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44 changes: 25 additions & 19 deletions matlab_exercises.mmd
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Try the exercises at [Quantitative Economics](www.http://quant-econ.net).

Some of the code for these exercises is in the `examples` folder on `github`.
Some of the code for these exercises is in the `examples` folder in [my github repo][githubLH].

Some of my general purpose reusable code is posted in the `shared` folder in [my github repo][githubLH].

## Read the Matlab documentation on the following subjects:

* Data types: matrices, strings, structures.
* [Data types](www.mathworks.com/help/matlab/data-types_data-types.html): matrices, strings, structures.

* Indexing to access matrix elements.
* [Indexing](www.mathworks.com/.../matrix-indexing-in-matlab.html) to access matrix elements.

* Operators: `*,.*,/,./`
* [Operators](http://www.mathworks.com/help/matlab/operators-and-elementary-operations.html): `*,.*,/,./`

* String handling: `sprintf`
* [String handling](http://www.mathworks.com/help/matlab/ref/strings.html): [sprintf](www.mathworks.com/help/matlab/ref/sprintf.html)

* Loops: while, for.
* [Loops](http://www.mathworks.com/help/matlab/matlab_prog/loop-control-statements.html): while, for.

* Functions: input/output arguments, global/local variables.
* [Functions](http://www.mathworks.com/help/matlab/function-basics.html): input/output arguments, global/local variables.

* Optimization: fzero, fsolve, optimset.
* [Optimization](www.mathworks.com/products/optimization): fzero, fsolve, optimset.

* Graphics: plot.
* [Graphics](www.mathworks.com/products/matlab/matlab-graphics): plot.

## Getting Started ##

Expand All @@ -30,19 +32,19 @@ Some of the code for these exercises is in the `examples` folder on `github`.

Output: mean and standard deviation for each column

My solution: `statsLH.std_w`
[My solution](https://github.com/hendri54/shared/blob/master/%2BstatsLH/std_w.m)

1. Compute the cdf for weighted data.

Input: data and weights

Output: cumulative percentiles and their values

My solution: `distrib_lh.cdf_weighted`

[My solution](https://github.com/hendri54/shared/blob/master/%2BdistribLH/cdf_weighted.m)
1. Compute points at the percentile distribution for weighted data (e.g. median)

My solution: `distrib_lh.pcnt_weighted`
[My solution](https://github.com/hendri54/shared/blob/master/%2BdistribLH/pcnt_weighted.m)

## Root finding

Expand All @@ -66,6 +68,8 @@ Consider a growth model given by the Bellman equation

\\(V(k) = \max u(c) + \beta V(f(k) - c) \\)

with \\(f(k) = k^\alpha\\).

Solve the growth model by value function iteration.

Steps:
Expand All @@ -78,23 +82,23 @@ Along the way, plot the value functions to show how they converge to the true on

![GrowthValueIter](figures/growth_value_iter.png "Value function iteration")

See [QuantEcon](http://quant-econ.net/jl/dp_intro.html).
See [QuantEcon](http://quant-econ.net/jl/dp_intro.html) and my [Matlab solution](https://github.com/hendri54/Econ821/blob/master/matlab_examples/growth_model_821.m).

## Useful exercises from [QuantEcon](http://quant-econ.net)

1. Draw discrete random numbers

Given: probability of each state

My solution: `random_lh.rand_discrete`
[My solution](https://github.com/hendri54/shared/blob/master/%2BrandomLH/rand_discrete.m)

1. Simulate a Markov Chain

See [QuantEcon](http://quant-econ.net/jl/finite_markov.html) for the setup and Julia code.

Also compute the stationary distribution

My solution: `markov_lh.markov_sim`
[My solution](https://github.com/hendri54/shared/blob/master/%2BmarkovLH/markov_sim.m)

Also try their Exercise 1 (illustrate the central limit theorem).

Expand All @@ -104,12 +108,14 @@ See [QuantEcon](http://quant-econ.net/jl/dp_intro.html).

You will see that the syntax is very similar.

My solution: `+ar1_lh/tauchen`

[My solution](https://github.com/hendri54/shared/blob/master/%2Bar1LH/tauchen.m)
2. Simulate random variables to illustrate the Law of Large Numbers and the Central Limit Theorem.

See [QuantEcon](http://quant-econ.net/jl/lln_clt.html) for the setup and Julia code.

My solution: `matlab_examples/clt_illustrate`
[My solution](https://github.com/hendri54/Econ821/blob/master/matlab_examples/clt_illustrate_821.m)

------------------------

[githubLH]: https://github.com/hendri54
192 changes: 124 additions & 68 deletions matlab_functions.mmd
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# Matlab: Functions and Scripts
# Functions and Scripts #

## Encapsulation ##

A key idea of [structured programming](programming.html#structured-programming):

* package code into self-contained functions
* avoid side effects

A function should only change the rest of the world through the outputs it explicitly returns. This is called [encapsulation][encaps].

