Question

1. Andrew has a constant elasticity of substitution (CES) utility function,

U(x₁, x₂) =  where  and

Determine Andrew’s optimal bundle (x₁, x₂) in terms of his income m and prices of the two goods, p₁ and p₂.

 

 

2. Lynn has a Cobb-Douglas utility function 

U(x₁, x₂) =

What share of her budget does she spend on x₁ (recorded music tracks) and x₂ (live music) in terms of her income m = $30, prices of the two goods, p₁ = $0.5 and p₂= $1?

3. Celine’s quasilinear utility function is

Her budget for these two goods is $10. Originally the prices are p₁ = p₂ = $1. However, the price of the first good rises to $2. Determine the substitution, income and total effect of this price change on the demand for x₁.

Answer 1

1.

Given in the question-

Income = M

Price of good 1 = P₁

Price of good 2 = P₂

Utility function $U(x_1, x_2) = x_1 x_2$

General Budget equation =

$P_1x_1 + P_2x_2 = M ................(1)$

At equilibrium -

$MRS = \frac{P_1}{p_2}\\ MRS =\frac{ x_2}{x_1}$

$\frac{x_2}{x_1 }=\frac{ P_1}{p2}\\ x_2 = \frac{p_1x_1}{p_2}\\ Putting \space value\space in\space equation \space "1"\\ p_1x_1 + p_2x_2 =M\\ p_1x_1 + p_2\ times (\ frac{p_1x_1}{p_2}) = M p_1x_1 + p_1x_1 = M\\ 2p_1x_1 = M\\ x_1 = \frac{M}{2P_1}\\ similarly,\\ x_2 = \frac{M}{2P_2}\\ optimal \space bundles \space will\space be \space "\frac{M}{2P_1 }, \frac{M}{2P_2}"$

2.

Given in the question-

Income = M

Price of good 1 = P₁ =0.5

Price of good 2 = P₂ =1

Utility function $U(x_1, x_2) = x_1 x_2$

General Budget equation =

$P_1x_1 + P_2x_2 = M ................(1)\\30=0.5X_1+X_2$

At equilibrium -

$MRS = \frac{P_1}{p_2}\\MRD=\frac{0.5}{1} =0.5\\ MRS =\frac{ x_2}{x_1}\frac{1}{0.5}=2$

3.

$\frac{x_2}{x_1 }=\frac{ P_1}{p2}\\ x_2 = \frac{p_1x_1}{p_2}\\ Putting \space value\space in\space equation \space "1"\\ p_1x_1 + p_2x_2 =M\\ p_1x_1 + p_2\ times (\ frac{p_1x_1}{p_2}) = M p_1x_1 + p_1x_1 = M\\ 2p_1x_1 = M\\ x_1 = \frac{M}{2P_1}\\ similarly,\\ x_2 = \frac{M}{2P_2}\\ optimal \space bundles \space will\space be \space "\frac{M}{2P_1 }, \frac{M}{2P_2}"$

Therefore