Question

Please answer questions 2 through 7

1. A deli raises the price of its deluxe cheeseburger from $9.50 to $10.50. The quantity sold falls from 125/day to 100/day. Calculate the arc price elasticity of demand.

2. Given your answer to (1), and given that the marginal cost of is $5, should the restaurant raise or lower its price of its deluxe cheeseburger to increase profits?

3. AutoClean does car detailing for $80 per car. Market research indicates that if the price was increased to $105 quantity demanded would fall to zero. Assuming that demand can be modeled with a linear demand curve, estimate the price elasticity of demand at $80.

4. The only thing that changes in Dullsville is the price of a stay at the Dullsville Inn. You've collected the following data on the rates charged (for a suite with 2 queen-sized beds and 'free' continental breakfast) and the number of rooms occupied. The Inn has 100 suites, and at no time were potential visitors turned away due to no vacancy. Use this data to estimate a 'constant elasticity' demand function. Estimate the price elasticity of demand.

Observation Rate per night Quantity (rooms rented)

1 $70 40

2 $65 50

3 $80 30

4 $52 62

5 $92 31

6 $64 41

7 $43 78

8 $74 35

9 $83 33

10 $54 52

11 $87 30

12 $84 28

13 $68 40

14 $43 69

15 $48 53

16 $78 34

17 $72 48

18 $58 53

19 $56 59

5 - 7. Next door to the Dullsville Inn is the Vagabond Hotel. Their rate for a single room is $50/night, with an average of 60 rooms occupied per night. Assume that the industry norm for the price elasticity of demand for hotels like the Vagabond Hotel is -1.6. Further assume that the demand function is reasonably approximated with a constant-price elasticity of demand functional form: Q = aP^b, where b is the price elasticity of demand.

5. Use the above information to calculate the value for 'a.'

6. Use the resulting demand function to estimate the number of rooms occupied if the price was increased to $60/night.

7. The marginal cost of providing a room at the Vagabond Hotel is $20. Use the markup rule for profit maximization to calculate the profit maximizing rate.

Answer 1

1)

In price elasticity, the negative sign is ignored. The Arc price elasticity of deluxe cheeseburger when the price increases from $9.50 to $10.50 is 1.9.

2)

PEd =1.9

MC= $5

Demand for cheeseburger is relatively elastic as PED coefficient is greater than 1 in absolute terms. So, given the Marginal cost and price elastic demand, a fall in price of cheeseburger will raise the total revenue.

3)

% rise in prices = 31.25%

% fall in quantity demanded = 100%

PEd = 100/31.25 = 3.2

4)

constant elasticity demand function :

Obs.	Rate per night	Quantity	change in price	change in qty	% chnge in price	% change in qty		PEd (absolute terms)
1	70	40	-
2	65	50	-5	10	-0.071428571	0.25		3.5
3	80	30	15	-20	0.230769231	-0.4		1.733333333
4	52	62	-28	32	-0.35	1.066666667		3.047619048
5	92	31	40	-31	0.769230769	-0.5		0.65
6	64	41	-28	10	-0.304347826	0.322580645		1.059907834
7	43	78	-21	37	-0.328125	0.902439024		2.75029036
8	74	35	31	-43	0.720930233	-0.551282051		0.764681555
9	83	33	9	-2	0.121621622	-0.057142857		0.46984127
10	54	52	-29	19	-0.34939759	0.575757576		1.647857889
11	87	30	33	-22	0.611111111	-0.423076923		0.692307692
12	84	28	-3	-2	-0.034482759	-0.066666667		1.933333333
13	68	40	-16	12	-0.19047619	0.428571429		2.25
14	43	69	-25	29	-0.367647059	0.725		1.972
15	48	53	5	-16	0.11627907	-0.231884058		1.994202899
16	78	34	30	-19	0.625	-0.358490566		0.573584906
17	72	48	-6	14	-0.076923077	0.411764706		5.352941176
18	58	53	-14	5	-0.194444444	0.104166667		0.535714286
19	56	59	-2	6	-0.034482759	0.113207547		3.283018868

5)

price of room (P) = $50/night

quantity demanded (Q) =  60 rooms

PEd(b) = -1.6 or

constant-price elasticity of demand : Q = aP^b

a=?

putting values in the functional form:

60=a(50)^-1.6

60=a(0.001912705)

a= 31,369.186571

6)

Price = $60/night

a= 31,369.186571

PEd(b) = -1.6

Q= aP^b

Q= (31,369.186571) (60)^-1.6

Q=44.81890 or 45 rooms approx.

7)

A markup rule is the pricing practice of a producer where firm charges a fixed mark-up over its marginal cost.

The markup rule can be derived for VAGABOND with price-setting power by maximizing the following expression for profit:

• · Q = quantity sold,
• · P(Q) = inverse demand function
• · C(Q) = total cost of producing Q.

Profit maximization means that the derivative of profit{\displaystyle \pi } with respect to Q is set equal to 0:

• · P'(Q) = the derivative of the inverse demand function.
• · C'(Q) = marginal cost

By definition is the reciprocal of the price elasticity of demand (1/E). Hence,

Letting "α" be the reciprocal of the price elasticity of demand,

• PEd= -1.6 or -8/5
• Reciprocal of PEd = -5/8=-0.625