • Fixed costs deter new firms from entering.
  • Long Run:
    • Firms can adjust fixed costs, making all costs variable.
    • Absence of fixed costs allows easy entry for firms seeking profits.

Profits

• Firm's profit is calculated as total revenue (TR) minus total cost (TC): $\pi = TR - TC$
• Economic profit refers to revenue minus opportunity cost.
• Business profit may differ due to tax or other accounting reasons.
• Total Revenue (TR) is calculated as Price (P) times Quantity (q): $TR = P \times q$

Profit Maximization

• Firms decide how much to sell to maximize profit.
• Two key decisions:
  • Output Decision: If operating, what output level maximizes profit?
  • Shutdown Decision: Is it more profitable to produce or shut down?

Output Decision

• To find the profit-maximizing output level, consider the profit curve.
• Marginalistic Analysis: Use a series of yes-or-no questions to determine optimal output.
• If producing $q_0 = 4$ units:
  • Is it worth producing one more unit?
  • If marginal profit is positive, producing more increases profits.
• If producing $q_0 = 7$ units:
  • Is it worth producing one more unit?
  • If marginal profit is negative, producing more decreases profits.
• Profit-maximizing output level $q^*$ is characterized by:
  • Marginal profit is positive for any quantity $q \leq q^*$.
  • Marginal profit is negative for any quantity $q \geq q^*$.
• Marginal Profit: The additional profit from producing one more unit of output.
• Marginal Revenue (MR): Additional revenue from producing one more unit.
• Marginal Cost (MC): Additional cost from producing one more unit.
• Marginal profit is calculated as: $Marginal Profit = MR - MC$
• Profit-maximizing output level $q^*$ is characterized by:
  • For any quantity $q \leq q^*, MR \geq MC$.
  • For any quantity $q \geq q^*, MR \leq MC$.
  • Therefore, $MR = MC$ at $q^*$.

Marginal Revenue

• Revenue equals price times quantity sold: $TR = P \times q$
• Total revenue depends on:
  1. Price set by the firm.
  2. Consumer willingness to buy at that price.
  3. Number of consumers willing to buy.
• Firms sell to consumers who haven't purchased the good elsewhere.
• Residual Demand ($Q<em>r$): Market demand ($Q</em>D$) minus supply of all other firms (QS^{\textasciitilde}) i.e. Qr = QD - QS^{\textasciitilde}.
• In a market with many firms, the supply of all other firms (Q_S^{\textasciitilde}) is approximately equal to the market supply ($S$).
• At the price where supply and demand intersect, residual demand is $Q_r = 0$.
• At higher prices, there is an excess of supply, so residual demand remains zero.
• At lower prices, there is an excess of demand, resulting in positive residual demand.
• Total revenue the firm receives is $TR = P \times Q_r$.

How does the firm’s revenue change from selling one additional unit?

• Quantity Effect: Selling one extra unit increases total revenue by the value of that unit.

• Price Effect: Selling one extra unit requires reducing the price, which decreases total revenue.

• The trade-off between price and quantity effects depends on the price-elasticity of the residual demand.

• To compute marginal revenue, we need to know the price-elasticity of residual demand.

• Price-elasticity of residual demand = $\frac{\% \Delta Q<em>r}{\% \Delta P} = \frac{\Delta Q</em>r}{\Delta P} \times \frac{P<em>1}{Q</em>{1r}}$

• The absolute change of the residual demand after the variation in price is
  \Delta Qr = \Delta QD - \Delta Q_S^{\textasciitilde}

• The relative change of the residual demand after the variation in price is
  \frac{\Delta Qr}{\Delta P} = \frac{\Delta QD}{\Delta P} - \frac{\Delta Q_S^{\textasciitilde}}{\Delta P}

• Suppose there are $n$ identical firms. Because all firms are identical, they all produce the same initial quantity, $q<em>1$. Therefore, we have that $Q</em>1^D = Q<em>1^S = Q</em>1 = n \times q<em>1$ and $Q</em>{1r} = Q<em>1^D - Q</em>1^S = n \times q<em>1 - (n-1) \times q</em>1 = q_1$

• The price-elasticity of the residual demand is:
  \frac{\Delta Qr}{\Delta P} \times \frac{P1}{Q{1r}} = \frac{\Delta QD}{\Delta P} - \frac{\Delta QS^{\textasciitilde}}{\Delta P} \times \frac{P1}{q1} = \frac{\Delta QD}{\Delta P} \times \frac{P1}{q1} \times \frac{Q1}{Q1} - \frac{\Delta QS^{\textasciitilde}}{\Delta P} \times \frac{P1}{q1} \times \frac{Q1^S}{Q1^S} = \frac{\Delta QD}{\Delta P} \times \frac{P1}{Q1} \times \frac{Q1}{q1} - \frac{\Delta QS^{\textasciitilde}}{\Delta P} \times \frac{P1}{Q1^S} \times \frac{Q1^S}{q1} = n \times \epsilonD - (n-1) \times \epsilon_S

• When there are $n$ (identical) firms in the market, the price-elasticity of a firm’s residual of demand is
  $n \times \epsilon<em>D - (n-1) \times \epsilon</em>S$

  • Price-elasticity of demand
  • Price elasticity of supply of all other firms

• When the number of firms is very large (as it is the case in a competitive market), the price-elasticity of the residual demand tends to $-\infty$ . In other words, the residual demand is “almost” perfectly elastic.

