Optimization

Author

Pablo Winant

Computational Economics (MIE37)

In this tutorial you will learn to code and use common optimization algorithms for static models.

A monopolist produces quantity \(q\) of goods X at price \(p\). Its cost function is \(c(q) = 0.5 + q (1-qe^{-q})\)

The consumer’s demand for price \(p\) is \(x(p)=2 e^{-0.5 p}\) (constant elasticity of demand to price).

Write down the profit function of the monopolist and find the optimal production (if any). Don’t use any library except for plotting.

Consider the function \(f(x,y) = 1-(x-0.5)^2 -(y-0.3)^2\).

Use Optim.jl to minimize \(f\) without constraint. Check you understand diagnostic information returned by the optimizer.

Now, consider the constraint \(x<0.3\) and maximize \(f\) under this new constraint.

Reformulate the problem as a root finding problem with lagrangians. Write the complementarity conditions.

Solve using NLSolve.jl

A consumer has preferences \(U(c_1, c_2)\) over two consumption goods \(c_1\) and \(c_2\).

Given a budget \(I\), consumer wants to maximize utility subject to the budget constraint \(p_1 c_1 + p_2 c_2 \leq I\).

We choose a Stone-Geary specification where

\(U(c_1, c_2)=\beta_1 \log(c_1-\gamma_1) + \beta_2 \log(c_2-\gamma_2)\)

Write the Karush-Kuhn-Tucker necessary conditions for the problem.

Verify the KKT conditions are sufficient for optimality.

Derive analytically the demand functions, and the shadow price.

Interpret this problem as a complementarity problem and solve it using NLSolve.

Produce some nice graphs with isoutility curves, the budget constraint and the optimal choice.