<p>MRS <span class="math-container">$\left(=\dfrac{\text{MU}_X}{\text{MU}_Y}\right)$</span> 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. <span class="math-container">$\text{MU}_X &gt; 0$</span> if and only if <span class="math-container">$\text{MU}_Y &lt; 0$</span>.</p>

CC BY-SA 4.0
2023-01-19T13:06:40.677Z
2023-01-19T13:17:04.103Z
2023-01-19T13:17:04.103Z
Amit
<p>I was doing a problem with the following data<br />
we have two utility functions which are as follows:
<span class="math-container">$$U_1(x_1,y_1)=\beta \ln(x_1y_1) \;,\; U_2(x_2,y_2)=(\frac{x_2}{y_2})^\alpha$$</span>
along with feasibility contraints
<span class="math-container">$$x_1+x_2=A \; \&amp; \; y_1+y_2=B$$</span>
Now the problem said to find the contract curve and the given answer is:</p>
<p><span class="math-container">$$Ay_1+Bx_1-2y_1x_1=0$$</span></p>
<p>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.,<span class="math-container">$$1. \; MRS_1 =MRS_2 \; \implies \frac{y_1}{x_1}=-\frac{y_2}{x_2}\\
2.\;x_1+x_2=A \\ \;\;\;y_1+y_2=B$$</span></p>
<p>but my main doubt is why is MRS of individual 2 coming out to be negative?</p>

CC BY-SA 4.0
2023-01-19T11:07:32.783Z
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
CC BY-SA 4.0
2023-01-19T12:47:37.263Z
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
mynameparv
CC BY-SA 4.0
2023-01-19T12:58:59.623Z
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
<general-equilibrium><pareto-efficiency>
Finding the Contract Curve
mynameparv
# Finding the Contract Curve

(Asked by: username_0 on 2023-01-19)

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?

**Tags:** general-equilibrium, pareto-efficiency
> **Comments:**
> **username_1**: 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 $y_1=1$.  
> **username_0**: 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  
> **username_1**: 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.  

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## 1 answers

### Answer by username_1 on 2023-01-19. Score: 3 | Accepted answer

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 $\text{MU}\_Y < 0$.


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