Endowment: Pareto efficient?

Question

How do I know if this endowment is pareto efficient?

Here is the problem:

Suppose there is an island with only two people on it, Pamela and Quentin. They have endowment of only 2 goods: bananas and fish. Assume bananas=good1, fish=good2. The endowment vectors for Pamela and Quentin are:

(Wp1, Wp2) = (10,5)

(Wq1, Wq2) = (5,20)

The utility fncs are Cobb-Douglas ones:

Up(Xp1, Xp2)= Xp1Xp2

Uq(Xq1, Xq2) = Xq1Xq2

so is this endowment (the vectors given which will coincide in one point)... Pareto efficient? why or why not?

i think i will have to draw the contract curve, but i don't understand how to do it.

please help??

Answer 1

Sorry I've never done this, so this may not help... the points on the contract curve are where both people's utility functions are tangential. In other words, when the gradient of the utility functions are equal. If you differentiate both utility functions, and set them equal, you should be able to come up with an expression in terms in good 1 and good 2, and then plot it. Alternatively, when you have this, just stick in the values from the vectors and they should fit in.

Nick

Answer 2

If the endowment is pareto efficient, the marginal rates of substitution will be equal.

dUp/dXp1=Xp2

dUp/dXp2=Xp1

MRS(p)=Xp2/Xp1

dUq/dXq1=Xq2

dUq/sXq2=Xq1

MRS(q)=Xp2/Xp1

Now set the MRS equal...

Xp2/Xp1=Xq2/Xq1

5/10=/=20/5

The endowment is not pareto efficient.

To find the contract curve, you have to find all the points where the MRS are equal

Xp2/Xp1=Xq2/Xq1

To draw the contract curve, you need to get it in terms of two variables, so we're going to go with Xp2 and Xp1

Xq2=the endowment minus whatever Pamela has

Xq2=25-Xp2

Xq1=endowment minus what Pamela has

Xq1=15-Xp1

so, substitute those in

Xp2/Xp1=(25-Xp2)/(15-Xp1)

cross multiply...

Xp2(15-Xp1)=Xp1(25-Xp2)

15Xp2-Xp1Xp2=25Xp1-Xp1Xp2

15Xp2=25Xp1

Xp2=25Xp1/15

Now you can draw the contract curve