### 经济代写|微观经济学代写Microeconomics代考|ECON2516

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## 经济代写|微观经济学代写Microeconomics代考|The CES Utility Function

It follows that $\varrho=(\xi-1) / \xi$ and therefore another way to write a CES function that makes its elasticity of substitution explicit is
$$\left(a_1 x_1^{(\xi-1) / \xi}+a_2 x_2^{(\xi-1) / \xi}\right)^{\xi /(\xi-1)} .$$
The CES function, invented to be used as a production function, is homogeneous of degree 1: the easy proof is left to the reader. When $a_1=a_2=1$, a monotonic transformation sometimes used which is not homogeneous of degree 1 but is very simple to use is
$$x_1^\delta / \delta+x_2^\delta / \delta .$$
Its elasticity of substitution is, again, constant and given by $\xi=1 /(1-\delta)$.
As one lets $Q$ vary, the CES approaches other well-known types of utility functions. For $\varrho$ tending to 1 from below, the elasticity of substitution tends to $+\infty$, that is, to the case of perfect substitutes. Indeed if one sets $Q=1$, one obtains the separable additive utility function of perfect substitutes, $u=a_1 x_1+a_2 x_2$. As $\varrho$ decreases, the elasticity of substitution decreases too, tending to 1 as $\varrho$ tends towards zero.

For $\varrho=0$ the CES function is not defined owing to division by zero, but if one lets $\varrho$ tend to 0 , the CES tends in the limit to have indifference curves identical to those of a Cobb-Douglas. Indeed, if $\varrho$ tends to zero the MRS tends to $-a_1 x_2 /\left(a_2 x_1\right)$ which is the same as for the generalized Cobb-Douglas $x_1{ }^\alpha x_2{ }^\beta$.

For $\varrho$ tending to $-\infty$ the indifference curves approach the L-shaped indifference curves of the case of perfect complementarity. ${ }^{35}$ The MRS, $-\left(a_1 / a_2\right) \cdot\left(x_2 /\right.$ $\left.x_1\right)^{1-e}$, tends to $-\left(a_1 / a_2\right) \cdot\left(x_2 / x_1\right)^{\infty}$ which has value $-\infty$ if $x_2>x_1$, zero if $x_2<x_1$; thus, indifference curves tend to become L-shaped with the corners on the $45^{\circ}$ straight line through the origin (which can always be obtained with an opportune choice of units for the goods).

## 经济代写|微观经济学代写Microeconomics代考|The Slutsky Equation

We finally tackle the issue of how the (Marshallian) demand for a good reacts to changes in prices. All the functions to appear below are assumed continuously differentiable, indifference curves are strictly convex, and the initial consumer choice is an interior basket, $x \gg 0$.

Let $\mathbf{p}^, m^$ be prices and income in the initial situation, and let $u^=v\left(\mathbf{p}^\right.$, $\left.m^\right)=u\left(\mathbf{x}\left(\mathbf{p}^, m^\right)\right), x_j^=x_j\left(\mathbf{p}^, m^\right)$. Differentiate both sides of $h_f(\mathbf{p}, u)=x_i(\mathbf{p}$, $e(\mathbf{p}, u))=x_i(\mathbf{p}, m)$ with respect to $p_j$
$$\frac{\partial h_i\left(\mathbf{p}^, u^\right)}{\partial p_j}=\frac{\partial x_i\left(\mathbf{p}^, m^\right)}{\partial p_j}+\frac{\partial x_i\left(\mathbf{p}^, m^\right)}{\partial e\left(\mathbf{p}^, u^\right)} \cdot \frac{\partial e\left(\mathbf{p}^, u^\right)}{\partial p_j},$$
where $\frac{\partial x_i\left(\mathbf{p}^, m^\right)}{\partial e\left(p^, u^\right)}$ can also be written $\frac{\partial x_i\left(\mathbf{p}^, m^\right)}{\partial m^}$, because initially $m=m^=e\left(\mathbf{p}^, u^\right)$ and under a balanced budget a variation in expenditure is the same thing as a variation in income. Furthermore, by Shephard’s Lemma $\partial e\left(\mathbf{p}^, u^\right) / \partial p_j=h_j\left(\mathbf{p}^, u^\right)$ $=x_j\left(\mathbf{p}^, e\left(\mathbf{p}^, u^\right)\right)=x_j^$. Hence
$$\frac{\partial h_i\left(\mathbf{p}^, u^\right)}{\partial p_j}=\frac{\partial x_i\left(\mathbf{p}^, m^\right)}{\partial p_j}+\frac{\partial x_i\left(\mathbf{p}^, m^\right)}{\partial m} \cdot x_j^* .$$
Let us indicate as $S(\mathbf{p}, m) \equiv\left[s_{i j}\right]$ the $n \times n$ Jacobian matrix of the vector of Hicksian demand functions for individual goods which form $\mathbf{h}(\mathbf{p}, u)$. Element $(i, j)$ of this matrix is
$$s_{i j}=\frac{\partial h_i\left(\mathbf{p}^, u^\right)}{\partial p_j},$$
for which equality (4.6) holds. Matrix $S(\mathbf{p}, m) \equiv\left[s_{i j}\right]$ is called Slutsky matrix or also matrix of Hicksian substitution effects. It yields
$$\mathrm{dh}=S \mathbf{d p}^T$$
as the variation of compensated demands for infinitesimal variations $\mathrm{d} p$ of the price vector (the superscript $T$ indicates transposition because we treat $\mathbf{p}$ as a row vector).

The usefulness of (4.6) is that, rearranging so as to isolate on one side of the equality sign the price effect on Marshallian demand, we obtain the Slutsky equation for Marshallian demand.

## 经济代写|微观经济学代写Microeconomics代考|The CES Utility Function

$$\left(a_1 x_1^{(\xi-1) / \xi}+a_2 x_2^{(\xi-1) / \xi}\right)^{\xi /(\xi-1)} .$$

$$x_1^\delta / \delta+x_2^\delta / \delta .$$

## 经济代写|微观经济学代写Microeconomics代考|The Slutsky Equation

$$\mathrm{dh}=S \mathbf{d} \mathbf{p}^T$$

(4.6) 的用处在于，重新排列以便在等号的一侧隔离价格对马歇尔需求的影响，我们得到马歇尔需求的斯卢茨基方 程。

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