Microeconomics lab 1

Zihan Zhang, 09/06/2024

0. Warm-up

Some math concepts here.

• Quadratic form

  A quadratic form on $R^n$$R^n$ is a real-valued function of the form $Q(x_1,\dots ,x_n)=\sum _{i\le j}{a}_{ij}{x}_i{x}_j$$Q\left(x_1, \ldots, x_n\right)=\sum_{i \leq j} a_{i j} x_i x_j$, in which each term is a monomial of degree two. It can be expressed in a matrix form: $Q(\mathbf{x})={\mathbf{x}}^T\cdot A\cdot \mathbf{x}$$Q(\mathbf{x})=\mathbf{x}^T \cdot A \cdot \mathbf{x}$, where $A$$A$ is a symmetric matrix. (${\mathbf{x}}^T$$\bold x^T$ means the transpose of $\mathbf{x}$$\bold x$).

  Examples of the quadratic form in two dimensions:

  • $Q_1(x_1,x_2)=x_1^2+x_2^2$$Q_1\left(x_1, x_2\right)=x_1^2+x_2^2$ (positive definite);

  $$\begin{array}{}\text{(1)}& \begin{array}{c}A=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]\end{array}\end{array}$$

  • $Q_1(x_1,x_2)=-x_1^2-x_2^2$$Q_1\left(x_1, x_2\right)=-x_1^2-x_2^2$ (negative definite);

  • $Q_1(x_1,x_2)=x_1^2-x_2^2$$Q_1\left(x_1, x_2\right)=x_1^2-x_2^2$ (indefinite);

  • $Q_1(x_1,x_2)={(x_1+x_2)}^2$$Q_1\left(x_1, x_2\right)=\left(x_1+x_2\right)^2$ (positive semidefinite);

  • $Q_1(x_1,x_2)=-{(x_1+x_2)}^2$$Q_1\left(x_1, x_2\right)=-\left(x_1+x_2\right)^2$ (negative semidefinite).

• Definite symmetric matrices

  • (a) positive definite if ${\mathbf{x}}^T\cdot A\cdot \mathbf{x}>0$$\mathbf{x}^T \cdot A \cdot \mathbf{x}>0$ for all $\mathbf{x}\ne 0$$\mathbf{x} \neq 0$ in $R^n$$R^n$,

  • (b) positive semidefinite if ${\mathbf{x}}^T\cdot A\cdot \mathbf{x}\ge 0$$\mathbf{x}^T \cdot A \cdot \mathbf{x} \geq 0$ for all $\mathbf{x}\ne 0$$\mathbf{x} \neq 0$ in $R^n$$R^n$,

  • (c) negative definite if ${\mathbf{x}}^T\cdot A\cdot \mathbf{x}<0$$\mathbf{x}^T \cdot A \cdot \mathbf{x}<0$ for all $\mathbf{x}\ne 0$$\mathbf{x} \neq 0$ in $R^n$$R^n$,

  • (d) negative semidefinite if ${\mathbf{x}}^T\cdot A\cdot \mathbf{x}\le 0$$\mathbf{x}^T \cdot A \cdot \mathbf{x} \leq 0$ for all $\mathbf{x}\ne 0$$\mathbf{x} \neq 0$ in $R^n$$R^n$, and

  • (e) indefinite if ${\mathbf{x}}^T\cdot A\cdot \mathbf{x}>0$$\mathbf{x}^T \cdot A \cdot \mathbf{x}>0$ for some $\mathbf{x}$$\mathbf{x}$ in $R^n$$R^n$ and $<0$$<0$ for some other $\mathbf{x}$$\mathbf{x}$ in $R^n$$R^n$.

• Principal minors

  Let $A$$A$ be an $n\times n$$n \times n$ matrix. A $k\times k$$k \times k$ submatrix of $A$$A$ formed by $A$$A$ formed by deleting $n-k$$n-k$ columns, say columns $i_1,i_2,\dots ,i_{n-k}$$i_1, i_2, \ldots, i_{n-k}$ and the same $n-k$$n-k$ rows, from $A$$A$ is called a $k$$k$ th order principal submatrix of $A$$A$. The determinant of a $k\times k$$k \times k$ principal submatrix is called a $k$$k$ th order principal minor of $A$$A$.

  For example,

  $$\begin{array}{}\text{(2)}& \begin{array}{c}A=\left(\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)\end{array}\end{array}$$

  there is one third order principal minor: $\det (A)$$\operatorname{det}(A)$. There are three second order principal minors: (1) $\begin{array}{c}\left|\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|\end{array}$$\left|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right|$, formed by deleting column 3 and row 3 from $A$$A$; (2) $\begin{array}{c}\left|\begin{array}{ll}{a}_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|\end{array}$$\left|\begin{array}{ll}a_{11} & a_{13} \\ a_{31} & a_{33}\end{array}\right|$, formed by deleting column 2 and row 2 from $A$$A$; (3) $\begin{array}{c}\left|\begin{array}{ll}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|\end{array}$$\left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|$, formed by deleting column 1 and row 1 from $A$$A$.

  There are three first order principal minors: (1) $\left|a_{11}\right|$$\left|a_{11}\right|$, formed by deleting the last 2 rows and columns, (2) $\left|a_{22}\right|$$\left|a_{22}\right|$, formed by deleting the first and third rows and the first and third columns, and (3) $\left|a_{33}\right|$$\left|a_{33}\right|$​, formed by deleting the first 2 rows and columns.

