Question

Question [3]: Derive the Hicksian demand function for 1. (Cobb-Douglas) u(x) = 24 m-a; a € (0,1). 2. (Leontief) u(x) = min{aT1, X2}, a > 0. 3. (perfect substitute) (2) = ax1 +22; a > 0

Answer 1

1) Cobb Douglas u(x) = x1ax21-a

Maximize U = xax21-a

Subject to PX.x + Px2.x2 = I (budget equation)

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Above figure is derived demand function.

2) u(x ) = min (ax1, x2), a>0

This is demand equation of complementary goods:

X1 = x2( since both the goods are consumed together no matter the price of each good. So we can denote x1& x2 by x

And budget line equation= p1x + p2 x= I

So the demand for x= I / p1+p2

Here the demand function for optimal choice is quite intuitive because two goods are consumed together, it is just as if the consumer were spending all of their money on a single good that had price of p1+ p2.

3) perfect substitute u(x ) = ax1+ x2 a>0

Slope of indifference curve will be -a

There can be three possible case. If p2 > p1 then the slope of budget line is flatter than the slope of indifference curve. In this case the optimal bundle where the consumer spends all of his income on good 1. If p1> p2 then the consumer purchase oy good 2. Finally if p1= p2 there is whole range of optimal choice - any amount that satisfies the budget constraint is optimal in this case. The demand function for good 1 will be :

X1= {I /p1 when p1< p2

{Any no between o and I /p1 when p1= p2

{0. When p1>p2