Question

A good can be produced in a competitive industry at a cost of \(\$ 10\) per unit. There are 100 consumers are each willing to pay \(\$ 12\) each to consume a single unit of the good (additional units have no value to them.) What is the equilibrium price and quantity sold? The government imposes a tax of \(\$ 1\) on the good. What is the dead weight loss of this tax?

Short answer

The equilibrium price is \(\$12\), and quantity sold is 100 units. The dead weight loss is \(\$0\).

Step-by-step solution

Determine Equilibrium Price and Quantity without Tax

In a competitive market, the equilibrium price is where the supply equals demand. Here, 100 consumers each willing to pay \(\\(12\) per unit, which is also the market price since the marginal cost is \(\\)10\) per unit. Thus, firms will produce and sell one unit to each consumer at \(\\(12\). Equilibrium price is \(\\)12\), and quantity is 100 units.

Calculate the New Equilibrium with Tax

The government imposes a \(\\(1\) tax, making the cost for producers \(\\)11\) per unit (\(\\(10\) cost + \(\\)1\) tax). The new equilibrium price accounting for the tax remains at \(\\(12\) to consumers. Producers will receive \(\\)11\) (price minus tax), maintaining profitability. Quantity remains at 100 units, since each consumer will still pay \(\$12\). Thus, the tax does not alter the quantity or consumer price, as only producers bear the tax burden.

Analyze Changes to Surplus from the Tax

The imposition of the tax means \(\\(1\) unit of surplus shifts from producers to the government tax revenue for each unit sold. As the equilibrium quantity is unaffected (100 units), the tax does not change the consumer or total quantity. However, the entire \(\\)1\) tax per unit transfers producer surplus to government revenue, with no dead weight loss.

Calculate the Dead Weight Loss

Dead weight loss occurs when taxes result in a loss of economic efficiency, usually if the equilibrium quantity decreases. In this case, since the quantity remains at 100 units and price to consumers stays \(\\(12\), there is no unfulfilled demand or production. Therefore, the dead weight loss of the tax is \(\\)0\).

Key concepts

Competitive Market

A competitive market is characterized by many buyers and sellers, where none of the participants can influence the market price. Consumers and producers are price takers. In such a market, the interaction of supply and demand determines the equilibrium price and quantity.
In our exercise, each consumer is willing to pay $12 per unit of the good, which matches their willingness to pay with the marginal cost of $10 per unit. The equilibrium price in this scenario is $12, with the quantity being 100 units, as there are 100 consumers.
At this price and quantity, the market operates efficiently because all goods produced are sold, and no excess or shortage exists.

Consumer Surplus

Consumer surplus is a measure of the benefit consumers receive when they pay less for a product than what they are willing to pay. It represents the difference between what consumers are willing to pay and what they actually pay.
In this scenario, each consumer is willing to pay $12, and the equilibrium price is also $12, meaning they pay exactly what they are willing to pay. Consequently, the consumer surplus in this market is technically zero because there's no difference between the maximum price a consumer is willing to pay and the market price. However, since there are no other purchasers willing to pay more, the market efficiently allocates goods to those who value them highest.

Producer Surplus

Producer surplus is the difference between what producers are willing to accept for a good and what they actually receive. It represents the additional benefit or profit producers gain from selling at the market price.
Without any taxes, producers receive $12 per unit, which includes a $2 surplus over their production cost of $10 per unit. With 100 units sold, the total producer surplus is $200.
After the tax, producers receive $11 per unit due to the $1 tax cut, reducing their surplus by $1 per unit. As a result, the total producer surplus decreases to $100 (since $11 minus $10 equals $1 per unit).
This tax shifts some surplus from producers to the government through tax revenue.

Taxation Impact

The impact of a tax in any market can shift the surplus between consumers, producers, and the government. It also has potential to introduce a dead weight loss, which occurs when the tax reduces the quantity traded below the market equilibrium.
In the presented exercise, the $1 tax on the good makes the cost to producers $11 per unit while keeping the price to consumers at $12. Because both the price consumers pay and the quantity remain unchanged, there is no dead weight loss.
This scenario is unique because the market equilibrium quantity stays the same, maintaining economic efficiency and displaying how the burden of the tax is entirely on the producers who now receive less for each unit sold. The government collects the resulting tax revenue, equivalent to $100 in total (from $1 on each of the 100 units).
Ultimately, the taxation impact in this situation is a reallocation of producer surplus to government revenue with no loss in market efficiency.