### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    #! format: off
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
    #! format: on
end

# ╔═╡ 36610294-87dc-11eb-21e8-5b1dd06c6020
begin
	using PlutoUI
	using Interpolations
	using Roots
	using LaTeXStrings
	using Plots
	using Optim
end

# ╔═╡ d367efe3-f2e1-40b2-8bd1-47bb827424d0
html"""
<style>
	main {
		margin: 0 auto;
		max-width: 2000px;
    	padding-left: max(160px, 10%);
    	padding-right: max(160px, 10%);
		font-size: x-large
	}
</style>
"""

# ╔═╡ 5fab5e80-87ce-11eb-111a-d5288227b97c
html"<button onclick='present()'>present</button>"

# ╔═╡ 705e96f2-87ce-11eb-3f5a-eb6cdb8c49d4
md"
# Dynamic Programming 2

> Florian Oswald, Carlo Alberto 2025


* In this notebook we'll add uncertainty to our simple setup from the last time.
* We'll stay in the finite time framework, but everything we say will be easily portable to infinite time. 
* We'll write code (of course). We won't worry about performance so far, but try to be as explicit as possible.
* We'll do something that looks reasonable but works terribly. 
* Then we'll fix it and celebrate! 🎉


"

# ╔═╡ 34403942-87d2-11eb-1172-7bd285bf7d75
md"
#

* A good place to start is a so called decision tree. 
![](https://upload.wikimedia.org/wikipedia/commons/4/48/DecisionCalcs.jpg)"

# ╔═╡ b9ed265e-87d2-11eb-1028-05b4c4ccbc74
md"

## Markov Decision Problems (MDPs)

* Assume a finite set of states $S$
* A finite set of actions $A$
* For state $s\in S$ and action $a \in A$ we can compute the *cost* or the *contribution* $C(s,a)$
* There is a *transition matrix* $p_t(S_{t+1} | S_{t}, a_t)$ which tells us with which probability we go from state $S_{t}$ to $S_{t+1}$ upen taking decision $a_t$
* Even though many problems have *continuous* state and action spaces, this is a very useful class of problems to study.
"

# ╔═╡ 07992abe-87d4-11eb-0fd7-1bef7fc2ada8
md"
## Optimality (Bellman) Equations

* But let's start with _no uncertainty_. If we are at state $S_t$, we know how $S_{t+1}$ will look like with certainty if we choose a certain action $a_t$ 
* Then we just choose the best action:
$$a_t^*(S_t) = \arg \max_{a_t \in A_t} \left(C_t(S_t,a_t) + \beta V_{t+1}(S_{t+1}(S_t,a_t)) \right)$$
* The associated *value* of this choice is then the same Bellman Equation we know from last time,
$$\begin{align}V_t(S_t) &= \max_{a_t \in A_t} \left(C_t(S_t,a_t) + \beta V_{t+1}(S_{t+1}(S_t,a_t)) \right) \\ &= C_t(S_t,a^*_t(S_t)) + \beta V_{t+1}(S_{t+1}(S_t,a^*_t(S_t)))\end{align}$$
"

# ╔═╡ 93148b06-87d2-11eb-2356-fb460acf8e66
md"

## Enter Uncertainty

* Uncertainty comes into play if we *don't know exactly what state* $S_{t+1}$ we'll end up in.
* We may have partial information on the likelihood of each state occuring, like a belief, or prior information.
* We have our transition matrix $p_t(S_{t+1} | S_{t}, a_t)$, and can define the *standard form of the Bellman Equation*:

$$V_t(S_t) = \max_{a_t \in A_t} \left(C_t(S_t,a_t) + \beta \sum_{s' \in S} p_t\left( S_{t+1} = s' | S_t, a_t \right)V_{t+1}(s') \right)$$
"

# ╔═╡ 3d97787a-87d2-11eb-3713-ab3dad6e5e8b
md"
#

* we can also write it more general without the summation
$$V_t(S_t) = \max_{a_t \in A_t} \left(C_t(S_t,a_t) + \beta \mathbb{E} \left[ V_{t+1}(S_{t+1}(S_t,a_t)) | S_t \right] \right)$$"

# ╔═╡ a41aff3a-87d8-11eb-00fe-e16c14870ba2
md"

## Let's Do it!

* Let's look at this specific problem where the *action* is *consumption*, and $u$ is a *utility function* and the state is $R_t$:
$$V_t(R_t) = \max_{0 \leq c_t \leq R_t} \left(u(c_t) + \beta \mathbb{E} \left[ V_{t+1}(R_{t+1}(R_t,c_t)) |R_t \right] \right)$$

* Here $R_t$ could be available cash on hand of the consumer.
* Let us specify a *law of motion* for the state as 
$$R_{t+1} = R_t - c_t + y_{t+1}$$ where $y_t$ is a random variable.
* We have to compute expectations wrt to $y$

"

# ╔═╡ c6ed3824-87d9-11eb-3e0a-b5a473fdb952
md"

## I.I.D. Shock

* Let's start with the simplest case: $y_t \in \{\overline{y}, \underline{y}\}, 0 < \overline{y} < \underline{y}$
* There is a probability π associated to $P(y = \overline{y}) = \pi$
* There is *no state dependence* in the $y$ process: the current $y_t$ does not matter for the future $y_{t+1}$

"

