Chapter 6, Problem 6.1P

a)

To determine

To know:

Impact on quantity of goat’s milk due to increase in price.

Explanation of Solution

Given:

Utility function: $U(m,s)=m.s$

m denotes goat’s milk and s denotes strudel.

Lagrange multiplier is used to find consumer’s equilibrium.

Budget constraint: $P_{M}M+P_{S}S=I$

Substitute utility function and budget constraint in Lagrange multiplier:

$L=U(M,S)+\lambda (I-P_{M}M-P_{S}S)$

Taking first order condition:

$\frac{\partial L}{\partial M}=S-\lambda P_M=0$

$\frac{\partial L}{\partial S}=M-\lambda P_S=0$

$\frac{\partial L}{\partial \lambda }=I-P_{M}M-P_{S}S=0$

Solving the above equations:

$\begin{array}{l}S-\lambda P_M=0\\ S=\lambda P_M\end{array}$

$\begin{array}{l}M-\lambda P_S=0\\ M=\lambda P_S\end{array}$

Ratio of given equations are as follows:

$\frac{S}{M}=\frac{\lambda P_M}{\lambda P_S}$

$\frac{S}{M}=\frac{P_M}{P_S}$

$M=\frac{S.P_s}{P_M}$

Substitution of M value in above budget constraint:

$\begin{array}{l}I=P_{M}M+P_{S}S\\ I=P_M\frac{S{P}_S}{P_M}+P_{S}S\\ I=P_{S}S+S{P}_S\\ S=\frac{1}{2P_S}\end{array}$

$\frac{\partial S}{\partial P_M}=0$

As value is 0, it shows no impact on quantity due to increase in price of milk.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction that will be achieved after the consumption of certain goods and services. It is an estimated utility.

b)

To determine

To prove: $\frac{\partial M}{\partial P_S}=0$

Explanation of Solution

The lagragian function for the given problem is shown below:

$L=U(M,S)+\lambda (I-P_{M}M-P_{S}S)$

Taking first order derivative, the following result is obtained which is shown below:

$\begin{array}{c}\frac{\partial L}{\partial M}=S-\lambda P_M=0\\ \frac{\partial L}{\partial S}=M-\lambda P_S=0\\ \frac{\partial L}{\partial \lambda }=I-P_{M}M-P_{S}S=0\end{array}$

By using above function, the following result is obtained which is shown below:

$\begin{array}{c}S-\lambda P_M=0\\ S=\lambda P_M\\ M-\lambda P_S=0\\ M=\lambda P_S\end{array}$

Taking ratio of above equations, the following result is shown below:

$\begin{array}{c}\frac{S}{M}=\frac{\lambda P_M}{\lambda P_S}\\ \frac{S}{M}=\frac{P_M}{P_S}\\ S=\frac{M.P_M}{P_S}\end{array}$

Substitute the M value in budget constraint:

$\begin{array}{c}I=P_{M}M+P_{S}S\\ I=P_{M}M+P_S\frac{M{P}_M}{P_S}\\ I=P_{M}M+P_{M}M\\ M=\frac{1}{2P_M}\end{array}$

Thus, from above value of M, the following result is obtained which is shown below:

$\frac{\partial M}{\partial P_S}=0$

The above result shows that increase in the price of strudel does not affect the quantity of goat’s milk.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction that will be achieved after the consumption of certain goods and services. It is an estimated utility.

c)

To determine

To show: Income effect in above parts is identical.

Explanation of Solution

If a two goods case is considered, then the income and substitution effects that arise due to the change in the price of one good on the demand for another good work in opposite directions. The substitution has a positive effect whereas income has a negative effect.

Slutsky equation:

$\frac{\partial d_x}{\partial P_y}=\frac{\partial h_x}{\partial P_y}+\frac{\partial d_x}{\partial I}$

Uncompensated demand = $d_x$

Compensated demand = $h_x$

To generate $h_x$ , create the expenditure function and take the first derivative:

$\begin{array}{l}v=\frac{I}{2P_S}.\frac{I}{2PM}\\ v=\frac{I^2}{4P_s{P}_M}\end{array}$

$E=2{(v{P}_{S}PM)}^{0.5}$

The Hickson demand function is obtained by taking the derivative of expenditure function with respect to their prices as shown below:

$\begin{array}{l}hs=\frac{\partial E}{\partial P_S}\\ $ ={(v{P}_M/P_S)}^{0.5}$\end{array}$

$\begin{array}{l}{h}_M=\frac{\partial E}{\partial P_M}\\ $ ={(v{P}_S/P_M)}^{0.5}$\end{array}$

Now taking the derivative of $h_s$ and $h_M$ with respect to their prices the following result is obtained which is shown below:

$\frac{\partial h_s}{\partial P_M}=0.5{(v/P_M{P}_S)}^{0.5}$

$\frac{\partial h_s}{\partial P_S}=0.5{(v/P_M{P}_S)}^{0.5}$

Since,

$\frac{M\partial d_S}{\partial I}=\frac{S\partial d_M}{\partial I}$ and $\frac{\partial d_S}{\partial P_S}=0$

It shows that $\frac{M\partial d_S}{\partial I}=\frac{S\partial d_M}{\partial I}$

Hence proved.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction that will be achieved after the consumption of certain goods and services. It is an estimated utility.

d)

To determine

To show:

Marshallian demand function

Explanation of Solution

The Marshallian demand function shows changes in the price of y do not affect x purchases.

That is,

$\frac{\partial x}{\partial P_y}=0$

Thus, by using this following result is obtained.

$\begin{array}{l}\frac{\partial s}{\partial I}=\frac{1}{2P_s}\\ \frac{{\partial }_M}{\partial I}=\frac{1}{2P_M}\end{array}$

$\begin{array}{l}\frac{\partial s}{\partial I}=\frac{1}{2P_s}\\ \frac{Mds}{\partial I}=\frac{1}{2P_M{P}_S}\\ \frac{S{d}_M}{\partial I}=\frac{1}{2P_S{P}_M}\end{array}$

Hence proved.

Economics Concept Introduction

Introduction:

Expected utility is the satisfaction that will be achieved after the consumption of certain goods and services. It is an estimated utility.