In an isolated mountain village, the harvest this year is 6,000 bushels of grain and the harvest next year will be 900 bushels. The villagers all have utility functions U(c₁, c₂) = c₁c₂, where c₁ is consumption this year and c₂ is consumption next year. Rats eat 40% of any grain that is stored for a year. How much grain could the villagers consume next year if they consume 1,000 bushels of grain this year? A) 5,850 bushels B) 3,000 bushels C) 3,900 bushels D) 6,900 bushels E) 1,000 bushels

Question

Quizplus · Accepted Answer

If the villagers consume 1,000 bushels of grain this year, then they will have (1-0.4) * 1,000 = 600 bushels of grain next year. Using the utility function, we want to maximize the consumption next year, subject to the constraint that the total grain consumed over the two years cannot exceed the total harvest of 6,900 bushels. Let x be the amount of grain consumed next year. Then, the problem can be formulated as:

maximize U(1000, x)
subject to 1000 + x - 0.4 * (1000 + x) <= 6,900

Simplifying the constraint, we get:

x <= 3,900

Therefore, the villagers can consume at most 3,900 bushels of grain next year, and this is the amount that maximizes their utility.

In an Isolated Mountain Village, the Harvest This Year Is