Question

Q: A consumer's utility function is given by the expression: U = (0.6X^0.5+ 0.4Y^0.5)².

• Determine the marginal utility functions for each commodity. Does marginal utility decrease when consumption increases?
• Assuming that the price of good X is Rs 15 and the price of Y is Rs 6, write the equation of the budget line and plot it when income is Rs 450. What is its slope? What does it indicate?
• Calculate the marginal rate of substitution of Y for X and interpret its economic meaning. Write the equation showing the consumer's equilibrium condition.
• Obtain the equilibrium values of X and Y.
• Find the expressions for change in MU_x due to an increase in Y and change in MU_y due to an increase in X.

Answer 1

a) U=$(0.6X$^0.5+0.4Y^0.5)

MUx= $\delta$U/$\delta$x=$2(0.6X$^0.5+0.4Y^0.5)$*0.3X$^-0.5 =0.6X^-0.5(0.6X^0.5+0.4Y^0.5)

MU_y= $\delta$U/$\delta$y=$2(06X$^0.5+0.4Y^0.5)*0.2Y^-0.5 = 0.4Y^-0.5 (0.6X^0.5+0.4Y^0.5)

The marginal utility decreases as the consumption increases.

b) Budget Line

P_xX+P_yY=M

15X+6Y=450

Slope =P_y/P_x = $6/15 =0.4$

An increase in consumption of good Y leads to a decrease of good X by 0.4 units.

c) Marginal rate of Substitution MRS=($\delta$U/$\delta$_y)/($\delta$U/$\delta$_X)

=0.4Y^-0.5(0.6X^0.5+0.4Y^0.5)/0.6X^0.5(0.6X^0.5+0.4Y^0.5)

=0.4Y^-0.5/0.6X^0.5

⁼0.67X^0.5Y^-0.5

It implies that one good is substituted at a rate 0.67X^0.5Y^-0.5 such that if one increases, the other one will decrease at the same rate.

Consumer's equilibrium conditions

Y^* = (0.67X^0.5$Py/Px)$^0.5 X^*=(0.67Y^0.5$Py/Px)$^0.5