Problem 15 The demand for ice cream is give... [FREE SOLUTION]

Chapter 3: Problem 15

The demand for ice cream is given by $Q^{D}=20-2 P$, measured in gallons of ice cream. The supply of ice cream is given by $Q^{S}=4 P-10$. a. Graph the supply and demand curves, and find the equilibrium price and quantity of ice cream. b. Suppose that the government legislates a $\$ 1$ tax on a gallon of ice cream, to be collected from the buyer. Plot the new demand curve on your graph. Does demand increase or decrease as a result of the tax? c. As a result of the tax, what happens to the price paid by buyers? What happens to the price received by sellers? How many gallons of ice cream are sold? d. Who bears the greater burden of the tax? Can you explain why this is so? e. Calculate consumer surplus both before and after the tax. f. Calculate producer surplus both before and after the tax. $\mathrm{g}$. How much tax revenue did the government raise? h. How much deadweight loss does the tax create?

Short Answer

Expert verified

Equilibrium without tax: $P = 5$, $Q = 10$. With tax: Sellers get $4.67$, buyers pay $5.67$, $Q = 8.67$. Tax revenue is $8.67$, deadweight loss is $0.665$.

Step by step solution

Finding the Equilibrium Price and Quantity

To find the equilibrium, set the quantity demanded equal to the quantity supplied: $Q^D = Q^S$. From the given equations, $20 - 2P = 4P - 10$. Solving for $P$, we get:$20 + 10 = 4P + 2P$ $30 = 6P$ $P = 5$.Substitute $P = 5$ back into either equation to find $Q$: $Q^D = 20 - 2(5) = 10$.Thus, the equilibrium price is $5$ and the equilibrium quantity is $10$ gallons.

Graphing Supply and Demand Curves

The demand curve $Q^D = 20 - 2P$ has a downward slope, starting from the point where $P = 0$ and $Q = 20$.The supply curve $Q^S = 4P - 10$ is upward sloping, beginning from the point where $P = 2.5$ and $Q = 0$.Graph both curves on the same axes, and mark the equilibrium point at $(P = 5, Q = 10)$.

Analyzing Impact of a $1 Tax

A $1 tax on buyers shifts the demand curve leftwards. The new demand curve becomes $Q^D = 20 - 2(P + 1)$ or $Q^D = 20 - 2P - 2 = 18 - 2P$.The demand decreases due to the tax as the effective price increases.

Determining New Equilibrium With Tax

Set the new demand equation equal to the supply equation to find equilibrium:$18 - 2P = 4P - 10$$28 = 6P$$P = \frac{28}{6} = 4.67$.The effective price for buyers is $4.67 + 1 = 5.67$. Sellers receive $4.67$, and the new equilibrium quantity is $4(4.67) - 10 = 8.67$ gallons.

Assessing Tax Burden

The buyers' price increases by $0.67$, while sellers receive $0.33$ less than before. Since buyers are shouldering a $1$ unit increase, compared to the decrease sellers face, buyers bear the greater tax burden.

Calculating Consumer Surplus Before Tax

Consumer surplus is the area of the triangle above the price level and below the demand curve:$\frac{1}{2} \times 10 \times (20 - 5) = 75$.

Calculating Consumer Surplus After Tax

With a new price $P = 5.67$ for buyers, consumer surplus becomes:$\frac{1}{2} \times 8.67 \times (20 - 5.67) = 62.22$.

Calculating Producer Surplus Before Tax

Producer surplus is the area of the triangle below the price level and above the supply curve:$\frac{1}{2} \times 10 \times (5 - 2.5) = 12.5$.

Calculating Producer Surplus After Tax

With a new price $P = 4.67$ for sellers, producer surplus becomes:$\frac{1}{2} \times 8.67 \times (4.67 - 2.5) = 9.42$.

Calculating Government Revenue from Tax

Government revenue from the tax is the tax amount per unit times the quantity sold:$1 \times 8.67 = 8.67$.

