A small economy has only two consumers, Boris and Vanessa. Boris's utility function is U(x, y) . Vanessa's utility function is U(x, y) = x + 2y. Boris is endowed with 160 units of x and 60 units of y. They make trades to reach a Pareto optimal allocation of resources in which both persons consume positive amounts. How much y does Boris consume?

Question

A small economy has only two consumers, Boris and Vanessa. Boris's utility function is U(x, y)  . Vanessa's utility function is U(x, y) = x + 2y. Boris is endowed with 160 units of x and 60 units of y. They make trades to reach a Pareto optimal allocation of resources in which both persons consume positive amounts. How much y does Boris consume?

Quizplus · Accepted Answer

To find the Pareto optimal allocation, we need to find the point where the indifference curves of Boris and Vanessa are tangent to each other, subject to Boris's endowment constraint.

Boris's utility function is quasi-concave, and so we know that his indifference curves are downward sloping and never intersect. Vanessa's utility function is linear, so her indifference curves are straight lines with a slope of -0.5.

We can start by drawing a diagram with Boris's endowment:

```
     |
60   |        A
     |
     |
     |
     |
     |
     |
     |
     |
     |
     |_____________________
            160
```

Point A represents Boris's endowment of (160,60). We know that Boris's optimal consumption bundle must lie on the highest possible indifference curve that is tangent to one of Vanessa's indifference curves. Let's call this indifference curve I1. We can draw indifference curve I1 as a downward sloping curve passing through point A.

```
     |
60   |        A   I1
     |
     |
     |\            .
     | \          .
     |  \        .
     |   \      .
     |    \    .
     |     \  .
     |______\._____________
            160
```

The point of tangency, B, is where the slope of indifference curve I1 equals the slope of Vanessa's indifference curve. We know the slope of Vanessa's indifference curve is -0.5. To find the slope of indifference curve I1, we need to use the marginal rate of substitution (MRS):

MRS = -dU/dy / dU/dx

For Boris, we have MRS = -11x/y. Setting this equal to -0.5, we get:

-11x/y = -0.5

x/y = 0.0455

We also know that Boris must consume all of his endowment, so:

x + y = 220

Substituting x/y = 0.0455, we can solve for x and y:

x = 160y/(1 + 0.0455)
y = 220/(1 + 0.0455)

x = 154.48
y = 16.52

So Boris consumes 16 units of y. The answer is D.

A Small Economy Has Only Two Consumers, Boris and Vanessa