I am trying to figure out the following:
I know I have not shown my attempt but trust me I have used everything I can think of to get it right. For example, first expanding the numerator in the fraction on RHS in the top equation. Taking it on the other side and factoring out $$\frac{1}{{c_1}^a*}$$ from both sides. However get stuck from there onwards.
Hint: Clear denominators by multiplying both sides by $c_1^{a*}(1-\theta c_1^{a*})$. Then solve the resulting linear equation for $c_1^{a*}$.