# Finding the Contract Curve

I was doing a problem with the following data
we have two utility functions which are as follows: $$U_1(x_1,y_1)=\beta \ln(x_1y_1) \;,\; U_2(x_2,y_2)=(\frac{x_2}{y_2})^\alpha$$ along with feasibility contraints $$x_1+x_2=A \; \& \; y_1+y_2=B$$ Now the problem said to find the contract curve and the given answer is:

$$Ay_1+Bx_1-2y_1x_1=0$$

The only way I could come to this answer was by equating the MRS of two individuals and solving using the feasibility constraints i.e.,$$1. \; MRS_1 =MRS_2 \; \implies \frac{y_1}{x_1}=-\frac{y_2}{x_2}\\ 2.\;x_1+x_2=A \\ \;\;\;y_1+y_2=B$$

but my main doubt is why is MRS of individual 2 coming out to be negative?

• Firstly, what is $u_2(0,0)$? Secondly, assuming $\beta > 0$ and $\alpha > 0$, commodity Y is a bad for individual 2, and is good for individual 1. So, equating MRS is a bad choice to find efficient allocations. Therefore, set of all Pareto efficient allocations in this economy is a subset of all feasible allocations that satisfy $y_2=0$ and $y_1=1$.
– Amit
Jan 19, 2023 at 11:07
• It would be better to solve it graphically, but the assumptions regarding the parameters and $$u_2(0,0)$$ are not given in the problem Jan 19, 2023 at 12:47
• As I have said utility function is not well defined. For example: $u_2(0,0) = ?$ . Irrepective of what the parameter values are, equating MRS is incorrect.
– Amit
Jan 19, 2023 at 12:58

MRS $$\left(=\dfrac{\text{MU}_X}{\text{MU}_Y}\right)$$ of individual 2 is coming out to be negative because ICs of individual 2 are upward sloping as one of the two commodities is necessarily a bad while the other is good i.e. $$\text{MU}_X > 0$$ if and only if $$\text{MU}_Y < 0$$.