Assignment Question
Suppose that demand in a market can be represented by the following equation: P=8−Q and that supply can be represented by the equation: P=Q What is the equilibrium price and quantity in this market? Now suppose that a sales tax of $2 per unit is imposed on the product in this market. How would you now express the supply curve with this tax included? What are the new equilibrium price and quantity in this market? How much tax revenue is raised? Why is it less than the original equilibrium quantity multiplied by the $2 tax? What is the effect of this tax on the price consumers pay and on consumer surplus? What price do producers receive net of the tax (without the tax added)? How did this influence the quantity supplied? Calculate the deadweight loss associated with this tax.
Assignment Answer
The Impact of a Sales Tax on Market Equilibrium: An Analysis Using Supply and Demand Equations
Introduction
The interaction between supply and demand in a market is a fundamental concept in economics. When supply and demand are in equilibrium, it represents an efficient allocation of resources, where the quantity demanded equals the quantity supplied, and a market-clearing price is established. However, the introduction of taxes can disrupt this equilibrium, affecting prices, quantities, and overall welfare in the market. In this essay, we will explore the impact of a $2 per unit sales tax on a hypothetical market, initially described by the equations P=8−Q for demand and P=Q for supply. We will determine the new equilibrium price and quantity, analyze the tax revenue generated, discuss the effect on consumer surplus and producer prices, and calculate the deadweight loss associated with the tax.
Equilibrium Price and Quantity in the Untaxed Market
To find the equilibrium price and quantity in the untaxed market, we need to solve for the point where the demand and supply curves intersect. The demand curve is represented by the equation P=8−Q, while the supply curve is represented by P=Q. Equilibrium is achieved when the quantity demanded (Qd) equals the quantity supplied (Qs), and the corresponding price (P) is the equilibrium price.
Equating the demand and supply equations:
8−Q = Q
Now, let’s solve for the equilibrium quantity:
8 = 2Q
Q = 8/2
Q = 4 units
Now that we have the equilibrium quantity, we can find the equilibrium price using the supply equation:
P = Q
P = 4 units
Therefore, in the untaxed market, the equilibrium quantity is 4 units, and the equilibrium price is $4 per unit.
Expressing the Supply Curve with the Tax Included
To account for the $2 per unit sales tax, we need to adjust the supply curve. The new supply curve with the tax included would be:
P + Tax = Q
Where Tax is the $2 per unit tax. Therefore, the supply equation with the tax included is:
P + $2 = Q
New Equilibrium Price and Quantity with the Tax
To find the new equilibrium price and quantity with the tax, we need to once again equate the adjusted supply equation (P + $2 = Q) with the original demand equation (P = 8 − Q). Solving for equilibrium:
8 − Q = P + $2
Now, let’s substitute the adjusted supply equation (P + $2) for P:
8 − Q = (P + $2) + $2
Now, simplify:
8 − Q = P + $4
We already know that P = 8 − Q, so we can substitute this expression for P:
8 − Q = (8 − Q) + $4
Now, let’s solve for the equilibrium quantity (Q) in the taxed market:
8 − Q = 8 − Q + $4
Next, we can subtract 8 − Q from both sides of the equation:
$4 = $4
This equation tells us that no matter the quantity (Q) exchanged, the price will always be $4 higher due to the tax. Therefore, in the taxed market, the equilibrium price is $4 + $2 = $6 per unit, and the equilibrium quantity remains at 4 units, just as it was in the untaxed market.
Tax Revenue Generated
Tax revenue is calculated by multiplying the tax rate by the quantity sold, which, in this case, is the $2 per unit tax times the equilibrium quantity. Therefore, the tax revenue generated is:
Tax Revenue = Tax Rate × Equilibrium Quantity
Tax Revenue = $2 × 4 units
Tax Revenue = $8
The tax revenue raised is $8. However, this amount is less than the original equilibrium quantity (4 units) multiplied by the $2 tax rate ($2 × 4 units = $8). This discrepancy can be explained by the fact that a tax creates a wedge between the price consumers pay and the price producers receive.
Effect on the Price Consumers Pay and Consumer Surplus
With a $2 per unit tax, consumers now pay the market price of $6 per unit, which includes the $2 tax. Prior to the tax, they paid $4 per unit, so the tax increases the price by $2. As a result, consumers bear the burden of the tax, and the price they pay has increased.
Consumer surplus represents the difference between what consumers are willing to pay (their reservation price) and what they actually pay for a product. The imposition of a tax reduces consumer surplus because consumers are now paying more for the same quantity of the product. To quantify this change, we can calculate the consumer surplus before and after the tax.
