Question

3.) Solve for the system of equations which define the solution to the following un- constrained optimization problems, and provide sufficient conditions to demonstrate the existence of a solution. When a functional form is not specified, state the necessary (first order) and sufficient (second order) conditions for the problem to generate the desired solu- tion (max or min). a.) $\max_x f(x) = \ln(x) - c(x)$ b.) $\max_{x,z} h(x, z) = x^\alpha z^\beta - \theta_1 x - \psi_2 z$ c.) $\max_{x,z} \Pi(x, z) = (x^\alpha z^{1-\alpha})^{1-\gamma} - (\psi_1 x + \psi_2 z)^2$ e.) $\max_{x,z} f(x, z) = \log(xz) - \rho(\theta_1 x + \theta_2 z)$

          3.) Solve for the system of equations which define the solution to the following un-
constrained optimization problems, and provide sufficient conditions to demonstrate the
existence of a solution. When a functional form is not specified, state the necessary (first
order) and sufficient (second order) conditions for the problem to generate the desired solu-
tion (max or min).
a.)
$\max_x f(x) = \ln(x) - c(x)$
b.)
$\max_{x,z} h(x, z) = x^\alpha z^\beta - \theta_1 x - \psi_2 z$
c.)
$\max_{x,z} \Pi(x, z) = (x^\alpha z^{1-\alpha})^{1-\gamma} - (\psi_1 x + \psi_2 z)^2$
e.)
$\max_{x,z} f(x, z) = \log(xz) - \rho(\theta_1 x + \theta_2 z)$

3.) Solve for the system of equations which define the solution to the following un-
constrained optimization problems, and provide sufficient conditions to demonstrate the
existence of a solution. When a functional form is not specified, state the necessary (first
order) and sufficient (second order) conditions for the problem to generate the desired solu-
tion (max or min).
a.)
f(x) = ln(x) - c(x)
b.)
maxx,z h(x, z) = x^α z^β - θ1 x - ψ2 z
c.)
maxx,zΠ(x, z) = (x^α z^1-α)^1-γ - (ψ1 x + ψ2 z)^2
e.)
maxx,z f(x, z) = log(xz) - ρ(θ1 x + θ2 z)

Added by Ryan P.

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