Problem 43 - Applied Calculus

Question

In Example \(8,\) the demand and supply curves are given by \(q=100-2 p\) and \(q=3 p-50,\) respectively; the equilibrium price is $$ 30\( and the equilibrium quantity is 40 units. A sales tax of \)5 \%$ is imposed on the consumer. (a) Find the equation of the new demand and supply curves. (b) Find the new equilibrium price and quantity. (c) How much is paid in taxes on each unit? How much of this is paid by the consumer and how much by the producer? (d) How much tax does the government collect?

Step-by-Step Solution

Answer

(a) New demand: \( q = 100 - 2.1p \). (b) Equilibrium: \( p \approx 29.41, q \approx 38.24 \). (c) Tax per unit: \( 1.47 \), consumer pays \( 0.88 \), producer \( 2.06 \). (d) Government collects \( 56.21 \).

Step 1: Adjust Demand Curve for Tax

The original demand curve is given by \( q_d = 100 - 2p \). When a 5% sales tax is imposed on the consumer, the price the consumer pays increases by 5%. Therefore, the effective price is \( p(1 + 0.05) = 1.05p \). Substituting back into the demand equation, we have \( q_d = 100 - 2(1.05p) = 100 - 2.1p \).

Step 2: Rewrite Supplier's Equation

The supply curve remains unchanged because the tax is imposed on the consumer. Hence, the supply curve is still \( q_s = 3p - 50 \).

Step 3: Find New Equilibrium

To find the new equilibrium, set the new demand equal to supply: \[ 100 - 2.1p = 3p - 50 \]Solving for \( p \), \[ 150 = 5.1p \] \[ p = \frac{150}{5.1} \approx 29.41 \].Substitute back to find \( q \): \[ q = 3(29.41) - 50 \approx 38.24 \].Therefore, the new equilibrium price is approximately \( 29.41 \) and the new equilibrium quantity is approximately \( 38.24 \) units.

Step 4: Calculate Tax per Unit

The tax per unit is 5% of the new consumer price, which is \( 0.05 \times 29.41 \approx 1.47 \).

Step 5: Determine Tax Division

The initial equilibrium price was \( 30 \), and the new price the consumer pays is \( 29.41 \), meaning the consumer pays less by \( 30 - 29.41 = 0.59 \) less per unit. The producer's price went from \( 30 \) to effectively \( 29.41 - 1.47 = 27.94 \), so the producer absorbs the tax difference of \( 30 - 27.94 = 2.06 \). Therefore, the consumer pays \( 1.47 - 0.59 = 0.88 \) and the producer pays \( 2.06 \).

Step 6: Calculate Total Tax Collection

Total tax collected by the government is the tax per unit multiplied by the number of units sold in the market. The tax per unit is \( 1.47 \), and the quantity sold is \( 38.24 \). Therefore, the total tax collected is\[ 1.47 \times 38.24 \approx 56.21 \].

Key Concepts

Demand and Supply Curves

The demand and supply curves are critical tools in analyzing any market. They show the relationship between the price of a good and the quantity demanded or supplied. In this scenario, the demand curve is represented by the equation \(q_d = 100 - 2p\), where \(q_d\) is the quantity demanded and \(p\) is the price. This equation suggests that as the price increases, the demand decreases.

• Demand curve: \(q_d = 100 - 2p\).
• The relationship is inverse: more expensive goods lead to lower demand.

On the other side, the supply curve is given by \(q_s = 3p - 50\), illustrating that as the price increases, the supply also increases. This direct relationship implies that producers are willing to supply more of the good when they can charge higher prices.

• Supply curve: \(q_s = 3p - 50\).
• Positive relationship: higher prices encourage greater supply.

Sales Tax Impact

A sales tax imposed on consumers alters the effective price they pay for a product. In this exercise, a 5% sales tax means consumers face a price that is 5% higher. As a result, the effective demand curve transforms. Originally, we had \(q_d = 100 - 2p\). With a 5% increase on price, the demand becomes \(q_d = 100 - 2(1.05p) = 100 - 2.1p\).

• This modifies the demand due to a perceived higher cost to consumers.
• The supply equation stays the same since the tax doesn't affect suppliers directly.

Remember, a sales tax tends to reduce the quantity demanded at every price point because consumers are now effectively paying more than the sticker price. This shift outlines a critical component in understanding how market equilibrium changes post-tax implementation.

Equilibrium Price and Quantity

The equilibrium price and quantity occur where the demand and supply curves intersect. Before tax, the equilibrium was at price \(30\) and quantity \(40\). With the tax adjusted demand \(q_d = 100 - 2.1p\) and unchanged supply \(q_s = 3p - 50\), we solve for equilibrium:
\[100 - 2.1p = 3p - 50\]Solving the equation gives a new equilibrium price \(p \approx 29.41\) and a quantity \(q \approx 38.24\). These values reflect the market adjustments to the sales tax.

• Adjusted demand decreases equilibrium price due to higher consumer cost.
• New equilibrium reflects reduced demand and supply volume.

Tax Burden Division

When taxes are introduced into the market, both consumers and producers share the burden, but not always equally. Here, the tax per unit is calculated as \(0.05 \times 29.41 \approx 1.47\). The consumer benefits as the price they pay decreases from \(30\) to \(29.41\), hence paying \(0.88\) in taxes after adjusting for their perceived savings.

• Consumer's tax contribution: \(0.88\).
• Producer's burden: \(2.06\), after adjusting for new revenue post-tax.

Getting the tax burden right is key in understanding market dynamics. The original prices, new equilibrium price, and how much of the tax each party absorbs help in perceiving the real impact of taxation policies on different market participants.