Question

11. Show your solutions/ calculations for each question (not just the final answer).

Dave’s demand for good X is given by the following equation:

q = 10 - 2p + 0.0004I

where q is the quantity demanded at price p when Dave’s income is I. His income is currently $85,000.

a. (3 points) At what price will Dave’s demand fall to zero?

b. (3 points) If the price of good X is $14, how many will be demanded by Dave?

c. (3 points) At a price of $14, calculate Dave’s price elasticity of demand for good X.

d. (3 points) In a scratch paper, draw a diagram showing the demand curve for good X and shade the consumer surplus region at a price of $14. Calculate consumer surplus. You do not need to submit your drawing.

e. (5 points) If the price of good X decreases to $12, how much consumer surplus is gained?

Answer 1

(a) When demand falls to zero, q=0.

Using q=0, I= 85000 in

q= 10-2p+0.0004I

Implies 0= 10-2p +(0.0004×85000)

2p = 10+ 34

p= 44÷ 2 = 22

So at price =$22, Dave's demand falls to zero.

(b) When p=$14, I = $85000

q= 10 - 2(14) + 0.0004(85000)

q= 10-28+34 = 16

So, at p=$14, Dave demands 16 units of good X.

(c) When p= 14,q=16 ( from part (b) )

Given - q= 10-2p+0.0004I

Differentiating q with respect to p gives

∆q/∆p = (-2)

Elasticity (e) = (∆q/∆p) × (p/q)

= (-2) × (14/16) = (-1.75)

Since |e| = 1.75>1, demand is elastic.

(d) when graphing q= 10-2p+ (0.0004×85000)

Y intercept = 22,

At p=14, consumer surplus is the area below the demand curve and above the price line.

Consumer surplus = (1/2) × base × height

= (1/2) × 16× (22-14)

=(1/2) ×16×8

=64

(e) When p=12,

q= 10-2p +34

q= 44-2(12)

q=20

New consumer surplus=(1/2) × 20 × (22-12)

= (1/2) × 20× 10= 100

Change in consumer surplus= new consumer surplus - old consumer surplus = (100-64) =36

So amount of consumer surplus gained = 36