I have a utility function of $U = B^{.67}Z^{.33}$ with Income $Y$, Price of Good B is $P_b$ and price of Z is $P_z$ I now need to derive an engel curve for this. I have no idea where to start.


It is quite simple. The engels curve is the change in demand for a good as a function of income, keeping prices fixed

Solve the constrained maximisation problem $$ \max B^{0.67}Z^{0.33}-\lambda(P_bB+P_zZ-Y) \Leftrightarrow \\ B=\frac{0.67Y}{P_b} \\ Z=\frac{0.33Y}{P_z} \\ \lambda = \frac{0.5303709372}{P_z^{33/100}P_b^{67/100}} $$

Fix prices at some level. I choose 1 for simplicity. Then the engels is simply $$ Y=\frac{1}{0.67}B \\ Y=\frac{1}{0.33}Z $$ enter image description here


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