QUESTIONS, PROBLEMS, ANSWERS, SOLUTIONS TO PROBLEMS
QUESTIONS
1. What does demand theory and analysis involve? List what is used in determining a firm`s revenue.
2. What are demand shifters? Give examples.
3. Define total revenue, average revenue, and marginal revenue. Explain the relationships among these.
4. What are the factors associated with demand elasticity?
5. Distinguish between arc elasticity and point elasticity.
6. What is the special feature of a linear demand curve in terms of price elasticity?
7. Describe the relationship between price elasticity of demand and total revenue.
8. Give two good examples of how price elasticity can be used for marketing decisions.
9. How do you know whether a product is a normal good or an inferior good?
10. What type of function form would give constant elasticity of demand?
11. Write down the relationship between the point price elasticity of demand, marginal revenue, and price.
PROBLEMS
1. A valve manufacturer is considering raising the price of its valves from the current average price of $2,000 per unit. It is currently selling 600 units per month, but trying to bring its sales down to its maximum output of 500 units per month. If the firm`s price elasticity of demand is 2 over the range of $2,000 to $2,500 per valve, what will the new price have to be in order to bring demand in line with capacity?
2. Lilex Watch, Inc. is considering lowering the price of its watches from the current $80 per unit to $70 per unit. It sells 2,000 units per month. If the watch`s price elasticity of demand is 2, what will be the new quantity sold?
3. The Sunhee Bags, Inc. has estimated the following demand function for its deluxe sleeping bag in the San Francisco Bay area:
P = 600  .3Q
(a) Determine the MR and TR functions.
(b) At what price would Sunhee Bags fail to sell any sleeping bags?
(c) What is the maximum quantity Sunhee Bags could sell?
(d) What is the maximum revenue that the firm could receive?
(e) For a given percentage change in price, what would be the percentage change in quantity demanded at the output level Q = 700?
(f) What is the arc price elasticity of demand for the quantity range of 700800 units?
4. Given the following demand curve:
Q = 250  .5P
Determine the corresponding total revenue (TR) function, marginal revenue (MR) function, and the average revenue (AR) function.
5. Given the following demand curve:
P = 400  3Q
Determine the corresponding total revenue (TR) function, marginal revenue (MR) function, and the average revenue (AR) function.
6. The marketing department of a compact automaker has determined the following demand functions for their cars:
Q = 150,000 52P + 80P_{C} + 0.3A + 0.5Y
where 
Q = the number of the firm`s cars sold weekly. 

P = the price of the firm`s car. 

P_{C} = the price of a close competitor`s car. 

A = weekly advertising dollars spent. 

