Question

The cross-price elasticity of demand DX for the utility function u = min (x, 2y) evaluated at m = 24, pX = 4 and pY = 4 is equal to __________.

Answer 1

The utility function is given as

u = min(x, 2.y)

This is a perfect complement utility function. The consumer consumes at fixed proportion i.e.

x = 2.y

or, y = x/2...........(1)

Now, pX and pY are the prices of x and y. m is the income. Hence, budget constraint is

x.pX + y.pY = m

Putting y = x/2 we get,

x.pX + (x/2).pY = m

or, x = 2.m/(2pX + pY)........(2)

Also we calculate,

dx/dpY = -[2m/(2pX + pY)²]........(3)

Now, for pX = 4, pY = 4and m = 24 we get

From equation (2),

x = (2×24)/(2×4 + 4)

or, x = 4

And, from equation (3),

dx/dpY = -[(2×24)/(2×4 + 4)²]

or, dx/dpY = -(48/144)

or, dx/dpY = -(1/3)

Hence, Cross Price Elasticity of Demand is

DX = (dx/dpY).(pY/x)

or, DX = (-1/3)×(4/4)

or, DX = -(1/3) or -0.33

The cross-price elasticity of demand is

DX = -(1/3) or (-0.33)

Hope the solution is clear to you my friend.