Tova’s utility function is u(c1, c2) = min{c1, c2}, where c1 is his consumption in period 1 and c2 is his consumption in period 2. He earns $200 in period 1 and $220 in period 2. Tova can borrow and lend at an interest rate of 10 percent, and there is no inflation. The number of dollars that Tova spends on consumption in the first period must be

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The problem uses Tova’s utility function u(c1, c2) = min{c1, c2}, which means his happiness is determined by the smaller of his two period consumptions. Therefore, to maximize utility, we want c1 and c2 to be as equal as possible subject to the intertemporal budget constraint.

First, compute the intertemporal budget constraint with the given interest rate i = 0.10. The present value of consumption across periods must equal the present value of income: c1 + c2/(1.1) = 200 + 220/(1.1). Since 220/(1.1) = 200, the right-hand side equals 400. Thus the constraint is c1 + c2/1.1 = 400.

To maximize min{c1, c2}, set c1 = c2 = c. Substituting into the constraint gives c + c/1.1 = 400. This is c * (1 + 1/1.1) = c * (1 + 0.90909) ≈ c * 1.90909 = 400, so c ≈ 400 / 1.90909 ≈ 209.5.

Therefore, the allocation that maximizes Tova’s utility has c1 ≈ 209.5 and c2 ≈ 209.5. This value is greater than 200 but less than 220, placing c1 in the interval (200, 220).

Considering the answer choices:
- 'more than 200 but less than 220.' aligns with c1 being around 209.5, so the candidate is plausible. 
- 'exactly 200.' would imply c2 is also 200, but that would yield total resources that do not exhaust the budget under the equality condition and would reduce min{c1, c2} below the maximum possible; hence not optimal.
- 'more than 220.' would violate the budget constraint, since you cannot have c1 exceed your intertemporal resources after properly accounting for the interest rate.
- 'exactly 180.' or 'more than 180 but less than 200.' would keep c1 below 200, which reduces the minimum further and cannot be the utility-maximizing point given the ability to borrow/lend to balance the two consumptions.

In summary, the optimal intertemporal bundle equalizes the two consumptions at about 209.5 each, which lies between 200 and 220. Thus the statement that the first-period consumption must be more than 200 but less than 220 correctly describes the required range.

BU.232.701.W1.FA24 Quiz 1 Practice

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