Question

Suppose the market for widgets can be described by the following equations:

Demand: P = 10 - Q

Supply: P = Q – 4

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? Whatamount 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 is 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

1. The equilibrium quantity is 7000 widgets, and the equilibrium price is $3 per unit.

2. The new equilibrium quantity would be 6500. The buyer will pay $3.5 per unit, and the seller will receive $2.5 per unit.

3. The equilibrium quantity would be 7500 widgets. The buyers will pay $2.5 per unit, and the seller will receive $3.5 per unit. The total cost to the government would be $7500.

Step-by-step solution

Step 1. Calculate the equilibrium price and quantity

The widget market would be in an equilibrium position where the demand for widgets and supply are equal. By using this concept of demand and supply, the equilibrium price and quantity are determined below:

$$\begin{gathered} Demand=Supply \\ 10-Q=Q-4 \\ 2Q=14 \\ Q=7 \end{gathered}$$

The equilibrium quantity of widgets is 7000. By putting the value of P in the supply equation, the equilibrium price is calculated below:

$$\begin{gathered} P\left(\$\right)=Q-4 \\ =7-4 \\ =3 \end{gathered}$$

The equilibrium price of the widget is $3 per unit.

Step 2. Calculate the equilibrium price and quantity after the taxation.

The taxation of $1 on each unit of widget sold would decrease the supply of widgets in the market. The supply curve would shift leftward, causing a new equilibrium point. The new equation for the supply of widgets after taxation is:

$$\begin{gathered} P=Q-4+1 \\ =Q-3 \end{gathered}$$

By equating the new equation of supply with the existing demand equation, the new equilibrium is calculated below:

$$\begin{gathered} Demand=Supply \\ 10-Q=Q-3 \\ 2Q=13 \\ Q=6.5 \end{gathered}$$

The new equilibrium quantity of widgets after taxation is 6500. By putting the value of P in the new supply equation, the equilibrium price is calculated below:

$$\begin{gathered} P\left(\$\right)=Q-3 \\ =6.5-3 \\ =3.5 \end{gathered}$$

The new equilibrium price of the widget after taxation is $3 per unit.

The new equilibrium price is the price at which the buyers will buy the widget.Thus, buyers would buy the widget at a price of $3.5 per unit.

The sellers will receive a price equal to the difference between the new equilibrium price and the taxed amount. The difference between the new equilibrium price and the taxed amount is 2.5 (3.5 - 1). Therefore, the sellers would receive a price of $2.5 per unit.

Step 3. Calculate the equilibrium price and quantity after the taxation

The subsidy of $1 on each unit of widget sold would increase the supply for widgets in the market. The supply curve would shift rightward, causing a new equilibrium point. The new supply equation for widgets after subsidy is:

$$\begin{gathered} P=Q-4-1 \\ =Q-5 \end{gathered}$$

By equating the new demand equation with the existing supply equation, the new equilibrium quantity is calculated below:

$$\begin{gathered} Demand=Supply \\ 10-Q=Q-5 \\ 2Q=15 \\ Q=7.5 \end{gathered}$$

The new equilibrium quantity of widgets after subsidy is 7500. By putting the value of Q in the new supply equation, the equilibrium price is calculated below:

$$\begin{gathered} P\left(\$\right)=Q-5 \\ =7.5-5 \\ =2.5 \end{gathered}$$

The new equilibrium price of the widget after subsidy is $2.5 per unit.

The price at which the buyers will buy the widget will be the new equilibrium price of it in the market.Thus, buyers would buy the widget at a price of $2.5 per unit.

The sellers will receive a price equal to the sum of the new equilibrium price and subsidy amount.The sum of the new equilibrium price and subsidy amount is 3.5 (2.5 + 1). Therefore, the sellers would receive a price of $3.5 per unit.

The total cost to the government would be the subsidy amount given to each seller for every unit of widget sold. It is calculated by multiplying the number of widgets sold with the subsidy amount. The total cost to the government is:

$$\begin{gathered} Government\cos t=Subsidyamount\times Quantitysold \\ =\$1\times 7500 \\ =\$7500 \end{gathered}$$