Y = average family disposable income. 
(a) If P = $12,000, P_{C} = $10,000, A = $310,000, and Y = $25,000, derive the demand function and determine the price elasticity of demand.
(b) Comment on the price elasticity obtained in part (a). If the firm raises price, what will happen to total revenue?
(c) Assuming the values given in part (a), calculate the income elasticity of demand. Interpret your answer.
(d) Determine the cross elasticity of demand between two competing cars. Interpret your answer.
7. The demand for your product has been estimated to be
Q_{A} = 8,500  4P_{A}  2P_{B} + P_{C}  .1Y, where P = price and Y = income. The relevant price and income data are as follows: P_{A} = 10, P_{B} = 50, P_{C} = 50, and Y = 40,000.
(a) Which goods are substitutes for product A? Which are complements?
(b) Is A an inferior or a normal good?
(c) How much A will be purchased?
8. (a) Derive the following formula:
(b) Explain the relationship between marginal revenue (MR) and the price elasticity of demand (e_{p})
9. Suppose the manager of a toy store notes a 2% increase in weekly sales following a 1% price discount on the Bamboo doll. The store`s wholesale cost per doll plus display and marketing expenses total $15 per unit.
(a) What is the store`s point price elasticity of demand?
(b) Calculate the optimal price.
(c) Suppose the manager can reduce, through quantity buying, marginal costs per unit by $1 to $14. What is the new optimal price?
10. Your company`s food product has its own price elasticity of demand of .8. If the price of food increased by 5 percent, what would happen to the quantity of food demanded and the total revenue for your company`s food product?
11. The following relations describe monthly demand and supply functions for dry cleaning services in the local area:
Q_{d} = 40,000  4,000P 
(demand) 
Q_{s} = 10,000 + 10,000P 
(supply) 
(a) At what average price level would demand equal zero?
(b) At what average price level would supply equal zero?
(c) Find the equilibrium priceoutput combination?
ANSWERS
1. Demand analysis include (a) quantitative expression of demand (specification of demand functions), (b) calculation of various elasiticities of demand, (c) estimation of demand functions, (d) forecasting sales, and (e) pricing decisions.
2. Demand shifters are factors that create increase or decrease in demand for a good brought about by a change in any factor other than the price of the product. They include (1) consumers` incomes, (2) the prices of substitute or complementary goods, and (3) consumers` tastes. Graphically, a shift in demand may be represented by a parallel movement of the demand curve.
3. Total revenue (TR) of a firm is directly related to the demand for the firm`s product or service. Average revenue (AR) is total revenue per unit of output, that is total revenue received divided by output., i.e., AR = TR/Q. Marginal revenue (MR) is the rate of change of total revenue with respect to quantity demanded, i.e., MR = dTR/dQ. The total revenue is maximized when MR is equal to zero.
4. The principal factors involved with demand elasticity are (1) the price of the good (in the case of price elasticity), (2) the price of a substitute product (in the case of cross elasticity), (3) income (in the case of income elasticity), and (4) advertising (in the case of advertising elasticity).
5. Arc elasticity of demand is the average responsiveness of quantity demanded to a change in price between two different values, while point elasticity is the elasticity measured at a given point.
6. In the case of a linear demand function, while the slope of a straight line demand curve is the same at all points, the elasticity for such a curve varies from one point to the next.
7. Economists have established the following relationships between price elasticity (e_{p}) and total revenue (TR), which can aid a firm in setting its price.
Price 
e_{p }> 1 
e_{p }= 1 
e_{p }< 1 
Price rises 
TR falls 
No change 
TR rises 
Price falls 
TR rises 
No change 
TR falls 
8. Price elasticities can be used to answer the following types of questions: (a) what will be the impact on sales of a 5% price increase? (b) how great a price reduction is necessary to increase sales by 20%?
9. If income elasticity (e_{Y}) is greater than zero, products are normal or superior goods whose demand varies directly with income, holding prices constant. If e_{Y} < 0, products are inferior goods.
10. The multiplicative form (or power function) gives constant elasticity. The form is found in power functions. An example is Q = aP^{b}A^{c}Y^{d}.
11. The relationship between the point price elasticity of demand, marginal revenue, and price is:
SOLUTIONS TO PROBLEMS
1.)
2 = [(500  600)/(P_{2}  2000)][(P_{2} + 2000)/(500 + 600)] = 100(P_{2} + 2000)/1100/(P_{2}  2000)
2(11P_{2}  22000) = P_{2}  2000
22P_{2} + 44000 = P_{2}  2000
21P_{2 }= 46000
P_{2} = $2190.48
2.
2 = [(Q_{2}  2000)/( 70  80)][(70 + 80)/(Q_{2} + 2000)]
2 = 150(Q_{2}  2000)/10(Q_{2} + 2000)
2(10Q_{2}  20000) = 150Q_{2}  300000
20Q_{2} + 40000 = 150Q_{2}  300000
130Q_{2} =340000
Q_{2} = 2615.38
3.
(a) Given P = 600  .3Q:
TR = PQ = (600  .3Q)Q = 600Q  .3Q^{2}
MR = dTR/dQ = 600  .6Q
(b) At P = $600, Q = 0 since P = 600  .3(0) = $600.
(c) P= 600  .3Q, so Q = 2,000  10/3P
At a price of 0, Q = 2,000  10/3(0) = 2,000 units.
(d) TR will be maximized when MR = 0.
MR = dTR/dQ = 600  .6Q = 0; Q = 1,000 units.
At Q = 1,000 units, TR = 600Q  .3Q^{2} = 600(1,000)  .3(1,000)^{2} = $300,000
(e)
First, dQ/dP= 10/3
Then, at Q = 700, P = 390
e_{ p} = (10/3) x (390/700) = 1.86. The point elasticity of demand is 1.86, so the percentage change in quantity demanded would be 1.86 times the percentage change in price.
(f) Since P= 600  .3Q,
At Q = 700, P= 600  .3(700) = $390
At Q = 800, P= 600  .3(800) = $360
Therefore,
Arc elasticity of demand
4.
Q = 250  .5P > .5P = 250  Q > P = 500  2Q
TR = P x Q = (500  2Q)Q = 500Q  2Q^{2}
MR = dTR/dQ = 500 4Q
AR = TR/Q = 500  2Q
5.
TR = P x Q = (400  3Q)Q = 400Q  3Q^{2}
MR = dTR/dQ = 400 6Q
AR = TR/Q = 400  3Q
6.
(a) The demand function is:
Q = 
150,000 52P + 80P_{C} + 0.3A + 0.5Y 
= 
150,000 52P + 80(10,000) + 0.3(310,000) + 0.5(25,000) 
= 
1,055.500 52P 
Hence, Q = 1,055,500  52P = 1,055,500 52(12,000) = 431,500
(b) The price elasticity of demand is
Demand for the car is elastic and thus if the price is raised, total revenue will drop sharply.
(c) The income elasticity is
Since the income elasticity is greater than 0 but less than 1, the car is normal but not superior. If income goes up, demand will rise but not at the same rate of increase as income.
(d) The cross elasticity is
Since the cross elasticity is greater than 0, these two cars are substitutes.
7.
(a) C is a substitute for A, while B is a complement for A.
(b) A is an inferior good, as indicated by a negative sign associated with the income variable.
(c) Q_{A} = 8,500  4P_{A}  2P_{B} + P_{C}  .1Y = 8,500 4(10)  2(50) + (50)  .1(40,000) = 4,410
8.
(a) By definition,
MR = dTR/dQ = d(P x Q)/dQ = d[f(Q)x Q)]/dQ =f(Q) + f`(Q) Q = P + Q (dP/dQ)
(b) We observe that
MR > 0 when e_{p }> 1
MR = 0 when e_{p} = 1
MR < 0 when e_{p} < 1
9.
(a)
(b) The profitmaximizing price is then:
P = $15/(1 + 1/2) = $30
(c) The new optimal price is:
P = $14/(1 + 1/2) = $28
Thus, the profitmaximizing price would fall by $2 following a $1 reduction in the store`s marginal costs.
10.
% change in Q = .8(5) = 4%. The demand for food will fall by 4% if the price increases by 5%. Since the price elasticity is less than 1, demand is inelastic. Thus, the price increase will lead to an increase in your total revenue.
11.
(a) Set Q_{d} = 0 and solve for P.
Q_{d} = 40,000  4,000P = 0; P = $10.
(b) Set Qs = 0 and solve for P.
Q_{s} = 10,000 + 10,000P = 0; P = $1
(c) At the market equilibrium, Q_{d} = Q_{s.}
40,000  4,000P = 10,000 + 10,000P
50,000 = 14,000P; P = $3.57, Substituting P = $3.57 into either the demand and the supply function yields:
Q_{d} = 40,000  4,000P = 40,000  4,000($3.57) = 25,720 units.