Problem 42 - Applied Calculus

Question

The demand and supply curves for a product are given in terms of price, \(p,\) by $$q=2500-20 p \quad \text { and } \quad q=10 p-500$$ (a) Find the equilibrium price and quantity. Represent your answers on a graph. (b) A specific tax of $$ 6\( per unit is imposed on suppliers. Find the new equilibrium price and quantity. Represent your answers on the graph. (c) How much of the $$ 6\) tax is paid by consumers and how much by producers? (d) What is the total tax revenue received by the government?

Step-by-Step Solution

Answer

Equilibrium price is $100, quantity is 500. Consumer pays $2, producer $4; tax revenue is $2760.

Step 1: Identify Equilibrium Conditions

To find the equilibrium price and quantity, set the demand and supply functions equal to each other since equilibrium occurs when quantity demanded equals quantity supplied: \(2500 - 20p = 10p - 500\).

Step 2: Solve for Equilibrium Price

Rearrange the equation to find the equilibrium price: \(2500 + 500 = 20p + 10p\), which simplifies to \(3000 = 30p\). Solve for \(p\) to find \(p = 100\).

Step 3: Determine Equilibrium Quantity

Substitute the equilibrium price back into either the demand or supply equation to find the equilibrium quantity. Using \(q = 2500 - 20p\), substitute \(p = 100\): \(q = 2500 - 2000 = 500\).

Step 4: Impose the Tax

When a tax of $6 per unit is imposed, the supply curve shifts. The new supply equation becomes \(q = 10(p - 6) - 500\), simplifying to \(q = 10p - 60 - 500\) or \(q = 10p - 560\).

Step 5: Find New Equilibrium with Tax

Set the new supply equation equal to the demand equation to find the new equilibrium with tax: \(2500 - 20p = 10p - 560\). Simplify to find \(3060 = 30p\), giving \(p = 102\).

Step 6: Determine New Equilibrium Quantity

Substitute the new equilibrium price \(p = 102\) into the demand function: \(q = 2500 - 20(102) = 2500 - 2040 = 460\).

Step 7: Consumer and Producer Tax Burden

The consumers pay \(p_{new} - p = 102 - 100 = 2\) per unit of the tax. The producers get \(p - (p_{new} - 6) = 4\) per unit less due to the tax.

Step 8: Calculate Total Tax Revenue

The total tax revenue is the tax per unit multiplied by the new equilibrium quantity. That is, \(6 \times 460 = 2760\).

Key Concepts

Demand and Supply Curves

In economics, demand and supply curves are fundamental tools used to illustrate how markets operate. They show the relationship between the price of a good and the quantity demanded or supplied. The demand curve typically slopes downwards from left to right, indicating that, as the price decreases, the quantity demanded increases. On the other hand, the supply curve usually slopes upwards, suggesting that higher prices incentivize producers to supply more of the product.

The demand equation provided is: \( q = 2500 - 20p \) This tells us that as the price \( p \) decreases, the quantity demanded \( q \) increases. For example, if the price is zero, the demand is at its maximum of 2500 units. Conversely, the supply equation: \( q = 10p - 500 \) shows that as the price \( p \) increases, the quantity supplied \( q \) rises as well.Combining and analyzing these curves provides insights into how prices and quantities fluctuate in a market, ultimately leading us to the concept of equilibrium.

Equilibrium Price and Quantity

Equilibrium in a market occurs where the demand and supply curves intersect. This intersection point indicates that the quantity demanded by consumers equals the quantity supplied by producers, establishing the market balance without shortages or surpluses.

In our exercise, to find the equilibrium price \( p \), we set the quantity demanded equal to the quantity supplied: \( 2500 - 20p = 10p - 500 \). By solving this equation, we find the equilibrium price \( p = 100 \). Once this price is known, we can substitute it back into either the demand or supply equation to find the equilibrium quantity. This yields a quantity \( q = 500 \). These values represent the point where market forces stabilize, resulting in no unplanned inventory and satisfied consumer demand at this price.

Tax Implications in Economics

Taxes modify market equilibrium and are a tool used by governments to generate revenue. When a specific tax is applied to suppliers, it affects their willingness to supply goods at previous prices, effectively shifting the supply curve upward by the tax amount.

In our case, a $6 unit tax imposed alters the original supply equation from \( q = 10p - 500 \) to \( q = 10(p - 6) - 500 \), simplifying to \( q = 10p - 560 \). To find the new equilibrium, set this adjusted equation equal to demand: \( 2500 - 20p = 10p - 560 \). Solving for \( p \) gives the new equilibrium price \( p = 102 \). This means consumers will now pay more, shifting some economic burden to them, and reducing the equilibrium quantity as seen in the calculation of the new quantity \( q = 460 \). This exercise demonstrates the shifting dynamics in response to economic policies like taxes.

Government Tax Revenue

When a government imposes a tax, it impacts both consumers and producers by altering their economic balances. The total government tax revenue can be calculated as the tax per unit multiplied by the number of units sold at the new equilibrium.

In the context of our problem, the imposed tax of \(6 per unit leads to a new equilibrium quantity of 460 units. Therefore, the government earns a total revenue of \( 6 \times 460 = 2760 \).

The burden of this tax is shared between consumers and producers. Consumers pay more than the previous market price, as the new market price with tax is \( 102 \) instead of \( 100 \). Therefore, consumers incur an additional \)2 per unit. Producers receive $4 less per unit than they would without the tax. This calculation shows how tax policies can influence market behavior and generate income for the government.