I need some help in solving this question:
Consider the following two-person, two-good economy. Persons $A$ and $B$ each consume two goods xylophones $(x)$ and yams $(y)$. Person $A$ enters the world with $14$ units of xylophones and $1$ unit of yams, person $B$ with $0$ units of xylophones and $4$ units of yams. Their utility functions are given by:
$U^A(x^A,y^A) = \frac{x^Ay^A}{x^A + y^A}$ and $U^B(x^B,y^B)=ln(x^B)+ln(y^B)$.
$(a)$ Construct an Edgeworth box diagram to illustrate the feasible consumption possibilites. Label the initial endowment point.
My solution: 
$(b)$ Find the equation of the contract curve, expressed in terms of goods consumed by person A. There is no need to solve for $y^A$ explicitly in terms of $x^A$.
My attempt: I have to solve the following equations:
- $MRS_{xy}^A = MRS_{xy}^B$
- $x^A + x^B =14$
- $y^A + y^B=5$
After some computations, I have found that $MRS_{xy}^A = \frac{y^{A^{2}}}{x^{A^{2}}}$ and $ MRS_{xy}^B= \frac{y^{B}}{x^{B}}$. So if $MRS_{xy}^A = MRS_{xy}^B$ we can manipulate equation 1. to write equation 4.: $y^{B}=\frac{y^{A^{2}}x^B}{x^{A^{2}}}$.
In equation 2.: $x^A + x^B =14\Rightarrow x^B=14 - x^A$.
In equation 3. $y^A + y^B=5\Rightarrow y^B=5 - y^A$.
Substitute in $x^B=14 - x^A$ and $y^B=5 - y^A$ in equation 4. to get: $5-y^A=\frac{y^{A^{2}}(14-x^{A})}{x^{A^{2}}} \Rightarrow y^{A(A-1)} = 5 - \frac{\left ( 14-x^A \right )}{x^{A^{2}}}$.
Is this the equation of the contract curve?
$(c)$ Describe the core of this economy.
$(d)$ Find the competitive equilibrium outcome, given the initial endowments. Is it on the contract curve? Explain.
The contract curve is where the MRS of both agents coincide. So compute FOC's for two agents. Set them equal.
The core are allocations which, assuming the given endowment, are both individual rational and coalition-rational. Individual rationality can be deduced from the respective indifference curves. You will see an olive-shaped region. Coalition-rationality in this case just means the contract curve.