# Deriving Equation for Engel Curve

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 $$Y=\frac{1}{0.67}B \\ Y=\frac{1}{0.33}Z$$

To derive the Engel curve follow the following steps:

1. Derive the demand functions by maximizing the Utility function subject to the budget constraint.
For this you can use either:

• The Lagrange method like in @Rud-Faden solutions;
• Or set $$MRS = P_b/P_z$$;
• Or even better since we have a Coub Douglas Utility let's use a common knowledge, for any utility function of type $$U = B^{\alpha}Z^{\beta}$$ $${\alpha}$$ and $${\beta}$$ shows the share of income spent on each good. Thus the demand functions will be $$B = \frac{{\alpha}Y}{P_B}$$ $$Z = \frac{{\beta}Y}{P_Z}$$ In your utility function $${\alpha}=0.67;{\beta} = 0.33$$ so $$B = \frac{0.67Y}{P_B}$$ $$Z = \frac{0.33Y}{P_Z}$$
2. On the demand functions derived 1, Treat the price as constant. The demand function itself is the Engel Curve but you need to treat it as a function of income not price because price is constant. $$B(Y), Z(Y)$$.

3. To plot the Engel curve we usually set the income on the $$y$$ axis and the quantity in the $$x$$ axis. Just arrange the demand functions so that you will have Income in the left and quantity of goods in the right. and fix prices to a certain constant.

$$Y(B) = \frac{B*P_B}{0.67}$$ $$Y(Z) = \frac{Z*P_Z}{0.33}$$

The price for both goods were fixed to 1.