Economics question pt. 2! HELP! (31.3)?

Question

I'm having problems with this one and was wondering if someone could walk me through it:

John and Mike exchange platitude and bromides. When John consumes Ti platitudes and Bi bromides, his utility is given by:

Ui(Bi,Ti)= Bi + 2 *square root symbol* Ti

When Mike consumes Tn platitudes and Bn bromide, his utility is given by:

Un(Bn,Tn)=Bn + 4 *square root* Tn

A.) If John consumes Ti platitudes and Bi bromides, his marginal rate of substitution will be _____

If Mike consumes Tn Platitudes and Bn bromides, his marginal rate of substitution will be _____

B.) On the contract curve, John's MRS equals Mike's. Write an equation that states this condition.

Answer 1

A) to calculate MRS, find U1 and U2. U1 is the derivative in terms of the first variable, in this case bi.

d(Ui)/d(bi)=1

U2 is the derivative in terms of the second variable. (Rt=square root symbol)

d(Ui)/d(Ti)=2/(2rt(ti))=1/rt(ti)

MRS=U1/U2=1/rt(ti)=rt(ti)

To find Mike's MRS, follow the same procedure

U1=d(Un)/d(bn)=1

U2=d(Un)/d(tn)=4/2tr(tn)=2/rt(tn)

MRS=1/2/rt(tn)=rt(tn)/2

B) Set their MRS equal.  rt(ti)=rt(tn)/2

If you want to get it so that you can draw it in the Edgeworth box, you need a total endowment number, though.

Does that help?