I have $U(x,y)=xy$, $p_1=4$ and $p_2=1$. Income is unknown. Where do I start?
closed as off-topic by Giskard, Kitsune Cavalry♦ Nov 9 '18 at 1:06
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Step 1, as it were, would be to write down the expenditure function (4x+y). Next, think about what happens if income is $0; the optimal choices of x and y are easy, as only (0,0) can be consumed. Then ask "what happens as income increases?" and you're off to the races!
The way I would attack this is to solve the utility function $u(x,y)=xy$ subject to the budget constraint: $4x+y=m$. This is using Lagrange method It turns out that $x=m/8$ and $y=m/2$.
The Engel curve show the relationship between income and the quantity of the good demanded, so substitute values for $m$ and find the corresponding $x$ or $y$ demanded and plot the Engel curve for good x or good y.