Tasha consumes two types of nuts: almonds and walnuts. Her utility function from eating nuts is
𝑈(𝑥𝑎, 𝑥𝑤) = 3𝑥𝑎 + 5𝑥𝑤,
where 𝑥𝑎 is the amount of almonds she consumes (in pounds) and 𝑥𝑤 is the amount of walnuts she
consumes (in pounds). Tasha has $72 to spend on the two types of nuts each month. Currently, the price
of walnuts is $6 per pound at the local grocery store. Tasha optimizes by buying only walnuts. Suppose
a membership-based warehouse club is running a promotion on almonds this month. If a consumer pays
a monthly membership fee, they can purchase almonds at a price of $3 per pound. The warehouse club
does not offer a discount on walnuts. It sells walnuts at the regular price of $6 per pound. What is the
maximum monthly membership fee that Tasha would be willing to pay to purchase almonds from the
club? (Assume that Tasha is not interested in any other products sold by the warehouse club.)

To find the maximum monthly membership fee that Tasha is willing to pay, we need to compare her utility when consuming only walnuts with her utility when she can optimize her consumption by buying both walnuts and almonds at the discounted price. Here's how we can proceed:

### Step 1: Tasha's current utility without the warehouse membership

Tasha currently buys only walnuts, and she has $72 to spend. Since the price of walnuts is $6 per pound, the number of pounds of walnuts she can buy is:

\[
x_w = \frac{72}{6} = 12 \text{ pounds of walnuts}.
\]

Substituting this into her utility function:

\[
U(x_a, x_w) = 3x_a + 5x_w.
\]

Since she buys only walnuts (i.e., $x_a = 0$):

\[
U(0, 12) = 3(0) + 5(12) = 60.
\]

So, her current utility is $60$.

### Step 2: Tasha's utility with the warehouse membership

Now, assume Tasha joins the warehouse club and can buy almonds for $3 per pound while walnuts still cost $6 per pound. Her budget is still $72, and she can allocate her spending between almonds ($x_a$) and walnuts ($x_w$).

The budget constraint is:

\[
3x_a + 6x_w = 72.
\]

To maximize her utility, we substitute her utility function $U(x_a, x_w) = 3x_a + 5x_w$ into the problem and solve.

#### Step 3: Maximization

We maximize the utility subject to the budget constraint by using the method of **marginal utility per dollar**:

- The **marginal utility per dollar** for almonds is:

\[
\frac{MU_a}{P_a} = \frac{3}{3} = 1.
\]

- The **marginal utility per dollar** for walnuts is:

\[
\frac{MU_w}{P_w} = \frac{5}{6}.
\]

Since $ \frac{MU_w}{P_w} < \frac{MU_a}{P_a} $, Tasha gets more utility per dollar by spending on almonds rather than walnuts. Therefore, she should spend all of her money on almonds.

If she spends her entire $72 on almonds, she can buy:

\[
x_a = \frac{72}{3} = 24 \text{ pounds of almonds}.
\]

Substituting this into the utility function:

\[
U(24, 0) = 3(24) + 5(0) = 72.
\]

### Step 4: Maximum membership fee

The maximum monthly membership fee that Tasha would be willing to pay is the difference between her utility when buying only walnuts (utility of 60) and her utility when she optimizes by buying only almonds (utility of 72). The increase in utility is $72 - 60 = 12$.

Thus, the **maximum membership fee** she would be willing to pay is the monetary value equivalent to the increase in utility. Since the price per pound of walnuts is $6, and 1 unit of utility corresponds to spending $1 on nuts, she would be willing to pay **$12** as the membership fee.

### Conclusion:
The maximum monthly membership fee that Tasha would be willing to pay is **$12**.

Would you like further details on this solution, or any specific step explained in more depth?

Here are 5 related questions you might consider:
1. How would the solution change if the price of walnuts increased to $7 per pound?
2. What if Tasha had a different utility function, such as $ U(x_a, x_w) = x_a^2 + 2x_w $?
3. How would Tasha's utility change if there was a limit to how many almonds she could buy at the discounted price?
4. What happens if the membership fee also provided a discount on walnuts?
5. How would the introduction of a third type of nut at a different price affect Tasha's optimal decision?

**Tip**: When comparing consumption choices, always look at the marginal utility per dollar to determine how to allocate your budget optimally.

Tasha consumes two types of nuts: almonds and walnuts. Her utility function from eating nuts is U(xa, xw) = 3xa + 5xw, where xa is the amount of almonds she consumes (in pounds) and xw is the amount of walnuts she consumes (in pounds). Tasha has $72 to spend on the two types of nuts each month. Currently, the price of walnuts is $6 per pound at the local grocery store. Tasha optimizes by buying only walnuts. Suppose a membership-based warehouse club is running a promotion on almonds this month. If a consumer pays a monthly membership fee, they can purchase almonds at a price of $3 per pound. The warehouse club does not offer a discount on walnuts. It sells walnuts at the regular price of $6 per pound. What is the maximum monthly membership fee that Tasha would be willing to pay to purchase almonds from the club? (Assume that Tasha is not interested in any other products sold by the warehouse club.)

Explore how Tasha optimizes her spending on almonds and walnuts to maximize utility, considering a membership fee discount. This problem involves utility maximization, budget constraints, and marginal utility per dollar. Ideal for college economics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation