Micro Lab 3

1. Understand Duality

In consumer theory, duality refers to the equivalence between two approaches in demand theory: utility maximization and expenditure minimization. These approaches provide different perspectives for analyzing consumer choice problems.

When ${\mathrm{x}}^{\ast }\gg 0$$\mathrm{x}^* \gg 0$

• In expenditure minimization: ${\gamma }^{\ast }\mathrm{\nabla }u\left({\mathbf{x}}^{\ast }\right)=\mathrm{p}$$\gamma^* \nabla u\left(\mathbf{x}^*\right)=\mathrm{p}$

• In utility maximization: $\mathrm{\nabla }u\left({\mathrm{x}}^{\ast }\right)={\lambda }^{\ast }\mathrm{p}$$\nabla u\left(\mathrm{x}^*\right)=\lambda^* \mathrm{p}$

• Both give $MR{S}_{ij}=\frac{p_i}{p_j}$$M R S_{i j}=\frac{p_i}{p_j}$

• In terms of demand functions (given price $p$$p$, income $w$$w$, and utility level $u$$u$)

• In terms of value functions

• Slutsky equation

  $$\begin{array}{}\text{(3)}& \frac{\partial x_j(p,w)}{\partial p_i}=\frac{\partial h_j(p,u)}{\partial p_i}-\frac{\partial x_j(p,w)}{\partial w}{x}_i(p,w)\end{array}$$

  Derivation:

  $$\begin{array}{}\text{(4)}& \begin{array}{rl}{h}_j(\mathbf{p},u)& \equiv x_j(\mathbf{p},e(\mathbf{p},u))\\ \frac{\partial h_j}{\partial p_i}& =\frac{\partial x_j}{\partial p_i}+\frac{\partial x_j}{\partial w}\frac{\partial e}{\partial p_i}\\ \frac{\partial h_j}{\partial p_i}& =\frac{\partial x_j}{\partial p_i}+\frac{\partial x_j}{\partial w}{x}_i\\ \frac{\partial x_j}{\partial p_i}& =\frac{\partial h_j}{\partial p_i}-\frac{\partial x_j}{\partial w}{x}_i\end{array}\end{array}$$

   

• Sheppard's lemma and Roy's identity

  Sheppard's lemma:

  $$\begin{array}{}\text{(5)}& h_i(p,u)=\frac{\partial e(p,u)}{\partial p_i}\end{array}$$

  Roy's identity:

  $$\begin{array}{}\text{(6)}& x_i(p,w)=-\frac{\partial v(p,w)}{\partial p_i}\cdot \frac{1}{\frac{\partial v(p,w)}{\partial w}}=-\frac{\frac{\partial v(p,w)}{\partial p_i}}{\frac{\partial v(p,w)}{\partial w}}\end{array}$$

• The Map (From MWG)

  

A Closer look at Duality in JR section 2.1

Theorem 2.1 tells us we can begin with an expenditure function $E(·)$$E(·)$ and use it to construct a direct utility function representing some convex, monotonic preferences.

Theorem 2.2 tells If we begin with them (the derived utility) and derive the associated expenditure function, we end up with the function $E(·)$$E(·)$ we started with!

These two theorems tell us that any time we can write down a function of prices and utility that satisfies properties 1 to 7 of Theorem 1.7, it will be a legitimate expenditure function for some preferences satisfying many of the usual axioms. We can of course then differentiate this function with respect to product prices to obtain the associated system of Hicksian demands. If the underlying preferences are continuous and strictly increasing, we can invert the function in u, obtain the associated indirect utility function, apply Roy’s identity, and derive the system of Marshallian demands as well. Every time, we are assured that the resulting demand systems possess all properties required by utility maximization. For theoretical purposes, therefore, a choice can be made. One can start with a direct utility function and proceed by solving the appropriate optimization problems to derive the Hicksian and Marshallian demands. Or one can begin with an expenditure function and proceed to obtain consumer demand systems by the generally easier route of inversion and simple differentiation.

Recall Theorem 1.7 - Properties of expenditure function

If $u(\cdot )$$u(\cdot)$ is continuous and strictly increasing, then $e(\mathbf{p},u)$$e(\mathbf{p}, u)$ defined in (1.14) is

1. Zero when $u$$u$ takes on the lowest level of utility in $\mathcal{U}$$\mathcal{U}$,

2. Continuous on its domain ${\mathbb{R}}_{++}^n\times \mathcal{U}$$\mathbb{R}_{++}^n \times \mathcal{U}$,

3. For all $\mathbf{p}\gg \mathbf{0}$$\mathbf{p} \gg \mathbf{0}$, strictly increasing and unbounded above in $u$$u$,

4. Increasing in $\mathbf{p}$$\mathbf{p}$,

5. Homogeneous of degree 1 in $\mathbf{p}$$\mathbf{p}$,

6. Concave in $\mathbf{p}$$\mathbf{p}$.

If, in addition, $u(\cdot )$$u(\cdot)$ is strictly quasiconcave, we have

7. Shephard's lemma: $e(\mathbf{p},u)$$e(\mathbf{p}, u)$ is differentiable in $\mathbf{p}$$\mathbf{p}$ at $({\mathbf{p}}^0,u^0)$$\left(\mathbf{p}^0, u^0\right)$ with ${\mathbf{p}}^0\gg \mathbf{0}$$\mathbf{p}^0 \gg \mathbf{0}$, and

$$\begin{array}{}\text{(7)}& \frac{\partial e({\mathbf{p}}^0,u^0)}{\partial p_i}=x_i^h({\mathbf{p}}^0,u^0),i=1,\dots ,n\end{array}$$

Theorem 2.3 tells the duality between the direct utility function and indirect utility function.

