Consider the problem of a rational consumer with an experienced utility function given by 8*x^(1/2)+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?
I have never taken business economics. I hope there is someone here who has but have not run into them.
To find the value of r that leaves the consumer indifferent between the two situations, we need to set up an equation that equates the utilities under the matching policy and the rebate policy.
Let's start by calculating the utility under the matching policy. The consumer's utility function is given by U = 8*x^(1/2) + m, where x is the quantity of goods purchased, and m is the income.
Under the matching policy, for every x units purchased, the consumer receives an additional sx units for free. This means the total quantity of goods received is (1 + s)x units.
The price of each unit of x is p = $1, so the total amount spent on x units is px. The income m is not affected by the matching policy.
Therefore, the utility under the matching policy can be expressed as:
U_matching = 8*(1 + s)x^(1/2) + m - px
Next, let's calculate the utility under the rebate policy. With the rebate policy, the consumer receives a rebate of size r per unit purchased. So, for every x units purchased, the consumer receives a total rebate of r * x units.
The effective price under the rebate policy would be p - r, as the consumer gets back r for each unit purchased. Therefore, the total amount spent on x units under the rebate policy would be (p - r) * x.
The utility under the rebate policy can be expressed as:
U_rebate = 8 * x^(1/2) + m - (p - r)x
To find the value of r that leaves the consumer indifferent between the two situations, we need to set U_matching equal to U_rebate and solve for r:
8 * (1 + s) * x^(1/2) + m - px = 8 * x^(1/2) + m - (p - r) * x
Simplifying the equation:
8 * (1 + s) * x^(1/2) - 8 * x^(1/2) = (p - r) * x - px
8 * s * x^(1/2) = (p - r) * x - px
8 * s * x^(1/2) = (r - p) * x
8 * s = (r - p) * x^(-1/2)
Now, x is the quantity of goods purchased, and it can vary. Therefore, for the consumer to be indifferent between the two situations, the coefficient of x on both sides of the equation should be the same (as they represent the same quantity of goods purchased). Therefore:
8 * s = (r - p) * x^(-1/2)
Simplifying this equation, we can solve for r in terms of s:
r = p + 8 * s * x^(-1/2)
Since x is arbitrary, we can assume a value for x. For simplicity, let's assume x = 1 (you can choose any other value):
r = p + 8 * s
Therefore, the value of r (as a function of s) that leaves the consumer indifferent between the two situations is given by:
r = p + 8 * s
In this case, r = $1 + 8s.