## Introductory Mathematics for Economists with Matlab

Chapter 10 inequality constrained optimization, 10.1 borrowing constrained profit maximization.

Go back to fan ’s MEconTools Package, Matlab Code Examples Repository ( bookdown site ), or Math for Econ with Matlab Repository ( bookdown site ).

In this problem, we solve the constrained firm’s profit maximization problem with decreasing returns to scale. This continues from the unconstrained profit maximization problem from Firm’s Profit Maximization Problem with Cobb Douglas Production Function (Decreasing Returns to Scale) .

## 10.1.1 Firm and Capital and Labor

The problem is the same as before, the profit maximization problem is:

The constraint is such that the firm can not borrow more than \(\bar{K}\)

To find optimal choices, we will assume that \(\alpha +\beta <1\)

## 10.1.2 Lagrangian and First Order Conditions

\(\displaystyle \lambda \left(K-\bar{K} \right)=0\)

\(\displaystyle \lambda \ge 0\)

## 10.1.3 Solving for Different Cases

Our problem here is simpler, we only have two cases:

## 10.1.4 Solution

With con denoting constrained, unc denoting unconstrained, we have:

## 10.2 Constrained Borrowing and Savings

## 10.2.1 What is the constrained borrowing problem?

Utility : \(U(c_{today} ,c_{tomrrow} )=\log (c_{today} )+\beta \cdot \log (c_{tomorrow} )\)

Budget Today : \(c_{today} +b=Z_1\)

Budget Tomorrow : \(c_{tomorrow} =b\cdot (1+r)+Z_2\)

## 10.2.2 Inequality Constraint

We can formulate the problem above as having 1 savings choice that is constrained.

specifically: \(\max_b \log (Z_1 -b)+\beta \cdot \log (Z_2 +b\cdot (1+r))\)

\(\displaystyle b\ge \bar{b}\)

We can think of the inequality constraint more generally as a function:

\(\displaystyle \bar{b} -b\le 0\)

## 10.2.3 Lagrangian with Inequality Constraint

For inequality constraint, we follow SB and use \(\lambda\) for the lagrange multiplier.

## 10.2.4 Derivative with Respect to \(b\)

## 10.2.5 First Order Conditions with Inequality Constraint

\(\displaystyle \frac{\partial \mathcal{L}}{\partial b}(b^* ,\lambda^* )=0\)

\(\displaystyle \lambda^* \cdot [g(b^* )-q]=0\)

\(\displaystyle \lambda^* \ge 0\)

\(\displaystyle g(b^* )\le q\)

## 10.2.6 Solving the Problem

Define the parameters and the equations

## 10.2.7 Effects of \(Z_2\) on optimal choices

How does optimal choice change if the household has more endowment tomorrow?

## 10.2.8 Effects of \(r\) on optimal choices

## 10.3 Leisure, Savings and Constrained Borrowing

## 10.3.1 What is the constrained asset choice problem with labor?

Budget Today : \(c_{today} +b=Z_1 +w\cdot \textrm{work}\)

\(w\) is the wage, and \(b\) can be positive or negative.

## 10.3.2 Single Inequality Constraint Problem

We can formulate the constrained problem as this:

solve analytically the unconstrained optimal choices by hand and using the symbolic toolbox

solve the optimal work time choice given binding borrowing constraint

solve numerically directly for the constrained optimal choices

## 10.3.3 Unconstrained Optimal Labor and Borrowing and Savings Choices Prlbme

We have two partial derivatives of the lagrangian, and at the optimal choices, these are true:

Unconstrained Choices–One Equation and One Unknown

We have two equations and two unknowns, from the two FOCs above, we have:

\(\displaystyle \frac{\beta \cdot (1+r)}{Z_2 +b\cdot (1+r)}=\frac{\psi }{w\cdot \left(T-H\right)}\)

\(\displaystyle H=T-\frac{Z_2 +b\cdot (1+r)}{\beta \cdot (1+r)}\cdot \frac{\psi }{w}\)

Then pluggint this back in to the first FOC, we have:

This is one equation and one unknown.

Unconstrained Choices–Analytical Optimal Borrowing and Savings Choice

We use \(\Omega\) and \(\chi\) to replace some terms above, and have:

\(\displaystyle \frac{1}{\Omega -\chi b}=\frac{\beta }{Z_2 \frac{1}{1+r}+b}\)

\(\displaystyle Z_2 \frac{1}{1+r}+b=\Omega \beta -\chi \beta b\)

\(\displaystyle b^{\ast } =\frac{\Omega \beta -\frac{1}{1+r}Z_2 }{1+\chi \beta }\)

Our optimal borrowing and savings choice is:

\[b^{\ast ,unc} =\frac{\left(Z_1 +wT\right)\beta -\frac{1+\psi }{1+r}Z_2 }{1+\beta +\psi }\]

Unconstrained Choices–Matlab Analytical Symbolic Solutions

Work Choice given Binding Borrowing Constraint–Matlab Analytical Symbolic Solutions

1em \(\displaystyle \frac{\psi }{H-T}+\frac{w}{z_1 -\textrm{bbar}+H\,w}\)

1em \(\displaystyle \frac{T\,w+\textrm{bbar}\,\psi -\psi \,z_1 }{w+\psi \,w}\)

## 10.3.4 Numerical Solution to the Inequality Constraint Problem

\(\displaystyle \textrm{work}\ge 0\)

\(\displaystyle \textrm{leisure}\ge 0\)

\(\textrm{work}+\textrm{leisure}\le T\) , where \(T\) is total time available

Formulating the Constraints as a System of Linear Equations

\(\displaystyle -\textrm{work}\le 0\)

\(\displaystyle -\textrm{leisure}\le 0\)

\(\displaystyle \textrm{work}+\textrm{leisure}\le T\)

This is actually a linear system, the equations above are equal to:

\(\displaystyle (-1)\cdot b+0\cdot \textrm{work}+0\cdot \textrm{leisure}\le -\bar{b}\)

\(\displaystyle 0\cdot b+(-1)\cdot \textrm{work}+0\cdot \textrm{leisure}\le 0\)

\(\displaystyle 0\cdot b+0\cdot \textrm{work}+(-1)\cdot \textrm{leisure}\le 0\)

\(\displaystyle 0\cdot b+1\cdot \textrm{work}+1\cdot \textrm{leisure}\le T\)

Which mean that we have a \(A\) matrix and \(q\) vector:

## IMAGES

## VIDEO

## COMMENTS

In a previous post, we introduced the method of Lagrange multipliers to find local minima or local maxima of a function with equality

This video shows how to solve a constrained optimization problem with inequality constraints using the Lagrangian function.

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Solving inequality Constrained Optimization Problems ... problem with inequality constraints, in which a specific problem to be. 109. 0022-247X/88 $3.00.