Microeconomics lab 2

1. Envelope theorem

let $f$$f$ be a continuously differentiable function of $n+k$$n+k$ variables. Define the function $f^{\ast }$$f^*$ of $k$$k$ variables by

where $\mathbf{x}$$\mathbf{x}$ is an $n$$n$-vector and $\mathbf{r}$$\mathbf{r}$ is a $k$$k$-vector. if the solution of the maximization problem is a continuously differentiable function of $\mathbf{r}$$\mathbf{r}$, then:

The meaning of the Lagrangian multiplier Suppose the utility function takes the form as:

We are going to maximize the utility given the constraints $\mathbf{p}\mathbf{x}\le c$$\bold p \bold x \leq c$. The Lagrange function:

At the optimal, we must have:

Define the indirect utility function,

By the envelope theorem,

Therefore, $\lambda$$\lambda$ measures the sensitivity the $u$$u$ change with respect to $c$$c$ change, which has a meaning of shadow price (implicit price) of income $c$$c$.

Derive the Roy's identity based on envelope theorem

From $\text{(7)}$$\eqref{eq7}$, we have $\frac{\partial V}{\partial c}=\lambda$$\frac{\partial V}{\partial c} = \lambda$

Similarly,

Combine $\text{(7)}$$\eqref{eq7}$ and $\text{(8)}$$\eqref{eq8}$, we can derive the Roy's identity as following:

2. Solve the Marshallian demand

Question - 2023 Midterm 1

On the assumption that the direct utility function is of the form $U(x_1,x_2)=x_1^{\alpha }{x}_2^{1-\alpha }$$U\left(x_1, x_2\right)=x_1^\alpha x_2^{1-\alpha}$ : (a) derive the Marshallian demand functions, (b) construct indirect utility function, (c) derive the Marshallian demand functions exploiting Roy's identity and verify your results.

3. Recover direct utility function from an indirect utility

Theory: Suppose that $u(\mathrm{x})$$u(\mathrm{x})$ is quasiconcave and differentiable on ${\mathbb{R}}_{++}^n$$\mathbb{R}_{++}^n$ with strictly positive partial derivatives there. Then for all $\mathrm{x}\in {\mathbb{R}}_{++}^n,v(\mathrm{p},\mathrm{p}\cdot \mathrm{x})$$\mathrm{x} \in \mathbb{R}_{++}^n, v(\mathrm{p}, \mathrm{p} \cdot \mathrm{x})$, the indirect utility function generated by $u(\mathrm{x})$$u(\mathrm{x})$, achieves a minimum in $\mathbf{p}$$\bold p$ on ${\mathbb{R}}_{++}^n$$\mathbb{R}_{++}^n$, and

Question - JR Example 2.1

Given $V(p_1,p_2,M)=M{(p_1^r+p_2^r)}^{-\frac{1}{r}}$$\quad V\left(p_1, p_2, M\right)=M\left(p_1^r+p_2^r\right)^{-\frac{1}{r}}$. Recover the corresponding direct utility function.

Sol.

Only three steps are needed for this types of question:

• Step 1: Let the expenditure (income) $M=1$$M = 1$

• Step 2: $\underset{p_1,p_2}{min}V(p_1,p_2)\text{ s.t. }{p}_1{x}_1+p_2{x}_2\le 1$$\min _{p_1, p_2} V\left(p_1, p_2\right) \text { s.t. } p_1 x_1+p_2 x_2 \leq 1$

• Step 3: Substitute $p_1^{\ast }$$p_1^*$ and $p_2^{\ast }$$p^*_2$ into indirect utility function then direct utility function is recovered.

Question

Derive the consumer's direct utility function given that his indirect utility function has the form $V(p_1,p_2,M)=max\{\frac{M^2}{p_1^2},\frac{M^2}{p_2^2}\}$$V\left(p_1, p_2, M\right)=\max \left\{\frac{M^2}{p_1^2}, \frac{M^2}{p_2^2}\right\}$.

4. CES utility function

Question - MWG 3.C.6

Suppose that in a two-commodity world, the consumer's utility function takes the form $u(x)={[{\alpha }_1{x}_1^{\rho }+{\alpha }_2{x}_2^{\rho }]}^{1/\rho }$$u(x)=\left[\alpha_1 x_1^\rho+\alpha_2 x_2^{\rho}\right]^{1 / \rho}$. This utility function is known as the constant elasticity of substitution (or CES) utility function. (a) Show that when $\rho =1$$\rho=1$, indifference curves become linear. (b) Show that as $\rho \to 0$$\rho \rightarrow 0$, this utility function comes to represent the same preferences as the (generalized) Cobb-Douglas utility function $u(x)=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}$$u(x)=x_1^{\alpha_1} x_2^{\alpha_2}$. (c) Show that as $\rho \to -m$$\rho \rightarrow-m$, indifference curves become "right angles"; that is, this utility function has in the limit the indifference map of the Leontief utility function $u(x_1,\mathrm{x},)=min\{x_1,x_2\}$$u\left(x_1, \mathrm{x},\right)=\min\left\{x_1, x_2\right\}$