Consider the problem of a rational consumer with an experienced utility function given by 8x√+m. Let p=$1 p/unit denote the market price of good x.

Suppose that, initially, the firm selling the good matches his purchases as follows: for every x units that he buys, he gets an additional sx units for free.

Based on customer feedback, the firm is considering eliminating the matching policy and introducing instead a price rebate of size r per-unit purchased. Note that under the rebate policy, the consumer gets back $r for every unit that he purchases

QUESTION: What is the value of r (as a function of s) that leaves the consumer indifferent between the two situations?

S/(S+1)

thanks

To find the value of r that leaves the consumer indifferent between the two situations, we need to set up an equation by equating the utilities of the two scenarios.

Let's first consider the utility under the matching policy:
Utility with matching policy = 8x√ + m

Now, let’s consider the utility under the rebate policy:
Utility with rebate policy = 8x√ + r + m

Since the customer is indifferent between the two situations, the utilities must be equal.
So, we have the equation:
8x√ + m = 8x√ + r + m

The m terms cancel out on both sides of the equation, giving us:
8x√ = 8x√ + r

Next, we want to find the value of r as a function of s. Since the customer gets additional sx units for free under the matching policy, we have to express the quantity of x in terms of the additional units received:

Quantity of x under matching policy = x + sx

Substituting this into the equation, we get:
8(x + sx)√ = 8x√ + r

Simplifying the equation:
8√(x + sx) = 8√x + r

Now, we need to solve for r. Rearranging the equation, we get:
r = 8√(x + sx) - 8√x

Since we want to express r as a function of s, we replace x with sx:
r = 8√(sx + ssx) - 8√(sx)

Simplifying further, we get the final expression for r as a function of s:
r = 8√(s + s^2) - 8s√s

Therefore, the value of r, as a function of s, that leaves the consumer indifferent between the two situations is:
r = 8√(s + s^2) - 8s√s

To find the value of r that leaves the consumer indifferent between the two situations, we need to set up and solve an equation.

Under the matching policy, for every x units purchased, the consumer receives additional sx units for free. This means that the total units received is x + sx = (1 + s)x.

Under the rebate policy, the consumer receives a rebate of r per unit purchased. So, for x units purchased, the total rebate received is rx.

Now, in order for the consumer to be indifferent between the two situations, the utility derived from both situations should be the same.

Under the matching policy, the utility function is 8x√ + m, where m is a constant representing the initial utility level.

Under the rebate policy, the utility function is 8(x + sx)√ + m - rx. Here, the first term represents the utility from the purchased units, the second term represents the utility from the free units received, and the third term represents the rebate cost.

Setting the two utility functions equal, we have:
8x√ + m = 8(x + sx)√ + m - rx

We can simplify this equation by canceling out the common terms:
8x√ = 8(x + sx)√ - rx

Now, we can solve for r:
rx = 8(x + sx)√ - 8x√
rx = 8(1 + s)x√ - 8x√
rx = 8x√(1 + s - 1)
rx = 8sx√
r = 8s√

Therefore, the value of r (as a function of s) that leaves the consumer indifferent between the two situations is r = 8s√.