Consider a rational consumer with a utility function given by U(x,m)=Aln(2x)+m. The consumer needs to decide how much of good x to buy given the following pricing rule: the first 10 units sell at a price of $2 p/unit, additional units sell at a price of $3 p/unit.
QUESTION. What is the maximum value of A at which the consumer buys 10 units or less?
To find the maximum value of A at which the consumer buys 10 units or less, we need to determine how much of good x the consumer will buy at different prices and utility levels.
To do this, we can set up an optimization problem. The consumer's objective is to maximize their utility function U(x, m) while subject to the budget constraint imposed by the pricing rule.
The budget constraint can be expressed as:
10x + (x - 10) * 3 = m
Here, 10x represents the cost of the first 10 units of good x at $2 per unit, (x - 10) represents the additional units of good x bought at $3 per unit, and m is the consumer's income.
Now, we can rewrite the consumer's utility function in terms of x and m:
U(x, m) = A * ln(2x) + m
To find the maximum value of A, we can differentiate U(x, m) with respect to x and set the derivative equal to zero:
dU(x, m)/dx = 0
Differentiating U(x, m), we get:
A * (1/x) * 2 = 0
Simplifying, we find:
(2A) / x = 0
Since x ≠ 0 (because we need to purchase at least some units), the only way this derivative can be zero is if 2A = 0. Therefore, A = 0 is the maximum value at which the consumer buys 10 units or less.
In simpler terms, the consumer will not buy any units or have a maximum value of A (i.e. any positive value of A will result in the consumer buying more than 10 units).