Question

As we shall see in Chapter \(15,\) a firm may sometimes seek to differentiate its product from those of its competitors in order to increase profits. In this problem we examine the possibility that such differentiation may be "spurious" that is, more apparent than real and that such a possibility may reduce the buyers' utility. To do so, assume that a consumer sets out to buy a flat screen television \((y) .\) Two brands are available. Utility provided by brand 1 is given by \(U\left(x, y_{1}\right)=x+500 \ln \left(1+y_{1}\right)\) (where \(x\) represents all other goods). This person believes brand 2 is a bit better and therefore provides utility of \(U\left(x, y_{2}\right)=x+600 \ln \left(1+y_{2}\right)\) Because this person only intends to buy one television, his or her purchase decision will determine which utility function prevails. a. Suppose \(p_{x}=1\) and \(I=1000,\) what is the maximum price that this person will pay for each brand of television based on his or her beliefs about quality? (Hint: When this person purchases a TV, either $$ \left.y_{1}=1, y_{2}=0 \text { or } y_{1}=0, y_{2}=1\right) $$ a. Suppose \(p_{x}=1\) and \(I=1000,\) what is the \(\operatorname{maxi}\) mum price that this person will pay for each brand of television based on his or her beliefs about quality? (Hint: When this person purchases a TV, either \(\left.y_{1}=1, y_{2}=0 \text { or } y_{1}=0, y_{2}=1\right)\) b. If this person does have to pay the prices calculated in part (a), which TV will he or she purchase? c. Suppose that the presumed superiority of brand 2 is spurious-perhaps the belief that it is better has been created by some clever advertising (why would firm 2 pay for such advertising?). How would you calculate the utility loss associated with the purchase of a brand \(2 \mathrm{TV} ?\) d. What kinds of actions might this consumer take to avoid the utility loss experienced in part (c)? How much would he or she be willing to spend on such actions?

Short answer

Answer: The utility loss associated with purchasing brand 2 if its presumed superiority is spurious is 100. To avoid this utility loss, the consumer could gather more information about the actual quality of the two brands or spend time and effort comparing the brands. The consumer's willingness to spend on such actions would be at most equal to the utility loss, which is 100.

Step-by-step solution

Calculate the maximum price for each brand

In order to calculate the maximum price for each brand, we must determine the consumer's utility function and constrain the income as given in the problem statement. We have been given two utility functions, one for each brand: 1. \(U(x,y_1)=x+500\ln(1+y_1)\) 2. \(U(x,y_2)=x+600\ln(1+y_2)\) Let \(p_{y_1}\) and \(p_{y_2}\) be the prices of brand 1 and brand 2, respectively. The consumer has an income of \(I=1000\), and the price of the other good is \(p_x=1\). Using the income as a constraint, we can say that: $$ x + p_{y_1}y_1 = 1000\ (when\ purchasing\ brand\ 1) $$ and $$ x + p_{y_2}y_2 = 1000\ (when\ purchasing\ brand\ 2) $$ Since the buyer only purchases one TV, either \(y_1=1, y_2=0\) or \(y_1=0,y_2=1\). Therefore, we write: 1. \(x+p_{y_1} = 1000\) 2. \(x+p_{y_2} = 1000\) Now we can find the maximum prices this person will pay for each brand by finding the maximum value of \(p_{y_1}\) and \(p_{y_2}\) subject to the constraints above.

Determine the purchase decision

Given the constraints, we need to find the maximum values of \(p_{y_1}\) and \(p_{y_2}\) for the consumer's utility to be equal. When the consumer is indifferent between the two brands, the utility from both brands should be the same: $$ U(x,y_1) = U(x,y_2) $$ Substitute the values of y and the income constraints for each utility function: $$ 1000-p_{y_1}+500\ln(1+1) = 1000-p_{y_2}+600\ln(1) $$ Solve for the maximum price values: $$ p_{y_1} = 500\ln(2) \approx 346.573 $$ $$ p_{y_2} = 500\ln(2) - 100 \approx 246.573 $$ If the consumer has to pay these amounts, they would opt for TV brand 2.

Calculate the utility loss if the superiority of brand 2 is spurious

If brand 2 does not provide any additional utility over brand 1, then their utility functions would be the same: $$ U(x,y_1)=x+500\ln(1+y_1)=x+500\ln(2) $$ The utility loss associated with purchasing brand 2 would be the difference in utility between the two brands: $$ Utility\ Loss = U(x,y_1) - U(x,y_2) = U(x,y_1) - (U(x,y_1) + 100) = 100 $$

Actions to avoid utility loss and willingness to pay

To avoid the utility loss of 100, the consumer could try to gather more information about the actual quality of the two brands or spend some time and effort comparing the brands. The consumer's willingness to spend on such actions would be at most equal to the utility loss, which is 100. If the cost of these actions is less than or equal to the utility loss, the consumer would be willing to undertake them.