Calculating Deadweight Loss Due to Tax

Deadweight loss is the reduction in total surplus due to the tax, calculated by:$\frac{1}{2} \times (10 - 8.67) \times (\text{price buyers minus price sellers}) = \frac{1}{2} \times 1.33 \times 1 = 0.665$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Consumer Surplus

Consumer surplus represents the benefit or value consumers receive when they can purchase a product for a price that is less than the maximum amount they are willing to pay. Visualize it as the triangular area under the demand curve and above the actual price level. In equilibrium, this is easily calculated using the formula for the area of a triangle:

Before tax, the equilibrium price is 5, and the equilibrium quantity is 10.
Therefore, the maximum price willing to be paid (at Q = 0) is 20.
The formula becomes: $\frac{1}{2} \times 10 \times (20 - 5) = 75$.

With this calculation, consumer surplus is 75 before the tax incidence. When a $1 tax is introduced, the demand curve shifts left, decreasing the quantity purchased and raising the effective price for consumers to 5.67. Re-calculating consumer surplus with this new price level:

The new quantity is approximately 8.67 gallons.
Recalculating using the new price, the surplus becomes: $\frac{1}{2} \times 8.67 \times (20 - 5.67) \approx 62.22$.

This demonstrates how consumer surplus decreases when taxes increase the price paid by consumers.

Delving into Producer Surplus

Producer surplus is the difference between what producers are willing to accept for a good and what they actually receive. Similar to consumer surplus, before tax, it can be represented as a triangle:

The equilibrium price producers receive is 5.
Supply curve starts at a price of 2.5 (where quantity supplied = 0).
Area calculation: $\frac{1}{2} \times 10 \times (5 - 2.5) = 12.5$.

This calculation shows the producer surplus before the imposition of the tax.After a $1 tax, the price producers effectively receive decreases to 4.67, reducing their surplus. New calculations are necessary:

The selling price drops to 4.67, while the new equilibrium quantity is 8.67.
The new producer surplus is the area: $\frac{1}{2} \times 8.67 \times (4.67 - 2.5) \approx 9.42$.

This illustrates how taxation can diminish producer surplus as sellers earn less per unit.

Exploring Deadweight Loss

Deadweight loss occurs whenever market transactions are less than the optimal quantity and lead to an inefficient market. It's the lost economic efficiency when the supply and demand equilibrium is not achieved. In this context, a $1 tax results in:

Reduced quantity from 10 gallons at equilibrium to about 8.67 gallons.
Price differences increase between what buyers pay (5.67) versus what sellers receive (4.67).

The deadweight loss is represented by the triangle formed between the previous and new quantities along the demand and supply curves. Calculating this involves the formula for a triangle:

Difference in quantities: $10 - 8.67 = 1.33$.
Width of the triangle (difference in price): 1 (5.67 - 4.67).
Deadweight loss: $\frac{1}{2} \times 1.33 \times 1 \approx 0.665$.

This number reflects the social cost of market inefficiencies caused by the tax.

Breaking Down Tax Incidence

Tax incidence determines how the burden of a tax is divided between consumers and producers. It depends on the relative elasticities of supply and demand. In simpler terms, it defines who actually pays the tax in the end. In the ice cream market scenario:

Consumers end up paying an additional 0.67 per unit as the price rises from 5 to 5.67.
Producers, on the other hand, receive 0.33 less per unit, dropping from 5 to 4.67.

Buyers face a higher increase than the decrease seen by sellers. This indicates that consumers are bearing a more significant portion of the tax burden. The greater tax burden on consumers can often be explained by their demand being relatively inelastic compared to supply, meaning they are less sensitive to price changes.

The Basics of Supply and Demand Curves

Supply and demand are the fundamental forces that define a market. The demand curve, which slopes downward, shows the relationship between the price of a good and the quantity demanded by consumers. Conversely, the supply curve trends upwards, demonstrating how price influences the amount producers are willing to supply. For this ice cream example:

Demand is represented by $ Q^D = 20 - 2P $.
Supply is given by $ Q^S = 4P - 10 $.

At equilibrium, these curves intersect. Solving $ Q^D = Q^S $ helps find the equilibrium price and quantity, where the market clears efficiently.Graphically representing these curves helps visualize the impact of various market changes, like taxes, on price and quantity. Such visualization is crucial for understanding dynamic market behavior and economic policy implications.

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