For this to work, all variables inside a function must be invisible to other code -- they are **local**.

### Example:

```matlab
function y = f(x)
a = 5;
y = x + a;
end

% Command line:
>> a = 3; y = f(0); disp(y)
5
>> disp(a)
3
% Now the converse
function y = g(x)
z = x + a;
end
>> g(0)
% Error: `a is not defined`
```

## Namespaces ##

A [namespace](https://en.wikipedia.org/wiki/Namespace) is a set of functions or scripts that can "see" the same objects.

Example:

* In the function `f(x)` above, `a` was in `f(x)`'s namespace, but not in `g(x)`'s.
* Conversely, the `a` inside `f(x)` is local and not in the command line's namespace.

In Matlab, there are only 2 namespaces:

1. Global: this is what you can access from the command line.
2. Local: inside a function.

Note: This is not exactly true because there are [nested functions][nested].

This means in particular that all global variables and all functions are visible everywhere. Their names must be unique.

Other languages have more control over namespaces (using "modules").

## Functions versus Scripts

[Scripts](http://www.mathworks.com/help/matlab/matlab_prog/create-scripts.html) are simply collections of commands that are run as if they were typed at the command prompt.

* They are rarely used b/c their contents is not [encapsulated](https://en.wikipedia.org/wiki/Encapsulation_(computer_programming)).
* Any variables defined in a script show up in the [global workspace](http://www.mathworks.com/help/matlab/ref/global.html).
* In the section on [structured programming](programming.html#structured-programming), we talk about why this is bad.
Scripts run in the [global][] workspace. This is, generally speaking, not good. They create side effects.

[**Functions**](http://www.mathworks.com/help/matlab/ref/function.html) are similar to scripts with one crucial difference:
Rule of thumb:

> Always use functions, never scripts!

[Functions](http://www.mathworks.com/help/matlab/ref/function.html) are similar to scripts with one crucial difference:

* All variables inside a function are **private**.
* Other functions cannot see the variables inside a function (*encapsulation*).
* Any variable a function should know must be passed to it as an input argument.
* Any variable to be retained after the function finishes must be passed out of it as a return argument.

> Always use functions, never scripts!

An important principle of structured programming:

>**Package a well-defined task into a function with a simple interface.**

Even built-in Matlab commands are often written in Matlab.
A function looks like this

```matlab
function [y1, y2] = fname(x1, x2)
% Do stuff with x1 and x2 to generate y1 and y2
end
```

This would be stored in a text file called `fname.m`.

A side note: Even built-in Matlab commands are often written in Matlab.

Try `open sub2ind`. You get to see (and you can modify) the source code for the built-in `sub2ind` command (`sub2ind.m`).

Now try `open zeros`. This opens `zeros.m`.

* you get an m-file that is blank except for comments

* this means: `zeros` is not written in Matlab (it's truly built-in)

## The Matlab Path <a name="mpath"></a> ##
## The Matlab Path <a Name="Mpath"></A> ##

The Matlab [path](www.mathworks.com/help/matlab/ref/path.html) determines which m-files are visible in the global namespace (and inside all functions).

Matlab commands are stored in text files with the extension `m`.
In contrast to other languages, this is an all or nothing affair: If a function is visible somewhere, it is visible everywhere. (Local functions and nested functions are the exceptions.)

When the user issues a command, such as `m = zeros([3,4]))`, Matlab searches
for a matching m-file, in this case `zeros.m` and runs it.
When the user issues a command, such as `m = zeros([3,4]))`, Matlab searches for a matching m-file, in this case `zeros.m` and runs it.

It does not matter whether `zeros` is a built-in command or an m-file written by a user.

Expand All @@ -53,9 +117,6 @@ Therefore, every time you write new code, you need to put the directories on the

There is no way to allow one function to access some code without changing the `path` globally. This differs from other languages. It makes it hard to organize code.

Every project therefore needs a **startup routine** that adds its directory to the Matlab path.
Or its program files need to be placed in the current working directory.

## Organizing Code ##

When you write new commands, you need to make sure Matlab can find them.
Expand All @@ -64,7 +125,7 @@ This is best accomplished in two ways:
1. For files that belong to the one particular project, place them in a project directory.
Then switch the current directory to that directory using `cd`.

Or put the directory on the `path`
Or put the directory on the `path` using a startup routine.

