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# Perfect competition profit maximization

Even though perfect competition in its pure form rarely exists (agricultural commodities are a close example), it is useful to study perfect competition for effective comparison with other market structures.

## Demand analysis in perfectly competitive markets

It is observed that quantity demanded increases with a decrease in price. This is due to substitution and income effects. Demand function shows the quantity demanded Q as dependent on price P. Inverse demand function expresses P as a function of Q.

If a product has demand function Q = 50 – 2P, its inverse demand function is P = 50 – 0.5Q. Total revenue (TR) is the product of Q and P, hence TR = Q × P = Q × (50 – 0.5Q) = 50Q – 0.5Q2. Average revenue equals TR/Q, which in turn equals the inverse demand function, i.e. price.

Marginal revenue, the additional revenue from selling one additional unit equals the first derivative of the TR function. The first derivative of TR equals 50 – Q, hence MR = 50 – Q. For an inverse demand function of the form P = a – b × Q, MR = a – 2b × Q.

The sensitivity of demand to a product’s price, price of substitutes and complements, income level, etc. is measured by calculating different elasticities.

Price elasticity of demand equals the percentage change in quantity demanded divided by the percentage change in the product price. Price elasticity is high if there are many close substitutes, a higher budget of the consumer is spent on the product, more time has passed since the price change, etc. Price elasticity is zero if the demand curve is vertical, and quantity does not respond to change in price at all. In perfect competition, price elasticity is infinite, the demand curve is horizontal, and even the slightest change in price changes quantity demanded infinitely. Whether a decrease in price increases revenue depends on the price elasticity. If demand is elastic, an increase in price decreases total revenue and vice versa, and if demand is inelastic, an increase in price increases revenue and vice versa. It is because when elasticity is greater than 1 (in absolute terms) percentage decrease in quantity as a result of a percentage increase in price is higher.

Income elasticity of demand measures change in quantity demanded in response to a change in income level. If income elasticity is positive, the good is a normal good, else it is an inferior good. Cross elasticity of demand measures how the quantity demanded of one product changes with change in the price of another product. If cross elasticity is positive, the goods are substitutes and if it is negative, the products are complements.

The relationship between marginal revenue MR, price P and elasticity of demand Ed.

$MR = P × (1 – \frac{1}{E_d})$

In understanding the effectiveness of the business strategy, it is important to understand the concept of consumer surplus, which is the difference between the value a consumer places on a unit and the price he ultimately pays for it. Since the demand curve plots the amount that a consumer is willing to pay for each additional unit of a good, it is also called the marginal value curve. However, in most cases, a product trades at a single price, which means that a consumer gets all the units at a single price P and hence total expenditure is the product of Q and P. Consumer surplus equals total value minus total expenditure.

## Supply analysis in perfectly competitive markets

A supply curve slopes upwards which means that at a higher price, a producer is willing to supply more output. It is because at the higher price, previously unprofitable production becomes profitable. The supply curve depends on the cost of production of the additional units which includes all opportunity costs and a normal profit.

A supply function provides a relationship between quantity supplied and price. It is of the form

$Q = a + b × P$ $P = 1/b × (Q – a)$

## Optimal price and output in perfectly competitive markets

Profit maximization in perfect competition occurs where marginal revenue is equal to marginal cost and the marginal cost curve is rising.

If a market faces an inverse demand curve, P = 50 – Q, total revenue TR = Q × (50 –Q) = 50Q – Q2. Since marginal revenue is equal to the first derivative of TR function, MR = 50 – 2Q.

If the market supply curve is of form P = 10 + 2Q, total costs TC = (10 + 2Q) × Q = 10Q + 2Q2. Since marginal cost is the first derivative of the TC curve, MC = 10 + 4Q. The supply curve depends on the cost of production, which includes opportunity costs and a normal profit. Productivity initially improves and marginal cost falls but at some point, the law of diminishing marginal return causes productivity to fall and marginal cost to rise.

To solve for perfectly competitive profit maximizing output level Q, we set MR = MC

$50 – 2Q = 10 + 4Q$ $Q = \frac{(50 – 10)}{(4 + 2)} = 6.67$

Price P corresponding to output 6.67 is 43.3 (= 50 – 6.67).

For an individual firm, the demand curve is horizontal. It is because there are such a large number of firms that if it attempted to increase price above the market price, all its consumers would switch to others. This intense competition ensures that for each individual firm, price = average revenue = marginal revenue. If the MR curve lies above the average total cost, the firm is earning a positive economic profit.

## Factors affecting long-run equilibrium in perfectly competitive markets

If a firm in a perfectly competitive market is earning positive economic profit i.e. its total revenues are greater than its total cost at the profit-maximizing output level, new firms will enter the market due to very low barriers to entry and homogenous nature of the products. With the entry of new firms, the long-run supply curve will shift outwards (due to an increase in supply) causing a decrease in market price and associated decrease in marginal revenue for individual firms.

Hence, in the long run, a firm in a perfectly competitive market operates at a point at which its average revenue (also marginal revenue and price) curve is tangent to the average total cost (ATC) curve. It is the point at which the marginal cost (MC) curve is intersecting the ATC curve and ATC is at its minimum. It implies that zero economic profit, i.e. just the normal profit, can exist in perfect competition in the long run.