Question

One consumer is choosing to maximize U(X,Y)=XY under a budget constraint of PxX+PyY=M.

(Px,Py,M)=(4,2,24).

(1) What does Px/Py=2. mean in this case?

(2) Draw a budget line.

(3) Draw an indiscriminate curve that conforms to a given utility function.

(4) Find out the optimal consumption (X*,Y*).

(5) Calculate the income elasticity of demand for X goods.

Answer 1

Utility function;

U(X,Y)=XY

Price of good X, P_X = 4
Price of good Y, P_Y = 2

Income, M = 24

The budget constraint ;

P_X X + P_Y Y = M
4X + 2Y = 24
2X + Y = 12

1) The price ratio;

P_X / P_Y = 4/2
P_X / P_Y = 2

The price ratio represents the slope of the budget line. It is the oppurtunity cost which consumer pays to gain some untis of one good. The slope represents the amount of good 2, i.e Y given up to have one more unit of good 1, i.e good X.

2) The budget line represents the bundles of goods X and Y which consumer can purchase given the prices and within his/her income.

2X + Y = 12

3) Indiscriminate curve or Indifference curve represents the various combinations of two goods which provide same level of satisfaction to the consumer.

Given the utility function;

U(X,Y)=XY

Let U = 10

10 = XY
X = 10/Y

X	10	5	2
Y	1	2	5

4) The optimal consumption level is achieved when slope of the budget line is equal to the slope of the indifference curve which means when indifference curve is tangent to the budget line.

Slope of indifference curve = MRS
Slope of the budget line = P_X / P_Y

MRS = P_X / P_Y

Marginal rate of substitution;

MRS = MUx/MUy
= d/dX (XY) / d/dY (XY)
MRS = Y / X

At optimal level,

MRS = P_X / P_Y
Y/X = 4/2
Y/X = 2
Y = 2X

Putting this in budget constraint;

2X + Y = 12
2X + 2X = 12
4X = 12
X* = 3

Y* = 6

5) Income elasticity for good X is ratio of percentage change in quantity demanded and the percentage change in income.

E = dX/dM * M/X

The demand function of good x will be;

Y = 2X

Putting in;

P_X X + P_Y Y = M
P_X X + P_Y 2X = M
X (P_X + 2P_Y) = M
X = M / (P_X + 2P_Y)

dX/dM = d/dM [M / (P_X + 2P_Y)]
dX/dM = 1 / (P_X + 2P_Y)

E = dX/dM * M/X
= 1 / (P_X + 2P_Y) * M/X

Given; P_X = 4
P_Y = 2
M = 24

and X = 3

E = 1 / (4+2*2) * 24/3
= 1/8 * 8
E = 1

Income elasticity of good X; E = 1