Consider the problem of a rational consumer with an experienced utility function given by 8√x+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 $x 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?

2750?

What about this one?

QUESTION: In this case, what is the difference between the optimal level of total consumption and the level of total consumption in equilibrium?

To find the value of r that leaves the consumer indifferent between the two situations, we need to equate the consumer's utility in both scenarios.

Let's start by calculating the consumer's utility function under the matching policy. The consumer's utility, U, can be expressed as U = 8√x + m, where x represents the units of the good purchased, and m is a constant representing the non-price-related utility.

Under the matching policy, for every x units of the good purchased, the consumer receives an additional x units for free. So, the total quantity of the good the consumer receives is 2x. However, the consumer only pays for x units at the market price of $1 per unit, resulting in a total expenditure of $x.

Therefore, the utility under the matching policy becomes U_m = 8√(2x) + m, where U_m represents the utility under the matching policy.

Next, let's consider the utility under the rebate policy. In this case, the consumer receives a rebate of size r for every unit of the good purchased. So, the consumer effectively pays a reduced price of $1 - r per unit.

The total expenditure under the rebate policy for x units of the good is (1 - r) * x. Therefore, the utility under the rebate policy becomes U_r = 8√x + m + r * x, where U_r represents the utility under the rebate policy.

To find the value of r that leaves the consumer indifferent, we need to equate U_m and U_r:

8√(2x) + m = 8√x + m + r * x

We can simplify this equation as follows:

8√2√x + m = 8√x + m + r * x
8√2√x - 8√x = r * x
8√x * (2 - 1) = r * x
8√x = r * x

Now, we can solve for r:

r = 8√x / x
r = 8/√x

Therefore, the value of r that leaves the consumer indifferent between the two situations is r = 8/√x, where x represents the units of the good purchased.