Question

1. The Ice Cream Parlor is the only ice cream parlor in Beautown. The son of the owner is just back from college, where he majors in economics. He has just studied demand analysis and he decides to apply what he has learned to estimate the demand for ice cream in his father’s parlor during his summer vacation. Using regression analysis, he estimates the following demand function:

Q = 80 -20P

(a) Find the point price elasticity at each dollar price, from P = $4 to P =$0

(b) Find the arc elasticity between consecutive one dollar price changes from price of $4 to price of $0 (i.e., between P = $4 and P = $3, P = $3 and P =$2, ……… P = $1 and P = $0 ) (15 points) •

Every elasticity coefficient; price, income, cross, supply, etc. is made-up of the product of two parts

1. A measure of the absolute rate of change

2. The ratio of size of the two variables involved; e.g, P/Q or I/Q, etc.

3. It is necessary to determine what the value of Q is to work the problem so the first step is to plug in the price and find Q.

• Point elasticities measure the elasticity at a particular point or set of coordinates of the two variables (e.g., a particular price and quantity, for price elasticity). The change (which is the first derivative of the equation) is measured by the slope coefficient of the equation. See pp. 131-133.

• Arc elasticities measures, which were called midpoints formulas in your principles course, are measuring of the elasticity across a certain arc or range of the curve. Changes are measured point to point and absolute size is measured by an average of two points; e.g., average price and average quantity. See pp.133-135.

Answer 1

a. Point price elasticity=(∆Q/∆P)(P/Q)

Q=80-20P

∆Q/∆P= -20

When P=0, Q=80

Elasticity= -20(0/80)=0

When P=1, Q=60

Elasticity= -20(1/60)=-0.33

When P=2, Q=40

Elasticity= -20(2/40)=-1

When P=3, Q=20

Elasticity= -20(3/20)=-3

When P=4, Q=0

Elasticity= -20(4/0)=infinity

b. Arc elasticity=Mid point method of elasticity=(Q2-Q1)/(Q2+Q1)÷(P2-P1)/(P2+P1)

When P=$4, Q=0

When P=$3, Q=20

Arc elasticity= (20-0)/(20+0)÷(3-4)/(3+4)= -7

When P=$3, Q=20

When P=$2, Q=40

Arc elasticity= (40-20)/(20+40)÷(2-3)/(3+2)= -1.6

When P=$2, Q=40

When P=$1, Q=60

Arc elasticity= (60-40)/(60+40)÷(2-1)/(2+1)= -0.6

When P=$1, Q=60

When P=$0, Q=80

Arc elasticity= (80-60)/(80+60)÷(0-1)/(0+1)= -0.14

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