Here is the question:
Consider a car-owning consumer with utility function $$u (x) = x_1x_2 + x_3 (x_4)^2 ,$$
where $x_1$ denotes food consumed, $x_2$ denotes alcohol consumed, $x_3$ denotes kms of city driving, and $x_4$ denotes kms of open road driving. Total fuel consumption (in litres) is given by $$h (x) = 2x_3 + x_4.$$
Food costs \$1 per unit, alcohol costs \$1 per unit and the consumer has income of \$10.
(a) Suppose fuel is provided free by the Government, but is rationed. This consumer is allowed to use c litres of fuel. Find her optimal choice of $x_1$, $x_2$, $x_3$ and $x_4$.
(b) Now suppose the consumer receives her quota of fuel ($c$ litres), but can buy or sell fuel on a fuel market for $p$ per litre. Find the value of $p$ at which this consumer will choose not to trade on the fuel market.
I tried writting the Lagrangian $L(x_1,x_2,x_3,x_4,\lambda,\mu) = x_1x_2 + x_3(x_4)^2 - \lambda(2x_3+x_4-c) - \mu(x_1+x_2-10)$. Is this correct? If so, how do I solve this equation? When I set the derivatives to $0$, I get $x_1=x_2=\mu=5$ but I can't solve for $x_3$, $x_4$ or $\lambda$.(resolved)
Now I have solved part (a), but how to solve part (b)? Is the price just equal to λ from part (a) or do I have to write a new Lagrangian?