# arc cross elasticity of demand

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1. The Potomac Range Corporation manufactures a line of microwave ovens costing $500 each. Its sales have averaged about 6,000 units per month during the past year. In August, Potomac's closest competitor, Spring City Stove Works, cut its price for a closely competitive model from $600 to $450. Potomac noticed that its sales volume declined to 4,500 units per month after Spring City announced its price cut.

a. What is the arc cross elasticity of demand between Potomac's oven and the competitive Spring City model?

b. Would you say that these two firms are very close competitors? What other factors could have influenced the observed relationship?

c. If Potomac knows that the arc price elasticity of demand for its ovens is -3.0, what price would Potomac have to charge to sell the same number of units it did before the Spring City price cut?

2. The Stopdecay Company sells an electric toothbrush for $25. Its sales have averaged 8,000 units per month over the last year. Recently, its closest competitor, Decayfighter, reduced the price of its electric toothbrush from $35 to $30. As a result, Stopdecay's sales declined by 1,500 units per month.

a. What is the arc cross elasticity of demand between Stopdecay's toothbrush and Decayfighter's toothbrush? What does this indicate about the relationship between the two products?

b. If Stopdecay knows that the arc price elasticity of demand for its toothbrush is -1.5, what price would Stopdecay have to charge to sell the same number of units as it did before the Decayfighter price cut? Assume that Decayfighter holds the price of its toothbrush constant at $30.

c. What is Stopdecay's average monthly total revenue from the sale of electric toothbrushes before and after the price change determined in part (b)?

d. Is the result in part (c) necessarily desirable? What other factors would have to be taken into consideration?

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#### Solution Preview

1 a Please note that the arc cross elasticity of demand between Potamac's oven and the competitive spring city is calculated by (Change in quantity demanded of Potamac/average quantity demanded of Potomac) divided by ( Change in price of Spring City/ average price of Spring City).

This translated into numbers means that (1500/5250)/(150/525) = 1. ( explanation: Quantities are 6000 and 4500, if we add these and divide it by 2 we get 5250, similarly we get the price average of 525)

b. Yes these are close competitors; the positive sign of 1 discloses this.

c. Potomac can find its own price elasticity (Change in quantity demanded of Potomac/average quantity demanded of Potomac) divided by (Change in price of Potomac/ average price of Potomac). In terms of figures we get (1500/5250)/(P - 500)/{P-500/(500 + P)/2} = -3. (Explanation: the new price is not known to us, however, the elasticity is given to us and the current price is known to us so we have a simple equation with one unknown.)

Please try and solve the equation. You will get $454.54.

2.a

Please note that the arc crosses elasticity of demand between Stopdecay's oven and the competitive spring city is calculated by (Change in quantity demanded of Stopdecay/average quantity demanded of Stopdecay) divided by (Change in price of Decay fighter/ average price of Decay fighter).

This translated into numbers means that (1500/7250)/(5/32.5) = +1.34. (Explanation: Quantities are 8000 and 6500, if we add these and divide it by 2 we get 7250, similarly we get the price average of 32.5).

b.

Stopdecay can find its own price elasticity (Change in quantity demanded ...

#### Solution Summary

Consider arc cross elasticity of demand.

Managerial Economics - Applying Concepts/Mathematics

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1. In 1998 the fare on Chicago's transit system was 60 cents per ride. This resulted in 711.6 million trips being taken on the system. In 1999 the fare was increased to 80 cents and ridership declined to 692.4 million trips.

a. Compute the arc price elasticity of demand for transit ridership in Chicago assuming that all other factors influencing demand remained constant during this period.

b. Based on your answer to part (a), do you believe the fare increase was a rational action for the Chicago Transit Authority?

c. What other factors do you feel may have had an impact on ridership during this period? Do you believe the decline in ridership experienced in 1999 tends to overstate or understate the actual impact of the fare increase?

d. In 2000 the fare increased to 90 cents and ridership declined to 640 million trips. Compute the arc price elasticity between 1999 and 2000. How can you account for the differences between the 1998-1999 elasticity coefficient and the 1999-2000 elasticity coefficient?

2. The demand for haddock has been estimated as28

log Q = a + b log P + c log I = d log Pm

where Q = quantity of haddock sold in New England

P = price per pound of haddock

I = a measure of personal income in the New England region

Pm = an index of the price of meat and poultry

If b = -2.174, c = .461, and d = 1.909,

a. Determine the price elasticity of demand.

b. Determine the income elasticity of demand.

c. Determine the cross price elasticity of demand

d. How would you characterize the demand for haddock?

e. Suppose disposable income is expected to increase by 5 percent next year. Assuming all other factors remain constant, forecast the percentage change in the quantity of haddock demanded next year.

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(See attached file for full problem description)

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