Question

Let someone have a utility function defined by u(x, y) = 5lnx + 4lny with an income of 100. Suppose the current price of the two goods are p1= 2 and p2=1.

a) What is their demand function for both goods? Note: This demand function is known as the Walarasian demand or Mashallian demand).

b) What can you say about the proportion of income spent on the two goods.

Answer 1

Solution:

u(x, y) = 5lnx + 4lny

Income, M = 100; p1 = 2 and p2 = 1

The budget line is: p1*x + p2*y = M

2*x + 1*y = 100

a) We have to find the demand function for both goods, x and y:

Demand function can be found using the optimality condition: Marginal rate of substitution, MRS = Price ratio, p1/p2

MRS = Marginal utility of x/Marginal utility of y

Marginal utility of x, MUx = = 5/x

Marginal utility of y, MUy = = 4/y

Thus, MRS = MUx/MUy = (5/x)/(4/y)

MRS = 5y/4x = 1.25*y/x

MRS = p1/p2

So, 1.25*y/x = p1/p2

This gives us y = 0.8*p1*x/p2 or x = 1.25*p2*y/p1

Using the budget line: p1*x + 1*y = 100

p1*x + 1*(0.8*p1*x/1) = 100

p1*x + 0.8*p1*x = 100

x = (100/1.8)*(1/p1)

This is the demand function for good x.

Similarly, using the budget line, we have 2*(1.25*p2*y/2) + p2*y = 100

2.25*p2*y = 100

y = (100/2.25)*(1/p2)

This is the demand function for good y.

b) The proportion of income spent on two goods depends on the preference weightage given to two goods in case of such Cobb-Douglas preferences.

Thus, we can say that proportion of income spent on good x = 5/(5 + 4) = 5/9 or 55.56%

And proportion of income spent on good y = 4/(5 + 4) = 4/9 or 44.44%