Optimization Methods

Pablo Winant

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

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Profit optimization by a monopolist

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).

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

using Plots

c(q) = 0.5 + q*(1-q*exp(-q))
x(p) = 2*exp(-0.5p)
π(p) = p*x(p) - c(x(p))

π (generic function with 1 method)

pvec = 0.1:0.01:4.0
# πvec = [π(e) for e in pvec] # vector comprhension: as in python but yields a vector
πvec = π.(pvec);   # vectorized call

plot(pvec, πvec)

# grid search
pvec = range(;start=0.1, stop=4.0, length=1000000)


# this version allocates πvec
# πvec = π.(pvec);   # vectorized call
# i_max = argmax(πvec)

# This one doesn't
i_max = argmax(map(π, pvec))
p_max = pvec[i_max]

2.5393003393003393

plot(pvec, πvec)
scatter!([p_max], [π(p_max)])

# derivatives of π


using ForwardDiff
π1(p) = ForwardDiff.derivative(π,p)
π2(p) = ForwardDiff.derivative(π1,p)

π2 (generic function with 1 method)

π2(0.4)

-2.079405263105777

# p0 = 2.0
function solve_profit(π, p0; tol_η = 1e-10)

    π1(p) = ForwardDiff.derivative(π,p)
    π2(p) = ForwardDiff.derivative(π1,p)

    for t = 1:100
        p1 = p0 - π1(p0)/π2(p0)
        η = abs(p1 - p0)
        println(t, η)
        if η < tol_η
            return p1
        end
        p0 = p1
    end

end

solve_profit (generic function with 2 methods)

solve_profit(π, 2.0)

10.41122344380439246
20.11939558775904713
30.008640374066304357
44.2346230528345075e-5
51.0117813253884833e-9
60.0

2.5393017528720536

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Constrained optimization

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

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

# your code here

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

# your code here

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

# your code here

1. Solve using NLSolve.jl

# your code here

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Consumption optimization

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

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

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

# your code here

1. Verify the KKT conditions are sufficient for optimality.

# your code here

1. Derive analytically the demand functions, and the shadow price.

# your code here

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

# your code here

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

# your code here