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# 金融代写|国际贸易理论代写Theory of International Trade代考|ECON4008 Expenditure minimization

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## 金融代写|国际贸易理论代写Theory of International Trade代考|Expenditure minimization

Expenditure minimization: Looking at the behavior of a household from a different angle, let us consider the following problem in which the household minimizes the expenditure on the purchase of a consumption vector that guarantees a specified utility level: given $p$ and $u$,
$$\min x p x \quad \text { subject to } U(x) \geq u \text {. }$$ The solution to the above problem depends on the pair of $(p, u)$ and is called the compensated demand function: $\bar{D}(p, u) \equiv\left(\bar{D}_1(p, u), \ldots, \widetilde{D}_n(p, u)\right) .^{13}$ By substituting $x=\widetilde{D}(p, u)$ into the objective function in (1.8), we obtain the expenditure function: $$E(p, u) \equiv p \widetilde{D}(p, u) .$$ The formal definitions of the compensated demand function and the expenditure function are parallel to those of the conditional factor demand function $\tilde{v}$ and the minimum cost function $C$. Therefore, similar to $\tilde{v}$ and $C$, we can show that the compensated demand function $\widetilde{D}(p, u)$ is homogeneous of degree zero in $p ;\left(p^{\prime}-p\right)\left{\widetilde{D}\left(p^{\prime}, u\right)-\widetilde{D}(p, u)\right} \leq 0$ for any $p$ and $p^{\prime}$; and $E(p, u)$ is monotonically increasing in $u$, concave and linearly homogeneous in $p$, and it satisfies Shephard’s lemma: $\partial E(p, u) / \partial p_i=\widetilde{D}_i(p, u)$ for $i=1,2, \ldots, n$. The $n \times n$ matrix $\left[E{i j}\right] \equiv\left[\partial^2 E / \partial p_i \partial p_j\right]{i, j=1, \ldots, n}$ consisting of the second-order derivatives of $E$ with respect to $p$ is negative semi-definite. Further, similar to Eq. (1.5), we can define the elasticity of substitution between good $i$ and good $j$ in consumption; with a slight abuse of notation, we shall denote it by $\sigma{i j}$. If $\sigma_{i j}>0\left(\sigma_{i j}<0\right)$, then good $i$ and good $j$ are mutually substitutes (complements, respectively) in consumption.

## 金融代写|国际贸易理论代写Theory of International Trade代考|Slutsky equation

Slutsky equation: As is obvious from the definitions of $D$ and $D$, we have $D_i(p, E(p, u)) \equiv \widetilde{D}_i(p, u)$ for any pair of $(p, u)$. By differentiating this identity with respect to $p_j$, we can show that the effect of a price change on the (ordinary) demand for good $i$ is decomposed into two effects. The decomposition is known as the Slutsky equation: for $i, j=1,2, \ldots, n$,
$$\frac{\partial D_i(p, I)}{\partial p_j}=\frac{\partial \widetilde{D}_i(p, u)}{\partial p_j}-x_j \frac{\partial D_i(p, I)}{\partial I},$$
where $x_j=D_j(p, I)=\widetilde{D}_j(p, u)$ is the quantity of demand for good $j$. The first term of the right-hand side of $(1.9), \partial \widetilde{D}_i(p, u) / \partial p_j$, is the substitution effect and the second term, $-x_j \partial D_i(p, I) / \partial I$, is the income effect. The substitution effect is positive (negative) if good $i$ and good $j$ are substitutes (complements, respectively). The substitution effect of its own price, $\partial \widetilde{D}_i(p, u) / \partial p_i$, is always negative. We say that good $i$ is a normal good (an inferior $g \circ o d)$ if $\partial D_i(p, I) / \partial I$ is positive (negative, respectively). Accordingly, the income effect in the Slutsky equation is negative (positive) if good $i$ is a normal good (inferior good, respectively). Therefore, the law of demand, which is equivalent to $\partial D_i(p, I) / \partial p_i<0$, holds true if good $i$ is a normal good.

## 金融代写国际贸易理论代写Theory of International Trade代考| Expenditure minimization

$\min x p x \quad$ subject to $U(x) \geq u$.

## 金融代写|国际贸易理论代写Theory of International Trade代考| Slutsky equation

$$\frac{\partial D_i(p, I)}{\partial p_j}=\frac{\partial \widetilde{D}_i(p, u)}{\partial p_j}-x_j \frac{\partial D_i(p, I)}{\partial I}$$

$-x_j \partial D_i(p, I) / \partial I$ ，是收入效应。如果良好，菖代效果为正 (负) $i$ 而且即好 $j$ 是菖代品 (分别是补品) 。自身价格的菖代效应， $\partial \widetilde{D}_i(p, u) / \partial p_i$ ，始终为负数。我们说好 $i$ 是正常商品 (忩质商品 $g \circ o d$ )如果 $\partial D_i(p, I) / \partial I$ 为阳性（分别为阴性) 。因此，斯卢 茨其方程中的收入效应是负的（正的），如果好的话 $i$ 是正常商品（分别为劣质商品）。因此，需求定律，其等价于 $\partial D_i(p, I) / \partial p_i<0$ ，如果良好，则成立 $i$ 是正常的好东西。

## MATLAB代写

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