Question

a) Alberto has utility function U (x,y)= 2*sqrt x +y. His income is $400, unit price of good y is Py= $20 and original unit price of good x is Px=$10. Compute X and Y, Alberto's uncompensated demands for good x and y. Compute level of utility that Alberto reaches under these prices.
b) assume unit price of good x becomes Px= $20. What's cheapest bundle (Xc, Yc) that allows Alberto to reach same level of utility found in a) under these new prices. How much does the bundle cost?
c) Compute compensating variation of that price change.

Answer 1

a) MRS(X,Y)=Px/Py, 2*1/2*x^(-1/2)/1=10/20, 1/sqrt(x)=1/2. So, X=4.
Y=(m-Px*X)/Py=(400-10*4)/20=360/20=18. So, Y=18.
The level of utility that Alberto reaches under these prices: U (4,18)= 2*sqrt(4) +18=22.
b) MRS(Xc,Yc)=20/20, 1/sqrt(Xc)=1, Xc=1.
Yc=(400+CV-20*1)/20=(380+CV)/20=19+0.05CV.
After price change the level of utility must be the same as before: U(Xc,Yc)=22; 2*sqrt(1)+(19+0.05CV)=22,
2+19+0.05CV=22, 0.05 CV=1, CV=20.
Yc=19+0.05*20=20. So, the cheapest bundle (Xc, Yc)=(1, 20).
This bundle costs 1*20+20*20=420.
c) The compensating variation of that price change CV=20 (was computed before at the paragraph b)).