Convergence of Sequences

Tutorial: Convergence

Solow Model

A representative agent uses capital \(k_t\) to produce \(y_t\) using the following production function:

\[y_t = k_t^{\alpha}\]

He chooses to consume an amount \(c_t \in ]0, y_t]\) and invests what remains:

\[i_t = y_t - c_t\]

He accumulates capital \(k_t\) according to:

\[k_{t+1} = \left( 1-\delta \right) k_{t} + i_{t}\]

where \(\delta\) is the depreciation rate and \(i_t\) is the amount invested.

The goal of the representative agent is to maximize:

\[\sum_{t\geq 0} \beta^t U(c_t)\]

where \(U(x)=\frac{x^{1-\gamma}}{1-\gamma}\) and \(\beta<1\) is the discount factor.

For now, we ignore the objective and assume that the saving rate \(s=\frac{c_t}{y_t}\) is constant over time.

Create a NamedTuple to hold parameter values \(\beta=0.96\), \(\delta=0.1\), \(\alpha=0.3\), \(\gamma=4\).

Write down the formula of function \(f\) such that \(k_{t+1}\): \(k_{t+1} = f(k_t)\).

Define a function f(k::Float64, p::NamedTuple)::Float64 to represent \(f\) for a given calibration

Write a function simulate(k0::Float64, T::Int, p::NamedTuple)::Vector{Float64} to compute the simulation over T periods starting from initial capital level k0.

Make a nice plot to illustrate the convergence. Do we get convergence from any initial level of capital?

Suppose you were interested in using f to compute the steady-state. What would you propose to measure convergence speed? To speed-up convergence? Implement these ideas.