2. For files that are **shared** between several projects, place them in a special directory (`blah/shared`).
Place this directory on the path: `addpath('blah/shared')` .
Expand All @@ -83,7 +144,7 @@ There are 2 ways of ensuring this:

Such as `plot_821.m`

2. Packages: If you place an m-file into a directory that starts with `+`, let's say `+econ821`, you can access it like this: `econ821.plot(x)`.
2. [Packages](http://www.mathworks.com/help/matlab/matlab_oop/scoping-classes-with-packages.html): If you place an m-file into a directory that starts with `+`, let's say `+econ821`, you can access it like this: `econ821.plot(x)`.

For this to work, the parent dir of `+econ821` must be on the `path`.

Expand Down Expand Up @@ -169,6 +230,28 @@ Now running `sum3.m` does not yield an error message.

>Globals create confusion.

## Nested Functions ##

If you need a function to see the local variables in another function, [nest][nested] it inside the other function.

Example:

```matlab
function f
a = 5;
disp(g(17));
% After calling g(), a == 52

% Nested function can see `a`
function z = g(x)
z = x + a;
a = 52;
end
end
```

The main use of this: optimization. See below.

## Passing by Value ##

In Matlab, all function arguments are passed **by value**.
Expand Down Expand Up @@ -217,13 +300,17 @@ Use the built-in function [`fzero`](http://www.mathworks.com/help/matlab/ref/fze
7.3891
```

What if I want to pass additional parameters to the objective function? Make it a [nested function][nested].

Example [here](https://github.com/hendri54/Econ821/blob/master/matlab_examples/root_find_821.m).

## Example: Two period household

Household solves \\(\max u\left(c,g\right)\\)

subject to \\(y=c+s \\) and \\(g=z+sR\\)

A solution: $c,g,s$
A solution: \\(c,g,s\\)

that solve 2 budget constraints and Euler equation

Expand Down Expand Up @@ -267,70 +354,39 @@ Utility function.

### Code

We write the code bottom up.
We write the [code](https://github.com/hendri54/Econ821/blob/master/matlab_examples/hh_example_821.m) bottom up.

Utility function:
Allow matrix inputs (cM, gM).
Parameters as arguments.

```matlab
function [ucM, ugM] = ces_util_x1(cM, gM, bb, sig);
% CES marginal utility when young, old
uM = (cM .^ (1-sig) + bb .* gM .^ (1-sig)) ./ (1-sig);
ucM = (1-sig) .* uM ./ cM;
ugM = (1-sig) .* uM ./ gM;
end
```
* Allow matrix inputs (cM, gM).
* Parameters as arguments.
* This should really be a general purpose function (my library contains an [OOP version](https://github.com/hendri54/shared/blob/master/UtilCrraLH.m)).

Deviation function:
Sample call:

```matlab
function dev = ee_dev_x1(c, y, z, R, bb, sig);
% EE deviation
% Get s, g from budget constraints
s = y - c;
g = z + R .* s;
if g <= 0
% c not feasible
dev = 1e8;
else
[uc, ug] = ces_util_x1(c, g, bb, sig);
dev = uc - ug .* R;
end
end
```

Main routine:

```matlab
function hh_x1(y, z, R, bb, sig)
% Solve 2 period hh problem
% Set up range of c values
ng = 50;
cGridV = linspace(0, y, ng);
% Calculate deviation for each grid point
devV = zeros(1,ng);
for ig = 1 : ng
devV(ig) = ee_dev_x1(cGridV(ig),y,z,R,bb,sig);
end
% Pick the smallest deviation
[devMin, idxMin] = min(abs(devV));
disp(sprintf('c = %f Dev = %f', cGridV(idxMin), devMin));
end
>> hh_example_821(2, 0.5, 1.04, 0.9, 1.5)
c = 1.224490 Dev = 0.034947
```

## Exercises ##

1. Write a CES utility function that computes $u'(c)$ and $u(c)$.
2. Write a function that computes the inverse of $u'(c)$.
1. Write a CES utility function that computes \\(u'(c)\\) and \\(u(c)\\).
2. Write a function that computes the inverse of \\(u'(c)\\).
3. Write a test function that checks properties of the utility function:
4. The inverse of the inverse equals $u'(c)$.
4. The inverse of the inverse equals \\(u'(c)\\).
5. Marginal utility is decreasing.

Extra credit:

1. Package all of that into an object (a user defined data type).
2. Now write all of this for $u(c)=e^{-\phi c}$.
2. Now write all of this for \\(u(c)=e^{-\phi c}\\).
3. In your test function, set things up so that you only need to change a single line of code to test both utility functions (the benefit of OOP in action).

----------
----------

[encaps]: https://en.wikipedia.org/wiki/Encapsulation_(computer_programming

[global]: http://www.mathworks.com/help/matlab/ref/global.html

[nested]: http://www.mathworks.com/help/matlab/matlab_prog/nested-functions.html?s_tid=gn_loc_drop

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