Question

Assume the following values for Figures 4.4a and 4.4b: Q1 = 20 bags; Q2 = 15 bags; Q3 = 27 bags. The market equilibrium price is \(45 per bag. The price at a is \)85 per bag. The price at c is \(5 per bag. The price at f is \)59 per bag. The price at g is $31 per bag. Apply the formula for the area of a triangle (Area = ½ × Base × Height) to answer the following questions.

a. What is the dollar value of the total surplus (= producer surplus + consumer surplus) when the allocatively efficient output level is produced? What is the dollar value of the consumer surplus at that output level?

b. What is the dollar value of the deadweight loss when output level Q2 is produced? What is the total surplus when output level Q2 is produced?

c. What is the dollar value of the deadweight loss when output level Q3 is produced?

Short answer

1. The total surplus is $800, and the consumer surplus is $400 at the Q1 level of output.
2. The deadweight loss is $50, and the total surplus is $750 at the Q2 level of output.
3. The deadweight loss is $98, and the total surplus is $702 at the Q3 level of output.

Step-by-step solution

Step 1. Explanation for part (a)

The total surplus is given by the sum of the triangle area above the price line formed by the demand curve and the area of the triangle below the price line formed by the supply curve.

Consider the following diagram:

The area of triangle abc gives the total surplus (sum of consumer and producer surplus).

$$\begin{gathered} areaof∆abc=\frac{1}{2}\times base\times height \\ =\frac{1}{2}\times \left(85-5\right)\times 20 \\ =\$800 \end{gathered}$$

The consumer surplus is given by the area of the triangle abh (above the price line and below the demand curve).

$$\begin{gathered} areaof∆abh=\frac{1}{2}\times base\times height \\ =\frac{1}{2}\times \left(85-45\right)\times 20 \\ =\$400 \end{gathered}$$

Thus, the total surplus is $800, and the consumer surplus is $400.

Step 2. Explanation for part (b)

Deadweight loss is the efficiency loss triangle formed due to overproduction or underproduction of a good.

To calculate the deadweight loss, you need to find the price at d and e. This can be done using the demand equation and supply equations.

The inverse demand equation is $P=a+b{Q}_d$where P is the price and Qd is the quantity demanded.

• a is the intercept term when Q=0, which is equal to $85.
• The slope b is measured by the ratio of change in quantity demanded and change in price (as this is an inverse demand function).

$$\begin{gathered} b=\frac{20-0}{45-85} \\ =-2 \end{gathered}$$

Thus, the function is$P=85-2Q_d$.

The inverse supply equation is$P=a+b{Q}_s$, where P is the price and Qs is the quantity supplied.

• a is the intercept term when Qs=0, which is equal to $5.
• The slope b is measured by the ratio of change in quantity demanded and change in price (as this is an inverse demand function).

$$\begin{gathered} b=\frac{20-0}{45-5} \\ =2 \end{gathered}$$

Thus, the function is$P=5+2Q_s$.

Thus, the price at d is $55, which is calculated below:

$$\begin{gathered} P=85-2Q_d \\ =85-2\left(15\right) \\ =\$55 \end{gathered}$$

The price at e is $35, which is calculated below:

$$\begin{gathered} P=5+2Q_d \\ =5+2\left(15\right) \\ =\$35 \end{gathered}$$

Now, you can calculate the deadweight loss, which is given by the area of triangle bde, as shown below:

$$\begin{gathered} areaof∆bde=\frac{1}{2}\times base\times height \\ =\frac{1}{2}\times \left(55-35\right)\times \left(20-15\right) \\ =\$50 \end{gathered}$$

The total surplus at Q2 is the difference between the total surplus at the efficient level of output and deadweight loss due to underproduction, that is, 800-50=$750.

Thus, the total surplus at Q2 is $750, and the deadweight loss is $50.

Step 3. Explanation for part (c)

The area of triangle bfg gives the deadweight loss, as shown below:

$$\begin{gathered} areaof∆bfg=\frac{1}{2}\times base\times height \\ =\frac{1}{2}\times \left(59-31\right)\times \left(27-20\right) \\ =\$98 \end{gathered}$$

The total surplus at the Q3 level of output is the difference between the total surplus at the efficient level of output and deadweight loss due to over-production until Q3. Thus, the total surplus at Q3 is 800-98=$702.

Thus, the total surplus at Q3 is $702, and the deadweight loss is $98.