Question

Using the results from question 2, answer the following questions:

1. Suppose p_x =5, p_z =2 and Y =50. What is the optimal bundle consumed? (Label this as the initial bundle)
2. Now suppose that the price of good x decreases to p_x = 2.5. Now what is the optimal bundle consumed? (Label this as the final bundle)
3. Calculate the decomposition bundle associated with this price change. (The answers won’t be integers)
4. Calculate the Income Effect, Substitution Effect, and Total Effect for good x.

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Results from question 2:
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Consider a market with two goods, x and z that has the following utility function U = x^0.2z^0.8

a) What is the marginal rate of substitution?

b) As a function of the price of good x (px), the price of good z (pz) and the income level (Y ), derive the demand functions for goods x and z.

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a) Marginal rate of substitution is the rate at which the consumer is willing to give up one good in order to obtain one more unit of the other good.

In this case, marginal rate of substitution = marginal utility of good x / marginal utility of good z

Marginal utility of the two goods can be calculated by differentiating the utility function with respect to that good keeping the other good constant.

Marginal utility of good x = 0.2x^-0.8z^0.8

Marginal utility of good z = 0.8x^0.2z^-0.2

So, marginal rate of substitution = 0.2x^-0.8z^0.8 / 0.8x^0.2z^-0.2

= 0.2x^-1/0.8z^-1 = 2z/8x = z/4x is the answer.

b) The budget line equation of the above question can be written as:

p_xx + p_zz = Y

The demand functions of the two goods can be calculated by using the optimal condition that is,

Marginal rate of substitution = p_x / p_z

So, z/4x = p_x / p_z

z = 4x ( p_x / p_z)

Putting the value of z in the budget line equation above,

p_xx + p_z [4x ( p_x / p_z)] = Y

p_xx + 4p_xx = Y

5p_xx = Y

x = Y / 5p_x is the answer.

Now, putting the value of x in the equation of z derived above,

z = 4x ( p_x / p_z)

z =  4 X (Y / 5p_x) X ( p_x / p_z)

z = 4Y / 5p_z is the answer.

Answer 1

a) U = x^0.2z^0.8

GIVEN : Y = 50 , Px = 5, Pz = 2

Thus our lagrange equation is :

  50 =5x + 2z is our Budget Constraint.

From the RESULT to question 2 supplied to me,

x = y /5Px = 50/5*5 = 50/25 = 2

x = 2

z = 4y/5pz = (4*50)/(5*2) = 200/10 = 20

z = 20

INITIAL BUNDLE : (x,z) = (2,20)

b) New price of good x is 2.5

So, new budget constraint is  50 =2.5x + 2z

accordingly,

x = y /5Px = 50/5*2.5 = 50/12.5 = 4

x = 4

z = 4y/5pz = (4*50)/(5*2) = 200/10 = 20

z = 20

NEW BUNDLE : (x,z) = (4,20)

c) demand of good x at new price 2.5 is 4. Total change in demand is (4-2) = 2

Y = xPx = 2*(2.5-5) = -5

Thus income has to fall by 5 to keep purchasing power constant. So, Y' = 50-5 = 45

Decomposition bundle is the bundle at adjusted income and new prices.

Hence,

x = y /5Px = 45/5*2.5 = 45/12.5 = 3.6 .

x = 3.6

z = 4y/5pz = (4*45)/(5*2) = 180/10 = 18

z = 18

Decomposition bundle : (x,z) = (3.6,18)

d) SUBSTITUTION EFFECT = x at new price new income - x at initial price initial income

= 3.6 - 2 = 1.6

x at new price old income = y /5Px = 50/5*2.5 = 50/12.5 = 4

INCOME EFFECT = x at new price old income - x at new price new income

= 4 - 3.6 = 0.4

TOTAL EFFECT = SUBSTITUTION EFFECT + INCOME EFFECT

= 1.6 + 0.4

= 2

SUBSTITUTION EFFECT = 1.6

INCOME EFFECT = 0.4

TOTAL EFFECT = 2