A consumer has an indirect utility function given by
$$v(p_1, p_2, R)=\frac{R}{p_{1}+p_2}$$
Where $p_1$ and $p_2$ denote the prices of the two goods consumed by the individual and $R$ the income.
How do I calculate this consumer's Hicksian demand function for goods 1 and 2?
To get the Hicksian demand function $h$, you can use Shephard's lemma, which says that
$$h_i(p_1,p_2,u)=\frac{\partial e(p_1,p_2,u)}{\partial p_i}$$
where $e$ is the expenditure function.
To get the expenditure function, use the equation:
$$v\Big(p_1,p_2,e(p_1,p_2,u)\Big)=u$$
i.e. solve
$$v(p_1,p_2,R)=\frac{R}{p_1+p_2}=u$$ for $R$ and call the resulting expression $e(p,u)$.
