2. The demand function for Fritz Reinhart premium energy drinks has been estimat

Question

2. The demand function for Fritz Reinhart premium energy drinks has been estimated as x 37,986.5- 4,476.9P 2,994.2P, 668.2A 849.7Ay where Qx is the demand for the Reinhart drink (in sixpacks) is the price of the Reinhart drink (in dollars); Pyis the price of the competing brand of premium energy drink (in dollars); A is the advertising expenditure for Reinhart (in thousands of dollars): and Ay is the advertising expenditure for the competing brand (in thousands of dollars). The current values of the independent variables are = 9.95; 8.95; Ax-36; and Ay-22. (a) Calculate the price elasticity of demand for Reinhart energy drinks. How would you interpret this? (b) Calculate the cross-price elasticity of demand for Reinhart energy drinks. How would you interpret this? (c) Calculate the cross-advertising elasticity of demand for Reinhart energy drinks. How would you interpret this? (d) If the marginal cost of producing Reinhart drinks is constant at $4.00 per sixpack, should the firm change its price to maximize profits? (e) Suppose instead that the Reinhart Company wishes to maximize sales revenue. What price should it set?

Explanation / Answer

(a) First Let us Find Q by plugging in the independent variables given in the Question. We get the value of Q as

Q= 37986.5-4476.9*Px+2994.2Py+668.2Ax-849.7Ay

Q=37986.5-4476*9.95+2994.2*8.95+668.2*36-849.7*22 = 25601.235

Price Elasticity of Reinhart Energy Drink =  4476.9* 9.95/25601.235 =1.73

This means that demand is relatively elastic. A small change in price brings about a very large change in Quantity.

(b) Cross Price Elasticity = (dQ/dPy*Py/Qx) = 2994.2* 8.95/25601.235 = 1.0467

This implies that as the price of competing  brand product increases the quantity demanded of Reinhart also increases i.e there is positive Cross Price Elasticity.

(c) Cross Advertising elasticity =  (dQ/dAy* A/Q) = -849.7*22/25601.235 = -0.730

-0.730 implies as the competing brand increases its promotional activity, the demand for Reinhart product falls.

(d) If the Marginal cost is 4 Reinhart Drinks should change its price from 9.95 to 12.009.

We Know that at Profit Maximization MC= MR

We Know Total Revenue = Price * Quantity

TR = 37986.5*Px-4476.9Px2 +1994.2Py *Px+668.2Ax*Px -849.7 Ay* Px

Differentiating Total Revenue we obtain Marginal revenue

MR =37986.5-4476.9*2*Px +2994.2 Py+668.2* Ax -849.7* Ay

Plugging the Value of Py Ax Ay (Since this is not determined in the system) We get

MR = 107533.19-8953.8Px

At profit Maximization MC=MR

107533.19-8953.8Px = 4

Px= 12.01

Thus Reinhart Drinks have to increase their price to 12.01.

(e) Sales Maximization occurs at AR (Average Revenue) = AC(Average Cost )

Average Cost, in this case, is equal to Marginal Cost AC= 4

AR = 37986.5-4476.9Px +2994.2Py +66802Ax-849.7*Ay

On substituting the value of Py , Ax, Ay we get

AR =70146.39 -4476.9Px = AC= 4

Px = 15.6676

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