• Example:

  • $n = 78$
  • price−elasticity of demand = $-1.1$
  • price−elasticity of supply = $3.1$
  • price−elasticity of a firm′s residual demand
    $78 \times -1.1 - 77 \times 3.1 = -324.5$
  • If a firm rises its price by 1%, the quantity falls by nearly 300%.

• Every producer perceives its residual demand as perfectly elastic at the market price.

• A firm’s marginal revenue is constant and equal to the competitive equilibrium price: $MR = P$.

• By selling 1 additional unit, total revenue increases by $P$.

• All firms set a price equal to the competitive price, $P$ because the residual demand is perfectly elastic at the competitive price.

  • At price $P$, all residual consumers are ready to buy an unlimited quantity.
  • Lowering the price does not increase the demand but reduces the total revenue.
  • A small rise in the price decreases the residual demand to zero.

• The optimal output decision then occurs when the competitive price, $P$, equals the marginal cost, $MC$.

Optimal Profits

• A firm produces the quantity $q$ that maximizes its profits when $MR = MC$.
• Optimal profits can be compouted as $\pi = P \times q - TC = P \times q - \frac{TC}{q} \times q = (P - ATC) \times q$
• If P < ATC, then the firm makes a loss.
• Shutdown rule: A firm shuts down if its gains from operating are less than the losses incurred in fixed costs:
  \pi < -FC \rightarrow TR - VC - FC < -FC \rightarrow TR < VC \rightarrow P \times q < VC \rightarrow P < AVC
• If $P < ATC$ (firm makes a loss) but $P > AVC$: The firm operates, obtaining negative profits (area A), but if it shuts down, it needs to pay fixed costs (area A+B), so operating is better.
• If P < ATC and P < AVC: Operating leads to negative profits (area A+B), but shutting down means the firm pays fixed costs (area A), so it's better to shut down.

Short-Run Individual Supply Curve

• When a firm chooses its output for a given market price, and By repeating this analysis at different prices, we can see how a firm’s supply varies with price.
• If price falls below minimum AVC, the firm shuts down.
• The short-run supply curve of a competitive firm is its MC curve above the minimum AVC.

Producer Surplus

• Consider the case of a seller of an indivisible good:
• Total Revenue (TR) is calculated as Price (P) times Quantity (q): $TR = P \times q$
• The variable cost is the sum of the unit costs of all units. Formally, $VC = MC<em>1 + MC</em>2 + …$
• Producer Surplus $Producer Surplus = TR - VC$ , that is, the profits differ from the individual producer surplus only by the value of the fixed costs.
  • The fixed costs are “sunk”, that is, they need to be paid regardless of whether the firm operates in the market or not.

Profit Maximization in the Long-Run

• Firms can vary inputs in the long run, so the long-run supply curve differs from the short-run curve.
• Long-run profit equals the difference between total revenue and long-run costs.
• The firm determines quantity to produce by maximizing long-run profit using the same rule as in the short-run: $MR = MC_{LR}$
• In the long-run all costs are variable, so the firm does not have to consider whether fixed costs are sunk or avoidable.
• A firm shuts down only if it would make a negative profit, i.e., if P < ATC.
• $ATC<em>{LR}$ curve is the “lower envelope” of $ATC</em>{SR}$ curves.
• The firm produces more in the LR because it can increase its installed capacity (i.e., tract of land), while reducing its production cost, resulting in larger profits, with $\pi<em>{LR} > \pi</em>{SR}$.
• A firm’s long-run supply curve is its long-run marginal cost curve above the minimum long-run average total cost.

Competition in the Long Run

• In the short run, fixed costs prevent new firms from entering or existing firms from leaving.
• In the long run, all costs are avoidable, so firms can enter or leave the market.
  • If market profits are positive, new firms will enter.
  • If market profits are negative, existing firms will leave.
• With $n$ firms, competitive price is $P<em>n$ and profits are \pin > 0. New firms are attracted, shifting the supply curve, and the competitive price drops, reducing individual profits but remaining positive.
• Attracted by profits, new firms enter again. If a sufficiently large number of firms enter, the price drops below min $ATC_{LR}$, and individual profits become negative.
• To avoid losses, firms leave the market until there is no economic profit, i.e., until $P = min ATC_{LR}$.
• In the long run the competitive equilibrium price equals min $ATC_{LR}$.
• No producer has incentives to either enter or exit the market.