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I was asked the following problem : for an individual, the ratio between the marginal utility of orange juice and marginal utility of apple juice is constant and equal to $0.5$. The two goods cost 3\$ per liter. Show that this individual consumes only apple juice.

I find it very weird since at the optimum we should have that the ratio of marginal utilities must be equal to the ratio of the prices of the goods but is $0.5\neq\frac 33=1$. I am also trying to find a way to link the data I have but I cannot see how to obtain such a conclusion since there are infinitely many possibilities for the ratio of marginal utilities to be $0.5$.

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MU of orange juice is always half that of apple juice, so the consumer always prefers additional apple juice over additional orange juice. Therefore, they will never increase the consumption of orange juice when they can increase the consumption of apple juice by an equal amount using the same resources. But that is precisely the case when the prices are equal. Thus, they end up consuming only apple juice until all their income is exhausted.

The ratio of MUs being equal to the price ratio at optimum is there when the preferences are strictly convex and we have an interior solution. But that is not always the case.

The constant ratio of MUs, being equal to 0.5, in your example, can be satisfied with the following utility function: $u(x, y) = 0.5x + y$, where $x$ and $y$ are quantities of orange juice and apple juice respectively. The utility function is linear and thus, not strictly convex.

If the income is $\\\$M$ and the prices are $\\\$3$ and $\\\$3$ respectively, then the budget constraint is $3x + 3y \leq M$. The other constraints are of course $x \geq 0$ and $y \geq 0$.

It is clear that we have a linear programming problem, as the utility function and the constraints are all linear. Therefore, we need to only check the corner points for optimality (if more than one corner yields the same result, then ofcourse any linear combination of the two points does as well).

$u(0, 0) = 0$, $u(0, M/3) = M/3$, $u(M/3, 0) = M/6$, so it is easy to see that the optimum is at $(0, M/3)$, where the consumer spends all the money on apple juice only.

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    $\begingroup$ Did you swap them ? Because we were supposed to show that he or she consumes only apple juice. Otherwise, great answer $\endgroup$
    – Kilkik
    Jan 15, 2023 at 13:14
  • $\begingroup$ Oh yes, I got confused, I'll chnge it $\endgroup$ Jan 15, 2023 at 13:19
  • $\begingroup$ I've made the chnges. If you find the answer satisfactory, you may consider marking it accepted (the tick mark). $\endgroup$ Jan 15, 2023 at 22:09

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