Question

Given the Utility Function is U = X

0.7Y

0.3 and Budget is taka 300. The original price was

(Px, Py) = (2, 2) and the new price is (Px’, Py) = (4, 2).

(a) Draw an angle curve for X by using the above information [Labelling is must]

(b) Calculate the value of Compensation variation (CV) and Equivalent variation (EV)? [10]

[ ordinary demand curve]

Answer 1

a)

Given

$U=X^{0.7}Y^{0.3}.......(1)$

Budget=300

$(P_X,P_Y)=(2,2) \space and \space (P_x^`,P_Y^`)=(4,2)$

Budget line:$P_XX+p_YY=300....(2)$

Slope of budget line $=\frac{-p_X}{p_Y}$

At price p_x=2,p_y=2

Slope of indifferent curve is MRS

$MRS=\frac{-MU_X}{MU_Y}$

Where MU_x is marginal utility of X and MU_y is marginal utility of y.

Slope of indifferent curve$=\frac{0.7X^{0.3}Y^{0.3}}{0.3X^{0.7}Y^{-0.7}}$

$MRS=\frac{-7Y}{3X}$

Utility optimizing condition

MRS=slope of budget line

$\frac{-7Y}{3X}=-1\\Y=\frac{3X}{7}.....(3)$

Equation 3 is the equation of the angle curve of income consumption curve.it shows the optimal consumption bundle chosen at various level of income.

b)

Put value of Y into equation 3 and 2

$P_XX+P_Y(\frac{3X}{7})=300\\X(P_X+\frac{3}{7}P_Y)=300\\X=\frac{300}{P_X+\frac{3}{7}P_Y}$

$X(P_X,P_Y)=(\frac{300}{P_X+\frac{3}{7}P_Y}).....(4)$

And

$Y(P_X,P_Y)=\frac{3}{7}×(\frac{300}{P_X+\frac{3}{7}P_Y})\\Y(P_X,P_Y)=(\frac{900}{7P_X+{3}P_Y}).....(5)$

Now

$v(P_X,P_Y)=(\frac{300×7}{7P_X+3P_Y})^{0.7}(\frac{300×3}{7P_X+3P_Y})^{0.3}$

$v(P_X,P_Y)=(\frac{300×7}{7P_X+3P_Y})×7^{0.7}×3^{0.3}$

Now compensating variation (CV)

Utility before price change

$v(P_X,P_Y)=v(2,2)=\frac{300}{20}×7^{0.7}×3^{0.3}\\v(2,2)=15×7^{0.7}×3^{0.3}.........(6)$

Indirect utility function is given by

$v(P_X,P_Y,m)=\frac{m}{7p_x+3p_y}×7^{0.7}×3^{0.3}$

Now we will find value of m for which we can get same level of utility equal to equation (6)

$\frac{M^`}{7P^`_X+3P_Y}{7^{0.7}×3^{0.3}}=15×7^{0.7}×3^{0.3}$

$\frac{M^`}{7×4+3×2}=15\\m^`=15(28+6)\\m^`=510$

So compensating variation

$CV=m^`-m\\=510-300\\=210$

Now

Equivalent variation

Utility after the price change with income equal to 300

$v(P_X,P_Y,300)=\frac{300}{28+6}(7^{0.7}×3^{0.3})$

To get utility equal to $v(P_X,P_Y,300)$

Income would have to be

$\frac{300}{34}(7^{0.7}×3^{0.3})=\frac{m^{"}}{7P_X+3P_Y}(7^{0.7}×3^{0.3})\\\frac{300}{34}=\frac{m^{"}}{7×2+3×2}\\m^{"}=300×\frac{20}{34}\\m^{"}=176.47$

Now equivalent variation

$EV=300-m^{"}\\=300-176.47\\=123.53$