How do prices and income jointly determine an interior optimum?
At an interior choice, willingness to trade equals the market trade-off.
Follow the cause and effect
Write pxx+pyy=m and find both intercepts.
Set MRS=px/py for an interior solution.
Solve with the budget and verify x,y are positive.
The model you are using
KEY MODEL
px x + py y = m
x* = αm/px
y* = (1−α)m/py
Conditions for this model
0 < α < 1, positive prices, divisible goods and no saving in this problem.
The exam trap
A tangency outside the budget set or with a negative quantity is infeasible.
Connect the reasoning to the graph
What changes
- A change in the price of x pivots the budget around the y-intercept while preferences stay fixed.
What stays fixed
0 < α < 1, positive prices, divisible goods and no saving in this problem.
Keep these conditions throughout the comparison; change only what the case above specifies.
What to inspect
Solve with the budget and verify x,y are positive.
Compare before and after, and locate the conclusion on the graph.
5 MINUTES · TRANSFER THE IDEA
Can you explain it without the lesson?
For U=x^0.4y^0.6, m=200, px=10, py=5, find and verify the optimum.