Question

Given U=f(x,y)=Xa Yb and the budget constraint as M=PxXpyY

Answer 1

Solution

Set up constrained utility maximization problem:

$Max. U=xy+x+y$

s.t $P_{x}X+P_{y}y=M$

Set up a Lagrangian function:

$L = x y + x + y − λ ( P_{ x} X + P_{ y} Y − M )$

Take the first derivative of L with respect to x and y:

$\frac{∂ L }{∂ x} : y + 1 − λ P_{ x} = 0..... ( i ) .$

$\frac{∂ L}{ ∂ y} : x + 1 − λ P_{ y} = 0........ ( i i ) .$

$\frac{∂ L}{ ∂ λ} : P_{ x} X + P_{ y} Y − M = 0.......... ( i i i ) .$

Solve equation (i) and (ii), and solve for x and y:

$\frac{y+ 1}{ P_{ x}} = λ , \frac{x + 1}{ P_{ y}} = λ$

$\frac{y + 1}{ P_{ x}} = \frac{x + 1}{ P_{ y}}$

$P_{ y} ( y + 1 ) = P_{ x} ( x + 1 )$

$P_{ y} Y + P_{ y} = P_{ x} X + P_{ x}$

$P_{ y} Y = P_{ x} X + P_{ x} − P_{ y}$

$Y = \frac{P_{ x} X + P_{ x} − P_{ y}}{ P_{ y}}$

$Y = \frac{P_{ x} ( X + 1 ) − P_{ y}}{ P_{ y}}$

$P_{ y} Y − P_{ x} + P_{ y} = P_{ x} X$

$X = \frac{P_{ y} Y − P_{ x} + P_{ y}}{ P_{ x}}$

Substitute both x and y in the budget constraint function:

$P_{ x} X + P_{ y} [\frac{ P_{ x} ( X + 1 ) − P_{ y}}{ P_{ y}} ] = M$

$P_{ x} X + P_{ x} ( X + 1 ) − P_{ y} = M$

$P x X + P x X + P x − P y = M$

$2 P X + P x − P y = M$

$2 P x X = M − P x + P y$

$X^ {M} =\frac{ M − P x + P y}{ 2 P x}$

The above $X^{M}$  is the Marshallian demand for good x.

$P x [\frac{ P y Y − P x + P y}{ P x }] + P_{ y} y = M$

$P_{ y} Y − P_{ x} + P_{ y} + P_{ y} y = M$

$2 P_{ y} y = M + P_{ x} − P_{ y}$

$Y^{ M} = M + P_{ x} − P_{ y} 2 P_{ y}$

The $Y^{M}$  is the Marshallian demand function for good Y.