Question

1. Clara's utility function is U(X,Y)= (x + 2)(Y +1). a) Write an equation for Clara's indifference curve that goes through the point(X,Y)-(2,8). b) Suppose that the price of each good is one and that Clara has an income of 11. Write an equation that describes her budget constraint. c) Find an equation the describes Clara's MRS for any given commodity bundle (X,Y). d) Use the equations in parts b) and e) to solve for Clara's optimal bundle Hint use the fact that at Clara's optimal choice = MRS) e) Find Clara's demand function for good X (i.e. find X(Px, Py, M). 2. Consider Bridget, who has preferences that can be described by the function U(X,Y) -5ln(x) + y. Find Bridget's demand functions for x and y.

Answer 1

a)

U(X,Y) = (X + 2)(Y +1)

U(2,8) = (2 + 2)(8 + 1)

= 36

Thus, 36 = (X + 2)(Y +1)

36/(X+ 2) = Y + 1

Y = 36/(X + 2) - 1

Hence equation of Indifference curve Y =   36/(X + 2) - 1

b) Px = 1 Py = 1 M = 11

Budget constraint  

XPx + YPy = M

X(1) + Y(1) = 11

X + Y = 11

c)   $\partial$U/$\partial$X = MUx = (Y + 1)

  $\partial$U/$\partial$Y = MUy = (X + 2)  

MRS = MUx/MUy

= (Y + 1)/(X + 2)

d) At optimal choice MRS = Px/Py  

(Y + 1)/(X + 2) = 1/1

   (Y + 1)/(X + 2) = 1  

Y + 1 = X + 2  

Y = X + 2 - 1  

Y = X + 1

Put    Y = X + 1 in budget constraint  

X + Y = 11  

X + X + 1 = 11  

2X = 11 - 1  

2X = 10   

X = 5  

Y = 5 + 1

= 6  

Thus optimal bundle is (5,6)

e) Again using   optimal choice MRS = Px/Py  

   (Y + 1)/(X + 2) = Px/Py  

Y + 1 =   (X + 2)(Px/Py)   

Y =   (X + 2)(Px/Py) - 1

Put     Y =   (X + 2)(Px/Py) - 1 in budget constraint

  XPx + YPy = M

XPx + Py[  (X + 2)(Px/Py) - 1] = M

XPx + (X+ 2)Px - Py = M

XPx + XPx + 2Px - Py = M  

2XPx = M - 2Px + Py

X = M/2Px - 1 + Py/2Px

Thus, demand function of x

X(Px, Py,M) =    M/2Px - 1 + Py/2Px