Q: u(x,y)=min(2x,3y), I=80 and Py=6. Find the equation of the demand for good x.
A: Given the utility function the optimal bundle is where 2x=3y or y=2x/3. Plugging this into the BC which is 80=PxX+6Y gives 80=PxX+4X. Solving for x we find that demand is X=80/(4+Px).
Q: If Py falls to Py=3, what will happen to demand for good x? What is the sign of the cross price elasticity of demand?
A: Again, we have y=2x/3. The BC is now 80=PxX+3Y. Through substitution and simplification we get X=80/(2+Px). Comparing this to the demand curve we found above, we see that the dominator is smaller and thus X is larger for any Px. So demand has shifted out as Py fell and these two goods are complements. Hence the cross price elasticity is negative.