   

• Leading principal minors

  Let $A$$A$ be an $n\times n$$n \times n$ matrix. The $k$$k$ th order principal submatrix of $A$$A$ obtained by deleting the last $n-k$$n-k$ rows and the last $n-k$$n-k$ columns from $A$$A$ is called the $k$$k$ th order leading principal submatrix of $A$$A$, denoted by $A_k$$A_k$. Its determinant is called the $k$$k$ th order leading principal minor of $A$$A$, denoted by $\left|A_k\right|$$\left|A_k\right|$.

  For example, for the previous matrix $A$$A$, the three leading principal minors are

  $$\begin{array}{}\text{(3)}& \begin{array}{c}\left|a_{11}\right|,\left|\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|,\left|\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|\end{array}\end{array}$$

• Definiteness of a matrix

  • $A$$A$ is positive definite if and only if all its $n$$n$ leading principal minors are strictly positive.

  • $A$$A$ is negative definite if and only if its $n$$n$ leading principal minors alternate in sign: the $k$$k$ th order leading principal minor should have the same sign as $(-1{)}^k$$(-1)^k$.

  • If some $k$$k$ th order leading principal minor of $A$$A$ is nonzero but does not fit either of the above two sign patterns, then $A$$A$ is indefinite.

  • $A$$A$ is positive semidefinite if and only if every principal minor of $A$$A$ is $\ge 0$$\geq 0$

  • $A$$A$ is negative semidefinite if and only if every principal minor of odd order is $\le 0$$\leq 0$ and every principal minor of even order is $\ge 0$$\geq 0$.

• Definiteness of a quadratic form under linear constraints

  To determine the definiteness of a quadratic form of $n$$n$ variables, $Q(\mathrm{x})={\mathrm{x}}^{T}A\mathrm{x}$$Q(\mathrm{x})=\mathrm{x}^T A \mathrm{x}$, when restricted to a constraint set given by $m$$m$ linear equations $B\mathbf{x}=0$$B \mathbf{x}=0$, we can construct the $(n+m)\times (n+m)$$(n+m) \times(n+m)$ symmetric matrix $H$$H$ by bordering the matrix $A$$A$ above and to the left by the coefficients $B$$B$ of the linear constraints. Check the signs of the last $n-m$$n-m$ leading principal minors of $H$$H$, starting with the determinant of $H$$H$ itself.

  • If $\det H$$\operatorname{det} H$ has the same sign as $(-1{)}^n$$(-1)^n$ and if these last $n-m$$n-m$ leading principal minors alternate in sign, then $Q$$Q$ is negative definite on the constraint set.

  • If $\det H$$\operatorname{det} H$ and these last $n-m$$n-m$ leading principal minors all have the same sign as $(-1{)}^m$$(-1)^m$, then $Q$$Q$ is positive definite on the constraint set.

  • If both of the above two conditions are violated by nonzero leading principal minors, then $Q$$Q$ is indefinite on the constraint set.

1. Unconstraint maximization

Consider the following optimization problem:

The first order conditions are:

Critical points are the solutions to the first-order conditions of a function. These points can indicate either a minimum or maximum, but they do not always guarantee either. This is because first-order conditions are necessary, but not sufficient, to determine whether a critical point is an extremum.

Sufficient second order conditions for unconstrained local maximum (minimum) is that Hessian matrix is negative (positive) definite at the critical points.

Define the Hessian matrix:

When $n=2$$n = 2$,

For a minimization problem, we need this Hessian matrix to be positive definite. Thus, we need the first order and second order leading principal minors to be non-negative,

For a maximization problem, we need this Hessian matrix to be negative definite. Thus, the first order leading principal minor has to be non-positive ($\le 0$$\leq 0$) and the second order leading principal minor has to be non-negative ($\ge 0$$\geq 0$).

Question 1

check the second order conditions at the critical points of $F(x,y)=x^3-y^3+9xy$$F(x, y)=x^3-y^3+9 x y$

2. Constraint maximization - one equality condition

Consider the following optimization problem,

For this type of problems, we can set up one Lagrangian function:

The FOCs are the necessary conditions:

Sufficient second order conditions for constrained local max(min) is that Hessian matrix is negative(positive) definite under the constraints at the critical points. To find this, we need to check bordered Hessian matrix.

For example, for the optimization problem above, we can derive the bordered Hessian matrix by:

A Hessian matrix is negative definite under the constraints means that

• the determinant of the bordered Hessian matrix has the same sign as $(-1{)}^n$$(-1)^n$

• the last $(n-1)$$(n-1)$ leading principal minors alternate in sign.

A Hessian matrix is positive definite under the constraints means that

• the determinant of the bordered Hessian matrix has the same sign as $(-1{)}^1=(-1)$$(-1)^1 = (-1)$

• the last $(n-1)$$(n-1)$ leading principal minors keep the same sign.

Question 2

Max $f=x_1^2+2x_1{x}_2-x_2^2$$f = x_1^2+ 2x_1 x_2 -x_2^2$ such that $g=x_1+x_2$$g = x_1 + x_2$ . check the sufficient conditions.

3. Optimization with inequality constraint

We need to set up a Kunn-Tucker Lagrangian.

Question 3

Solve the following constrained maximization problem using Kuhn Tucker conditions :

$$\begin{array}{}\text{(14)}& max{x}_1+x_2\end{array}$$

subject to

$$\begin{array}{}\text{(15)}& \begin{array}{rl}{x}_1+2x_2& \le M\\ x_1& \ge 0\\ x_2& \ge 0\end{array}\end{array}$$

Assume $M$$M$​ is positive.

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

The meaning of the Lagrangian multiplier

Suppose the production function takes the form as:

We are going to maximize the production given the constraints $g(x)\le c$$g(x)\leq c$.

Construct the Lagrange function:

At the optimal, we must have:

According to the envelope theorem,

Therefore, $\lambda$$\lambda$ measures the sensitivity the $f$$f$ change with respect to $c$$c$ change.