Answer to Question #269622 in Microeconomics for OBK

Question

Given the following utility function for two goods X and Y

U(X, Y) =

Where 0<α<1

i)            Establish the Mashallian demand functions for X and Y                                                                                                      [8 Marks]

ii)           From (i) above, derive the indirect utility function        [3 Marks]

iii)         Using the indirect utility function, derive the expenditure function                                                                                      [4 Marks]

iv)         Estimate the Hicksian demand function for X and Y by Shephard’s lemma                                                                          [ 5 Marks]

Using (i) and (iv) above for X, establish the Slutsky equation for good X.                                                                              [8 Marks

Answer 1

(i)

$U=(x,y)=[x^\alpha+y^\alpha]^\frac{1}{\alpha}$

suppose x unit and y unit are the quantity demanded for the two commodities X and Y respectively. The Lagrangian function is given as:

$\phi=(x^\alpha+y^\alpha)^\frac{1}{\alpha}-\lambda(M-xP_x-yP_y)$

According to the first order conditions at equilibrium,

$\frac {\delta \phi}{\delta x}=0$

$\frac{\delta \phi}{\delta y}=0$

$\frac{\delta \phi}{\delta \lambda}=0$

Thus, we get

$\frac{\delta \phi}{\delta x} =\frac{1}{\alpha}(px^{\alpha-1})(x^\alpha+y^\alpha)^{\frac{1}{\alpha}-1}$

$\frac {\delta \phi}{\delta x}=(x^{\alpha-1})(x^\alpha+y^\alpha)^{\frac{1}{\alpha}-1}=0$

$x^{\alpha-1}=0$

$\frac{\delta \phi}{\delta y}=\frac{1}{\alpha}(\alpha y^{\alpha-1})(x^\alpha+y^\alpha)^{\frac{1}{\alpha}-1}$

$\frac {\delta \phi}{\delta y}=(y^{\alpha-1})(x^\alpha+y^\alpha)^{\frac{1}{\alpha}-1}=0$

$y^{\alpha-1}=0$

$\therefore x^{\alpha-1}=y^{\alpha-1}$

$\implies x=y$

$\frac{\delta \phi}{\delta \lambda}=M-xP_x-yP_y$

$xP_x+yP_y=M$

The Marshallian demand function is:

$x=\frac{M}{P_x+P_y}$

$y=\frac{M}{P_x+P_y}$

(ii)

The law of demand states that there is an inverse relationship between quantity demanded an price of the commodity. It means that the higher the price of a commodity, then the lower will be the demand for the commodity. From the above expression of the demand for X commodity, it can be seen that there is an inverse relationship between price and commodity demanded X. Thus it can be said that the law of demand holds for the commodity X.

(iii)

Expenditure:

The Engel curve is the income consumption curve derived as the locus of all the points of contact between utility function and income function at different levels in commodity basket. It is as shown below:

The horizontal axis shows the demand from commodity Y and vertical axis show the price of Y. The joining of the points A,B and C shows the income consumption curve which is locus of all points of the utility curve and budget line. The slope of the curve is given by differentiating the income function:

$M=xP_x+yP_y$

$\delta M=P_x.\delta x+P_y.\delta y$

$\frac{\delta y}{\delta x}=-\frac{P_x}{P_y}$

(iv)

Marginal utility of x is:

$MU_x=\frac{1}{\alpha}(\alpha x^{\alpha-1})(x^\alpha+y^\alpha)^{\frac{1}{\alpha}-1}$

Marginal utility of Y is:

$MU_y=\frac{1}{\alpha}(\alpha y^{\alpha -1})(x^\alpha+y^\alpha)^{\frac{1}{\alpha}-1}$

Marginal rate of substitution:

$MRS_{xy}=\frac {MU_x}{MU_y}$

$=(\frac{x}{y})^{\alpha -1}$