Question

David N. gets \(\$ 3\) per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter (at \(\$ .05\) per ounce and jelly (at \(\$ .10\) per ounce). Bread is provided free of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions. a How much peanut butter and jelly will David buy with his \(\$ 3\) allowance in a week? b. Suppose the price of jelly were to rise to \(\$ .15\) an ounce. How much of each commodity would be bought? c. By how much should David's allowance be increased to compensate for the rise in the price of jelly in part (b)? d Graph your results in parts (a) to (c). e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity. f. Discuss the results of this problem in terms of the income and substitution effects in volved in the demand for jelly.

Short answer

Short Answer: With his $3 allowance, David can buy enough jelly and peanut butter to make 15 sandwiches. When the price of jelly increases to $0.15, David can only make 12 sandwiches. His allowance should be increased by $0.60 to compensate for the rise in the price of jelly. In this problem, only a single commodity, peanut butter and jelly sandwiches, is involved because David is only interested in maximizing the number of sandwiches he can make with his allowance. The income effect is demonstrated by the reduced number of sandwiches when the price of jelly increases, but the substitution effect is not present as David's preferences do not allow him to substitute jelly for peanut butter or vice versa.

Step-by-step solution

Determine the Constraint and Proportions

To find out how much peanut butter and jelly David will buy, we need to keep in mind that he has $3 to spend and his preferences for sandwich proportions: 1 ounce of jelly and 2 ounces of peanut butter.

Calculate How Much of Each Commodity David Can Buy

Given the prices of peanut butter and jelly, we can determine how much he can buy of each. We have to maximize the number of sandwiches David can make while keeping the proportions. Let x be the number of sandwiches. Then, the cost will be: \(3 = (JellyCost \times x) + (PeanutButterCost \times 2x)\) \(3 = (0.10x) + (0.05 \times 2x)\) \(3 = 0.10x + 0.10x\) \(x = \frac{3}{0.20} = 15\) With his $3 allowance, David can buy enough jelly and peanut butter to make 15 sandwiches. #b. Suppose the price of jelly were to rise to $0.15 an ounce. How much of each commodity would be bought?#

Update the Equation with the New Jelly Price

We need to update the equation with the new price of jelly and then solve for x. \(3 = (0.15x) + (0.05 \times 2x)\) \(3 = 0.15x + 0.10x\)

Calculate the New Number of Sandwiches

Now we solve for x: \(x = \frac{3}{0.25} = 12\) With the new price of jelly, David can buy enough jelly and peanut butter to make 12 sandwiches. #c. By how much should David's allowance be increased to compensate for the rise in the price of jelly in part (b)?#

Find the Initial Cost of 12 Sandwiches

First, we need to calculate how much it would have cost David to make 12 sandwiches before the price increase of jelly. InitialCostOf12Sandwiches = (JellyCost \times 12) + (PeanutButterCost \times 2 \times 12) InitialCostOf12Sandwiches = (0.10 \times 12) + (0.05 \times 2 \times 12) = $3.60

Find the New Cost of 12 Sandwiches

Now, we calculate the cost of 12 sandwiches with the increased price of jelly. NewCostOf12Sandwiches = (0.15 \times 12) + (0.05 \times 2 \times 12) = $4.20

Calculate the Necessary Allowance Increase

Finally, we find the difference between the new and initial costs to determine the necessary allowance increase. AllowanceIncrease = NewCostOf12Sandwiches - InitialCostOf12Sandwiches AllowanceIncrease = \(4.20 - \)3.60 = $0.60 Therefore, David's allowance should be increased by $0.60 to compensate for the rise in the price of jelly. #d Graph your results in parts (a) to (c).# Unfortunately, as an AI language model, I cannot create graphical representations. However, I encourage you to graph the results on your own, keeping in mind the differences in limits and costs. #e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity.# This problem involves only a single commodity because David is only interested in peanut butter and jelly sandwiches. All of his allowance goes towards making these sandwiches, and he is only concerned with maximizing the number of sandwiches given his allowance. The demand curve for this single commodity will be a downward-sloping straight line, indicating the relationship between the price of sandwiches and the number of sandwiches David buys. #f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.# When the price of jelly increased, the number of sandwiches David could make decreased due to the income effect, as his real purchasing power was reduced. However, his preference for sandwich proportions meant that he continued to consume the same ratio of peanut butter and jelly, even at a higher price. This showcases a lack of substitution effect, as David's preferences do not allow him to substitute jelly for peanut butter or vice versa.