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Chapter 9: Problem 2

Suppose the market for widgets can be described by the following equations: \\[ \begin{array}{cl} \text { Demand: } & P=10-Q \\ \text { Supply: } & P=Q-4 \end{array} \\] where \(P\) is the price in dollars per unit and \(Q\) is the quantity in thousands of units. Then: a. What is the equilibrium price and quantity? b. Suppose the government imposes a tax of \(\$ 1\) per unit to reduce widget consumption and raise government revenues. What will the new equilibrium quantity be? What price will the buyer pay? What amount per unit will the seller receive? c. Suppose the government has a change of heart about the importance of widgets to the happiness of the American public. The tax is removed and a subsidy of \(\$ 1\) per unit granted to widget producers. What will the equilibrium quantity be? What price will the buyer pay? What amount per unit (including the subsidy) will the seller receive? What will be the total cost to the government?

Short Answer

Expert verified
The original equilibrium price and quantity are $3 and 7,000 units. With a $1/unit tax, the equilibrium quantity drops to 6,500 units, buyers pay $3.50/unit and sellers receive $2.50/unit after subtracting the tax. With a $1/unit subsidy, the equilibrium quantity rises to 7,500 units and the market price falls to $2.50/unit, while the seller receives $3.50/unit after the subsidy. The total cost to the government for the subsidy is $7.5 thousand.

Step by step solution

01

Find the equilibrium price and quantity

First, to find the equilibrium price and quantity, set the demand equation \(P=10-Q\) equal to the supply equation \(P=Q-4\). This leads to the equation \(10-Q=Q-4\), which can be solved for \(Q\) by adding \(Q\) and 4 to each side: \(Q=7\). Substitute \(Q=7\) into either the demand or supply equation, getting \(P=3\). Thus, the equilibrium price is $3 and the equilibrium quantity is 7,000 widgets.
02

Calculate the new equilibrium with a tax

The tax of $1 per unit effectively raises the cost of production from the viewpoint of the suppliers. So, the new supply equation becomes \(P=(Q+1)-4\) or \(P=Q-3\). Again, set the supply and demand equations equal to each other: \(10-Q=Q-3\). This simplifies to \(Q=6.5\), so the equilibrium quantity falls to 6,500 units. Substituting \(Q=6.5\) into the demand equation yields \(P=3.5\), so the buyer now pays $3.50 per unit. However, the seller still receives $2.50 per unit after the $1 tax is subtracted.
03

Calculate the new equilibrium with a subsidy

To consider the $1-per-unit subsidy, this effectively decreases the cost of production, so the new supply equation becomes \(P=(Q-1)-4\) or \(P=Q-5\). Set the new supply and demand equations equal to each other: \(10-Q=Q-5\), which simplifies to \(Q=7.5\). Therefore, the equilibrium quantity rises to 7,500 units. Substitute \(Q=7.5\) in the demand equation to find \(P=2.5\), so the buyer now pays $2.50 per unit. However, the seller still receives $3.50 per unit after adding the $1 subsidy. The total cost to the government of the subsidies is equal to the subsidy per unit times the quantity, or $(1)(7.5) = $7.5 thousand.

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Most popular questions from this chapter

In Exercise 4 in Chapter 2 (page 84 ), we examined a vegetable fiber traded in a competitive world market and imported into the United States at a world price of \(\$ 9\) per pound. U.S. domestic supply and demand for various price levels are shown in the following table $$\begin{array}{|ccc|} \hline & \text { U.S. SUPPLY } & \text { U.S. DEMAND } \\ \text { PRICE } & \text { (MILLION POUNDS) } & \text { (MILLION POUNDS) } \\ \hline 3 & 2 & 34 \\ \hline 6 & 4 & 28 \\ \hline 9 & 6 & 22 \\ \hline 12 & 8 & 16 \\ \hline 15 & 10 & 10 \\ \hline 18 & 12 & 4 \\ \hline \end{array}$$ Answer the following questions about the U.S. market: a. Confirm that the demand curve is given by \(Q_{D}=40-2 P,\) and that the supply curve is given by \(Q_{s}=2 / 3 P\) b. Confirm that if there were no restrictions on trade, the United States would import 16 million pounds. c. If the United States imposes a tariff of \(\$ 3\) per pound, what will be the U.S. price and level of imports? How much revenue will the government earn from the tariff? How large is the deadweight loss? d. If the United States has no tariff but imposes an import quota of 8 million pounds, what will be the U.S. domestic price? What is the cost of this quota for U.S. consumers of the fiber? What is the gain for U.S. producers?

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