这是一份manchester曼切斯特大学ECON10172T作业代写的成功案例

If we assume $V=2$ is the initial utility level, this loss (because $P_{Y}=1$ ) is given by
$$
\text { loss }=4(1)^{5}-4(.25)^{5}=2,
$$
which is exactly what we found in Example 5.3-when $P_{X}$ rises to 1, expenditures must rise from 2 to 4 to keep this person from being made worse off. If the utility level experienced after the price rise is believed to be the more appropriate utility target for measuring the welfare loss, then $V=1$ (see Example 5.3) and the loss would be given by
$$
\text { loss }=2(1)^{5}-2(.25)^{5}=1 .
$$
If the loss were evaluated using the uncompensated (Marshallian) demand function
$$
X=d_{X}\left(P_{X}, P_{Y}, I\right)=\frac{I}{2 P_{X}}
$$
the computation would be
$$
\begin{aligned}
\text { loss } &=\int_{.25}^{1} \frac{I}{2 P_{X}} d P_{X} \
&=\left.I \frac{\ln P_{X}}{2}\right|_{.25} ^{1}=0-(-1.39)=1.39,
\end{aligned}
$$

ECON10172T COURSE NOTES ：

Moving the terms in $\lambda$ to the right and dividing the first equation by the second yields
$$
\begin{aligned}
\frac{1}{X} &=\frac{P_{X}}{P_{Y}} \
P_{X} X &=P_{Y} .
\end{aligned}
$$
Substitution into the budget constraint now permits us to solve for the Marshallian demand function for $Y$ :
$$
I=P_{X} X+P_{Y} Y=P_{Y}+P_{Y} Y
$$
Hence,
$$
P_{Y} Y=I-P_{Y} .
$$
This equation shows that an increase in $P_{Y}$ must decrease spending on good $Y$ (that is, $\left.P_{Y} Y\right)$. Therefore, since $P_{X}$ and $I$ are unchanged, spending on $X$ must rise. So
$$
\frac{\partial X}{\partial P_{Y}}>0,
$$
and we would term $X$ and $Y$ gross substitutes. On the other hand, Equation $6.18$ shows that spending on $Y$ is independent of $P_{X}$. Consequently,
$$
\frac{\partial Y}{\partial P_{X}}=0
$$