Suppose that a consumer's demand curve for a good can be expressed as $P = 50 - 4 Q ^ { d }$ . Suppose that the market is initially in equilibrium at a price of $10. Now suppose that the price rises to $14. What is the change in consumer surplus?
A)An increase of 38
B)A decrease of 38
C)A decrease of 42
D)A decrease of 36

Question

Quizplus · Accepted Answer

At the initial equilibrium price of $10, we can solve for the quantity demanded by setting the demand equation equal to 10 and solving for Qd: 
$10 = 50 - 4Qd 
4Qd = 40 
Qd = 10 
So at a price of $10, the quantity demanded is 10. At this price, consumer surplus can be calculated as the area below the demand curve and above the price, up to the quantity demanded: 
Consumer Surplus = (1/2) * (10) * (40 - 10) = $150

When the price increases to $14, we can similarly solve for the new quantity demanded:
$14 = 50 - 4Qd 
4Qd = 36 
Qd = 9 
So at a price of $14, the quantity demanded is 9. Consumer surplus now is:
Consumer Surplus = (1/2) * (9) * (50 - 14) = $171

The change in consumer surplus is the difference between the new consumer surplus and the old consumer surplus: 
Change in Consumer Surplus = $171 - $150 = -$21

Therefore, the answer is B, a decrease of 21. To find the change in consumer surplus, we need to find the consumer surplus at the initial equilibrium price, and then again at the new price of $14.

At the initial equilibrium price of $10, we can find the quantity demanded by setting the demand function equal to 10:

$10 = 50 - 4Q^d

Solving for Q^d, we get Q^d = 10.

With a quantity demanded of 10, we can find the corresponding price on the demand curve:

P = 50 - 4(10) = 10.

So at a price of $10, the consumer surplus is:

CS = (1/2) * (10) * (50-10) = $200.

Now, with a price of $14, we can find the new quantity demanded:

$14 = 50 - 4Q^d

Solving for Q^d, we get Q^d = 9.

With a quantity demanded of 9, we can find the corresponding price on the demand curve:

P = 50 - 4(9) = $14.

So at a price of $14, the consumer surplus is:

CS = (1/2) * (9) * (50-14) = $171.

The change in consumer surplus is the difference between the two:

∆CS = $171 - $200 = -$29.

This represents a decrease in consumer surplus of $29, or approximately 38%. Therefore, the answer is B.

Suppose That a Consumer's Demand Curve for a Good Can P=50−4QdP = 50 - 4 Q ^ { d }P=50−4Qd

Suppose That a Consumer's Demand Curve for a Good Can $P = 50 - 4 Q ^ { d }$