这是一份anu澳大利亚国立大学ECON2131作业代写的成功案例

$$
\varepsilon_{p_{j}}^{s_{j}} \equiv\left(\partial s_{i} / \partial p_{j}\right)\left(p_{j} / s_{i}\right)
$$
and the elasticity of the total consumption of equation
$$
\varepsilon_{p}^{x} \equiv(\partial x / \partial \bar{p})(\bar{p} / x)
$$
To see the relation between share elasticities and total elasticity of demand define
$$
w_{j}=\left(p_{j} x_{j}\right) / \bar{p} x
$$
the expenditure in period $j$ relative to the cost of total consumption at the index price. Then the full elasticity can be written
$$
\eta_{i j}=\varepsilon_{p_{j}}^{s_{i}}+\varepsilon_{\bar{p}}^{x} w_{j} \quad
$$

ECON2131 COURSE NOTES ：

$$
\mathrm{d} y_{i}=D_{i}^{o} \mathrm{~d} \mathbf{p}+\left(\mathrm{d} R-\mathbf{x}^{o} \mathrm{~d} \mathbf{p}\right) / H
$$
It follows immediately that
$$
\sum_{i=1}^{H} \mathrm{~d} y_{i}=\mathrm{d} R
$$
Moreover, Roy’s identity implies that
$$
V_{i}\left(\mathbf{p}^{o}+\mathrm{d} \mathbf{p}, \mathbf{q}, y_{i}+\mathrm{d} y_{i}\right)=V_{i}\left(\mathbf{p}^{o}, \mathbf{q}, y_{i}\right)+\left(\partial V_{i} / \partial y_{i}\right)\left(\mathrm{d} y_{i}-D_{i}^{o} \mathrm{~d} \mathbf{p}\right)
$$
Since marginal utility of money $\partial V_{i} / \partial y_{i}$ is positive, and since $\mathrm{d} y_{i}-D_{i}^{o} \mathrm{~d} p$ must be positive if holds, it can be easily seen that utility
$$
V_{i}\left(\mathbf{p}^{0}+\mathrm{d} \mathbf{p}, \mathbf{q}, y_{i}+\mathrm{d} y_{i}\right)>V_{i}\left(\mathbf{p}^{0}, \mathbf{q}, y_{i}\right)
$$