Before the tax, consumers paid $4 per unit, and they purchased 4 units. Therefore, the total consumer expenditure before the tax was:
Consumer Expenditure (Before Tax) = Price (Before Tax) × Quantity
Consumer Expenditure (Before Tax) = $4 × 4 units
Consumer Expenditure (Before Tax) = $16
Now, let’s calculate the consumer surplus before the tax. The demand curve equation, P=8−Q, helps us determine consumer willingness to pay. Before the tax, consumers are willing to pay $8 per unit when they purchase the first unit, and this willingness to pay decreases as they purchase more units. We can calculate consumer surplus using the formula for the area of a triangle:
Consumer Surplus (Before Tax) = 0.5 × Base × Height
The base of the triangle is 4 units (the quantity purchased), and the height is the difference between the maximum willingness to pay ($8) and the price paid ($4):
Consumer Surplus (Before Tax) = 0.5 × 4 units × ($8 − $4)
Consumer Surplus (Before Tax) = 0.5 × 4 units × $4
Consumer Surplus (Before Tax) = $8
Now, with the tax included, consumers pay $6 per unit and purchase 4 units, resulting in a total consumer expenditure of:
Consumer Expenditure (After Tax) = Price (After Tax) × Quantity
Consumer Expenditure (After Tax) = $6 × 4 units
Consumer Expenditure (After Tax) = $24
To calculate the consumer surplus after the tax, we use the same formula as before:
Consumer Surplus (After Tax) = 0.5 × Base × Height
The base is still 4 units, and the height is now the difference between the maximum willingness to pay ($8) and the price paid ($6):
Consumer Surplus (After Tax) = 0.5 × 4 units × ($8 − $6)
Consumer Surplus (After Tax) = 0.5 × 4 units × $2
Consumer Surplus (After Tax) = $4
Comparing consumer surplus before and after the tax, we can see that consumer surplus has decreased from $8 to $4 due to the tax.
Price Received by Producers (Net of Tax) and Its Effect on Quantity Supplied
Producers receive the market price minus the tax as their revenue. In this case, the price producers receive, net of the $2 tax, is $6 per unit. This is because they receive the market price of $6 per unit, but $2 of that price goes towards paying the sales tax. So, the price received by producers is $6 per unit.
The price received by producers significantly affects their willingness and ability to supply the product. In this scenario, producers receive a price that is lower than the equilibrium price in the untaxed market, which was $4 per unit. With the introduction of the $2 tax, the effective price received by producers increases by $2, providing an incentive for them to produce and supply more. This is why the quantity supplied in the taxed market remains the same as in the untaxed market, at 4 units.
Deadweight Loss Associated with the Tax
Deadweight loss is an economic concept that quantifies the efficiency loss caused by market distortions such as taxes. It represents the reduction in economic welfare that occurs when the market equilibrium is not reached due to factors like taxes. In this case, the imposition of a $2 per unit sales tax creates a deadweight loss.
To calculate the deadweight loss associated with the tax, we need to compare the economic surplus in the untaxed market (i.e., the market without the tax) to the economic surplus in the taxed market. The economic surplus consists of consumer surplus and producer surplus.
- Consumer Surplus (Before Tax) = $8
- Producer Surplus (Before Tax) = Area of the triangle between the supply and demand curves, which is also $8.
Now, let’s calculate the economic surplus in the untaxed market:
Economic Surplus (Before Tax) = Consumer Surplus (Before Tax) + Producer Surplus (Before Tax)
Economic Surplus (Before Tax) = $8 + $8
Economic Surplus (Before Tax) = $16
In the taxed market, the consumer surplus has decreased from $8 to $4, as we calculated earlier. However, the producer surplus remains unchanged at $8 because the price received by producers increases by $2 due to the tax.
Now, let’s calculate the economic surplus in the taxed market:
Economic Surplus (After Tax) = Consumer Surplus (After Tax) + Producer Surplus (Before Tax)
Economic Surplus (After Tax) = $4 + $8
Economic Surplus (After Tax) = $12
The deadweight loss is the difference between the economic surplus in the untaxed market and the economic surplus in the taxed market:
Deadweight Loss = Economic Surplus (Before Tax) – Economic Surplus (After Tax)
Deadweight Loss = $16 – $12
Deadweight Loss = $4
Therefore, the deadweight loss associated with the tax is $4. This represents the loss of economic welfare due to the tax, as it reduces the overall efficiency of the market. Deadweight loss occurs because the tax distorts market outcomes, leading to a reduction in the total surplus (consumer and producer surplus) relative to the untaxed market.
Conclusion
The impact of a sales tax on a market can be complex and far-reaching. In this analysis, we examined a hypothetical market with demand and supply equations, initially finding the equilibrium price and quantity in the untaxed market. We then introduced a $2 per unit sales tax, adjusted the supply curve, and determined the new equilibrium price and quantity in the taxed market. We calculated the tax revenue generated and explained why it was less than the product of the original equilibrium quantity and the tax rate.
The tax had noticeable effects on both consumers and producers. Consumers paid a higher price for the product, leading to a reduction in consumer surplus. Producers, on the other hand, received a price net of the tax that was greater than the original equilibrium price, which encouraged them to supply the same quantity as before.
Finally, we calculated the deadweight loss associated with the tax, which represents the reduction in economic welfare due to market inefficiencies created by the tax. Deadweight loss occurs because the tax distorts the market equilibrium, leading to a loss in overall economic surplus. This analysis illustrates the economic consequences of taxes on market dynamics and the importance of understanding their impact on consumer and producer behavior, as well as overall market efficiency.
References
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