Suppose that Sam allocates his income between milk and cereal. Milk costs $2.50/gallon and cereal costs $5.00/box. Sam has $50/week to spend on these two goods. The table shows Sam's preference for consumption bundles as well as how Sam's marginal utility ($MU$) for milk and cereal, respectively, varies as consumption varies.
Given the information provided here, how should Sam allocate his income between milk and cereal?
As this is his optimal consumption bundle, Sam should purchase
[ ] gallons of milk
| Milk (gallons) | $MU_{milk}$ | Cereal (boxes) | $MU_{cereal}$ |
|---|---|---|---|
| 0 | --- | 0 | --- |
| 2 | 16 | 1 | 135 |
| 4 | 15 | 2 | 120 |
| 6 | 13 | 3 | 105 |
| 8 | 9 | 4 | 90 |
| 10 | 8 | 5 | 75 |
| 12 | 7 | 6 | 60 |
| 14 | 6 | 7 | 45 |
| 16 | 5 | 8 | 30 |
| 18 | 4 | 9 | 12 |
| 20 | 3 | 10 | 8 |
Answer (2)
Calculate the marginal utility per dollar for milk and cereal.
Find the consumption bundle where the marginal utility per dollar is approximately equal for both goods.
Check if the consumption bundle is within Sam's budget.
Determine the optimal quantities of milk and cereal: 4 gallons of milk.
Explanation
Problem Analysis Sam needs to allocate his income of $50 between milk and cereal to maximize his utility. Milk costs $2.50 per gallon, and cereal costs $5.00 per box. The table provides the marginal utilities for different consumption levels of both goods. The goal is to find the consumption bundle where the marginal utility per dollar spent on each good is approximately equal, while staying within the budget.
Calculating Marginal Utility per Dollar First, we calculate the marginal utility per dollar for both milk and cereal. This is done by dividing the marginal utility of each good by its price.
For Milk:
Price = $2.50/gallon
For Cereal:
Price = $5.00/box
Calculating MU per Dollar for Different Levels Now, let's examine the provided table and calculate the marginal utility per dollar for a few consumption levels:
Milk (gallons)
M U mi l k
M U mi l k / P r i ce
Cereal (boxes)
M U cere a l
M U cere a l / P r i ce
2
16
16/2.50 = 6.4
1
135
135/5.00 = 27
4
15
15/2.50 = 6.0
2
120
120/5.00 = 24
6
13
13/2.50 = 5.2
3
105
105/5.00 = 21
8
9
9/2.50 = 3.6
4
90
90/5.00 = 18
10
8
8/2.50 = 3.2
5
75
75/5.00 = 15
12
7
7/2.50 = 2.8
6
60
60/5.00 = 12
14
6
6/2.50 = 2.4
7
45
45/5.00 = 9
16
5
5/2.50 = 2.0
8
30
30/5.00 = 6
18
4
4/2.50 = 1.6
9
12
12/5.00 = 2.4
20
3
3/2.50 = 1.2
10
8
8/5.00 = 1.6
Finding the Optimal Bundle We are looking for a combination where M U mi l k / P mi l k ≈ M U cere a l / P cere a l and the total cost is within Sam's budget of $50.
Let's consider the bundle of 4 gallons of milk and 8 boxes of cereal:
Cost of milk: $4 \times 2.50 = $10.00
Cost of cereal: $8 \times 5.00 = $40.00
Total cost: $10.00 + $40.00 = $50.00
At this bundle:
M U mi l k / P mi l k = 15/2.50 = 6.0
M U cere a l / P cere a l = 30/5.00 = 6.0
Since the marginal utility per dollar is equal for both goods and the total cost is exactly $50, this is the optimal consumption bundle.
Final Answer Therefore, Sam should purchase 4 gallons of milk and 8 boxes of cereal to maximize his utility given his budget constraint.
Examples
Imagine you're planning a balanced diet with a fixed budget. You need to decide how much of each food item to buy to get the most nutritional value for your money. This problem is similar to optimizing your spending on different food items to maximize your overall health benefits within your budgetary constraints. By understanding marginal utility and budget constraints, you can make informed decisions that lead to the best possible outcome, whether it's maximizing satisfaction or nutritional value.
Sam should purchase 4 gallons of milk and maximize his cereal intake under his budget, spending approximately equally on both goods to maximize utility. In particular, a good consumption bundle is 4 gallons of milk and 8 boxes of cereal. This allocation ensures that the marginal utility per dollar spent is equal and stays within his budget of $50.
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