Suppose that $u(\mathbf{x})$$u(\mathbf{x})$ generates the indirect utility function $v(\mathbf{p},y)$$v(\mathbf{p}, y)$. Then by definition, for every $\mathbf{x}\in {\mathbb{R}}_{+}^n,v(\mathbf{p},\mathbf{p}\cdot \mathbf{x})\ge u(\mathbf{x})$$\mathbf{x} \in \mathbb{R}_{+}^n, v(\mathbf{p}, \mathbf{p} \cdot \mathbf{x}) \geq u(\mathbf{x})$ holds for every $\mathbf{p}\gg \mathbf{0}$$\mathbf{p} \gg \mathbf{0}$. In addition, there will typically be some price vector for which the inequality is an equality. Evidently, then we may write

Utilizing this theorem, we are able to recover the direct utility based on an indirect utility. Detailed procedures about how to do this are displayed in the previous lecture notes.

Question JR 2.6

A consumer has expenditure function $e(p_1,p_2,u)=u{p}_1{p}_2/(p_1+p_2)$$e\left(p_1, p_2, u\right)=u p_1 p_2 /\left(p_1+p_2\right)$. Find a direct utility function, $u(x_1,x_2)$$u\left(x_1, x_2\right)$, that rationalizes this person's demand behavior.

 

Theorem 2.4 The Hoteling-Wold lemma

Let $u(\mathbf{x})$$u(\mathbf{x})$ be the consumer's direct utility function. Then the inverse demand function for good $i$$i$ associated with income $y=1$$y=1$ is given by

Question JR 2.7

Derive the consumer's inverse demand functions, $p_1(x_1,x_2)$$p_1\left(x_1, x_2\right)$ and $p_2(x_1,x_2)$$p_2\left(x_1, x_2\right)$, when the utility function is of the Cobb-Douglas form, $u(x_1,x_2)=A{x}_1^{\alpha }{x}_2^{1-\alpha }$$u\left(x_1, x_2\right)=A x_1^\alpha x_2^{1-\alpha}$ for $0<\alpha <1$$0<\alpha<1$.

 

Integrability problem

If $\mathbf{x}(\mathbf{p},y)$$\mathbf{x}(\mathbf{p}, y)$, satisfies budget balancedness (Walras' law), symmetry, and negative semidefiniteness. $\mathbf{x}(\mathbf{p},y)$$\mathbf{x}(\mathbf{p}, y)$ must be utility-generated.

Example and Question - JR 2.5

Consider the solution, $e(\mathbf{p},u)=u{p}_1^{{\alpha }_1}{p}_2^{{\alpha }_2}{p}_3^{{\alpha }_3}$$e(\mathbf{p}, u)=u p_1^{\alpha_1} p_2^{\alpha_2} p_3^{\alpha_3}$ at the end of Example 2.3. (a) Derive the indirect utility function through the relation $e(\mathbf{p},v(\mathbf{p},y))=y$$e(\mathbf{p}, v(\mathbf{p}, y))=y$ and verify Roy's identity. (b) Use| the construction given in the proof of Theorem 2.1 to recover a utility function generating $e(\mathbf{p},u)$$e(\mathbf{p}, u)$. Show that the utility function you derive generates the demand functions given in Example 2.3.

 

2. Homothetic function

Let $y=f(x_1,\dots ,x_n)$$y=f\left(x_1, \ldots, x_n\right)$ be homogenous of degree $k$$k$, and let $z=F(y)$$z=F(y)$, where $F^{\prime }(y)>0$$F^{\prime}(y)>0$ (then $F$$F$ is a monotonic transformation of $y$$y$ ), the function $z(x_1,x_2,\dots ,x_n)$$z\left(x_1, x_2, \ldots, x_n\right)$ is homothetic.

• Homothetic functions have MRS homogeneous of degree 0 :

• If $f(x_1,\dots ,x_n)$$f\left(x_1, \ldots, x_n\right)$ is first-order differentiable, then this is a sufficient and necessary condition for a function to be homothetic.

Question

Show that the following functions are homothetic. (a) $y=\log x_1+\log x_2$$y=\log x_1+\log x_2$ (b) $y=e^{x_1{x}_2}$$y=e^{x_1 x_2}$ (c) $y={\left(x_1{x}_2\right)}^2-x_1{x}_2$$y=\left(x_1 x_2\right)^2-x_1 x_2$ (d) $y=\log \left(x_1{x}_2\right)+e^{x_1{x}_2}$$y=\log \left(x_1 x_2\right)+e^{x_1 x_2}$ (e) $y=\log {(x_1^2+x_1{x}_2)}^2$$y=\log \left(x_1^2+x_1 x_2\right)^2$