Question

Quantitative Question

2 Consider a situation where a consumer demands two goods, x and z with the utility function

U¯ = x 0.2 z 0.8

(a) Derive the marginal rate of substitution

(b) Derive the demand functions for x and z as a function of income (Y ), the price of good x, (px) and the price of good z (pz)

(c) Let Y = 200, px = 4, and pz = 8. Find the equilibrium quantities demanded for this consumer. 1

(d) Now, good x gets more expensive, so p 0 x = 10, but nothing else changes. What are the new equilibrium quantities demanded?

(e) Calculate the decomposition bundle. Round to three decimal places.

(f) Calculate the total, income, and substitution effects for good x.

Please answer the E AND F ONLY.

Answer 1

U = x^0.2z^0.8  

Demand function for x is

x(p_x , p_z ,Y)    = 0.2Y/p_x  

Y = 200

p_x = 4

x = 0.2Y/p_x  

x = 0.2(200)/4

x = 10

e)

Now price of x goes up from p_x = 4 to p_x = 10

Y' = Y + Y

= Y + (p_x' - p_x)x  

= 200 + (10 - 4)(10)

= 200 + 60

= 260

x' (p_x' , p_z , Y') = 0.2Y'/p_x'  

x' = 0.2(260)/10  

x' = 5.2

x''(p_x' , p_z , Y) = 0.2Y/p_x'  

= 0.2(200)/10  

= 4

So decomposition bundles are

x' (p_x' , p_z , Y') = 5.2

x''(p_x' , p_z , Y) = 4

f)  

SE = x' (p_x' , p_z , Y') - x(p_x , p_z ,Y)  

= 5.2 - 10

= - 4.8

IE =  x''(p_x' , p_z , Y) - x' (p_x' , p_z , Y')  

= 4 - 5.2  

= - 1.2

TE = SE + IE  

= - 4.8 - 1.2

= - 6