Answer to Question #201739 in Microeconomics for Mavis

Question

David's utility function for good X and Y is given by U (X,Y) =X²y³. Where p_x,p_y and I are the price of good X, price of good Y and consumer income respectively.

A. Write the budget constraint of the consumer

B. Derive the demand function for good X and Y

C. What combination of X and Y maximizes the consumer at I=100,p_x=4,and p_y=5

D. Calculate the marginal rate of substitution between X and Y at equilibrium and interpret your results.

E. Suppose david faces a new utility such as U(X, Y)=25_x^0.2 Y^0.75 for consuming commodity X and Y, redo question b and c.

F. Suppose david faces another new utility function such as U (X, Y) =15xy² for consuming commodity X and Y,redo question b and c

,C. What combination of X and Y maximizes the consumer at I=100

Answer 1

A

The budget constraint is 

$Xp_x+ Y p_y = I$

B.

When p_x = 4 , p_y = 5 and I = 100, 

the budget constraint becomes, 4x + 5y = 100

Hence, we need to maximize U(x, y) = x² y³ subject to 4x + 5y = 100

We know, for two goods, utility maximizing condition is when the ratio of marginal utilities is equal to the price ratio i.e. 

$\frac{MUx }{ MUy} = \frac{px }{py}$

MU_x is obtained by partial differentiation of U(x, y) wrt x

Therefore, $MUx = 2 x y^3$

Similarly, $MUy = 3 x^2 y^2$

Hence,$\frac{MUx }{ MUy }= \frac{2 x y^3 }{3 x^2 y^2}= \frac{2}{3}(\frac{y}{x}) = \frac{2y }{ 3x}$

C.

Therefore, utility maximizing condition is:

$\frac{2y}{ 3x} = \frac{p_x }{p_y }= \frac{4 }{ 5}$

i.e. $5y = 6x$

From the budget equation, replacing 6x in place of 5y we get,

 $4x + 6x = 100\\ 10x = 100\\ x = 10\\ y = 12$

Hence, the combination of X and Y that maximizes consumer's utility is x = 10 units and y = 12 units.

D.

marginal rate of substitution between X and Y at equilibrium is actually the value of (MU_x / MU_y) at equilibrium. Obviously at equilibrium this is same as price ratio which is (4/5) or 0.8

MRS is the slope of the indifference curve which, at equilibrium is going to be same as the slope of the budget line which is the price ration. The reason behind this is that, at equilibrium, budget line is tangent to the IC. 

Hence, the value of MRS at equilibrium is going to be 0.8

E.

$MU_X=\frac{\delta U}{\delta X}=25(0.2)X^{0.2-1}Y^{0.75}$

$MU_Y=\frac{\delta U}{\delta Y}=25X^{0.2}(0.75)Y^{0.75-1}$

$MRS=\frac{MU_X}{MU_Y}=\frac{0.2X^{-0.8}Y^{0.75}}{0.75X^{0.2}Y^{-0.25}}$

$MRS=\frac{P_X}{P_Y}$

or $MRS=\frac{4Y}{5X}$ gives,

$\frac{4Y}{5X}=\frac{P_X}{P_Y}$

$Y=\frac{5X.P_X}{4P_Y}$

$X=\frac{4Y.P_Y}{5P_X}$

These are the generalized demand functions for good X and Y.

Given:

I=100, P_x=4,and P_y=5

So the budget constraint becomes 4X+5Y=100

 

Substituting these values in Y^* we get

$Y=\frac{5X}{4}\times\frac{4}{5}$

or

Y=X

thus

$4X+5X=100\\X=Y=11.11$

F.

$MU_X=\frac{\delta U}{\delta X}=15 Y^2$

$MU_Y=\frac{\delta U}{\delta Y}=30XY$

$MRS=\frac{MU_X}{MU_Y}=\frac{15Y^2}{30XY}$

$\frac{Y}{2X}=\frac{P_X}{P_Y}=\frac{4}{5}$

$8X=5Y\\4X+8X=100\\12X=100\\X=8.333\\Y=13.333$

Hence, the combination of X and Y that maximizes consumer's utility is x = 8 units and y = 13 units.