Sam likes Pepsi (P) twice as much as he likes Coke (C). Assume the price of Pepsi is $3 and the price of Coke is $1 and that Sam’s income is $150. What is Sam’s optimal consumption bundle?

Cans of Pepsi: 0

Cans of Coke: 150

Question

Sam likes Pepsi (P) twice as much as he likes Coke (C). Assume the price of Pepsi is $3 and the price of Coke is $1 and that Sam’s income is $150. What is Sam’s optimal consumption bundle?

Cans of Pepsi: 0

Cans of Coke: 150

BlackTom AI · Accepted Answer

We restate the scenario to ensure clarity: Sam’s utility from consumption is determined by his liking for Pepsi (P) relative to Coke (C), with Pepsi valued twice as much as Coke. The prices are Pepsi $3 and Coke $1, and his income is $150. The provided answer pair is Pepsi: 0, Coke: 150.

First, translate the budget constraint: 3P + 1C = 150. Since cans are discrete, we’ll treat P and C as nonnegative integers.

Next, express Sam’s utility given the stated preference: if Sam likes Pepsi twice as much as Coke, a simple additive utility model would be U = 2P + 1C. This captures the idea that each Pepsi can contributes twice as much to utility as each Coke can.

Now, substitute the budget constraint into the utility function to see how utility changes with P. From 3P + C = 150, we can write C = 150 − 3P. Then U(P) = 2P + (150 − 3P) = 150 − P.

Observe how U(P) changes with P: U(P) = 150 − P decreases as P increases. Therefore, to maximize utility, we should choose the smallest feasible P, which is P = 0.

With P = 0, the budget constraint gives C = 150. This yields U = 150 − 0 = 150, which is the maximum achievable utility under the given linear model and prices.

If you tried any P > 0, say P = 1, then C = 147 and U = 2(1) + 147 = 149, which is lower than 150. Larger P would further decrease utility because each additional Pepsi reduces Coke by 3 units but only adds 2 units of Pepsi utility, resulting in a net drop of 1 utility per extra P.

Thus, the optimal bundle occurs at P = 0 and C = 150. Any deviation from this allocation (positive P) would yield strictly lower utility given the budget and the stated valuation.

ECON3200001.1251 Quiz 3: Utility Functions

Sam likes Pepsi (P) twice as much as he likes Coke (C). Assume the price of Pepsi is $3 and the price of Coke is $1 and that Sam’s income is $150. What is Sam’s optimal consumption bundle? Cans of Pepsi: 0 Cans of Coke: 150

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