concave preference

Question

The utility function of a consumer is as follows: U (x; y) = (1 + x) y The prices of goods are P for good X and Q for Y, R the consumer's income

a.Calculate the optimum quantities that maximize consumer satisfaction. Is this a maximum or a minimum, P1 = 20, P2 = 8, R = 4

b. Does this utility function admit a wedge equilibrium?

Answer 1

MUx=y MUy =1+x MRS = Y/(1+X). Note than when X= 0 MRS = Y so that all of the indifference curve will intersect the Y For interior solution Y/(1+X) = P/Q or P(1+x) =Qy. for an interior solution we need this tangency condition along with R=pX+Qy or R = PX +P(1+X) or R=2Px +P or X= (R-P)/2P = X = 0 when R=P

So we write X=(R-P)/2P for P<=R and 0 elselwhere. and Y = (P/Q)(X) when X>0 an Y= R/Q elsewhere.

When R=4 ,P=20 and Q=8 the slope of the budget line is -20/8. sinc P>R we have a corner solution (X=0)

Hence, Y = R/Q = 4/8 = 1/2. When Y= 1/12 and X=0 MRS = 1/2 <P/Q.. The indifference passing through (0,1/2) is give by U*=(1+0)*(1/2) =1/2=(1+x)Y or y=1/(2*(1+X)).. You can verify that these are indeed convex preferences