# ╔═╡ 51983542-87f9-11eb-226a-81fe383f0763
md"
#
"

# ╔═╡ 6ddfc91c-87da-11eb-3b5c-dfe006576f29
"""
	Bellman(grid::Vector,vplus::Vector,π::Float64,yvec::Vector)

Given a grid and a next period value function `vplus`, and a probability distribution
calculate current period optimal value and actions.
"""
function Bellman(grid::Vector,vplus::Vector,π::Float64,yvec::Vector{Float64},β::Float64)
	points = length(grid)
	w = zeros(points) # temporary vector for each choice or R'
	Vt = zeros(points) # optimal value in T-1 at each state of R
	ix = 0 # optimal action index in T-1 at each state of R
	at = zeros(points) # optimal action in T-1 at each state of R

	for (ir,r) in enumerate(grid) # for all possible R-values
		# loop over all possible action choices
		for (ia,achoice) in enumerate(grid)
			if r <= achoice   # check whether that choice is feasible
				w[ia] = -Inf
			else
				rlow = r - achoice + yvec[1] # tomorrow's R if y is low
				rhigh  = r - achoice + yvec[2] # tomorrow's R if y is high
				jlow = argmin(abs.(grid .- rlow))  # index of that value in Rspace
				jhigh = argmin(abs.(grid .- rhigh))  # index of that value in Rspace
				w[ia] = sqrt(achoice) + β * ((1-π) * vplus[jlow] + (π) * vplus[jhigh] ) # value of that achoice
			end
		end
		# find best action
		Vt[ir], ix = findmax(w) # stores Value und policy (index of optimal choice)
		at[ir] = grid[ix]  # record optimal action level
	end
	return (Vt, at)
end

# ╔═╡ aeb2b692-87db-11eb-3c9a-114acdc3d159
md"
#
"

# ╔═╡ b3f20522-87db-11eb-325d-7ba201c66452
begin
	points = 500
	lowR = 0.01
	highR = 10.0
	# more points towards zero to make nicer plot
	Rspace = exp.(range(log(lowR), stop = log(highR), length = points))
	aT = Rspace # consume whatever is left
	VT = sqrt.(aT)  # utility of that consumption
	yvec = [1.0, 3.0]
	nperiods = 10
	β = 1.0  # irrelevant for now
	π = 0.7
end

# ╔═╡ 6e8aec5e-87f9-11eb-2e53-bf5ba90fa750
md"
#
"

# ╔═╡ f00890ee-87db-11eb-153d-bb9244b46e89
# identical
function backwards(grid, nperiods, β, π, yvec)
	points = length(grid)
	V = zeros(nperiods,points)
	c = zeros(nperiods,points)
	V[end,:] = sqrt.(grid)  # from before: final period
	c[end,:] = collect(grid)

	for it in (nperiods-1):-1:1
		x = Bellman(grid, V[it+1,:], π, yvec, β)	
		V[it,:] = x[1]
		c[it,:] = x[2]
	end
	return (V,c)
end

# ╔═╡ 4da91e08-87dc-11eb-3238-433fbdda7df7
V,a = backwards(Rspace, nperiods, β, π, yvec);

# ╔═╡ 7a97ba6a-87f9-11eb-1c88-a999bcff4647
md"
#
"

# ╔═╡ ca3e6b12-87dc-11eb-31c6-8d3be744def0
let
	cg = cgrad(:viridis)
    cols = cg[range(0.0,stop=1.0,length = nperiods)]
	pa = plot(Rspace, a[1,:], xlab = "R", ylab = "Action",label = L"a_1",leg = :topleft, color = cols[1])
	for it in 2:nperiods
		plot!(pa, Rspace, a[it,:], label = L"a_{%$(it)}", color = cols[it])
	end
	pv = plot(Rspace, V[1,:], xlab = "R", ylab = "Value",label = L"V_1",leg = :bottomright, color = cols[1])
	for it in 2:nperiods
		plot!(pv, Rspace, V[it,:], label = L"V_{%$(it)}", color = cols[it])
	end
	plot(pv,pa, layout = (1,2))
end

# ╔═╡ 027875e0-87e0-11eb-0ca5-317b58b0f8fc
PlutoUI.Resource("https://i.ytimg.com/vi/wAbnNZDhYrA/maxresdefault.jpg")

# ╔═╡ 48cc5fa2-87dd-11eb-309b-9708941fd8d5
md"
#

* Oh wow! That's like...total disaster!
* 😩
* This way of interpolating the future value function does *not work very well*
* Let's use proper interpolation instead!
* Instead of `vplus[jlow]` we want to have `vplus[Rprime]` where `Rprime` $\in \mathbb{R}$!

#

"

# ╔═╡ 04b6c4b8-87df-11eb-21dd-15a443042bd1
"""
	Bellman(grid::Vector,vplus::Vector,π::Float64,yvec::Vector)

Given a grid and a next period value function `vplus`, and a probability distribution
calculate current period optimal value and actions.
"""
function Bellman2(grid::Vector,vplus::Vector,π::Float64,yvec::Vector,β::Float64)
	points = length(grid)
	w = zeros(points) # temporary vector for each choice or R'
	Vt = zeros(points) # optimal value in T-1 at each state of R
	ix = 0 # optimal action index in T-1 at each state of R
	at = zeros(points) # optimal action in T-1 at each state of R
	
	# interpolator
	vitp = interpolate((grid,), vplus, Gridded(Linear()))
	vitp = extrapolate(vitp, Interpolations.Flat())

	for (ir,r) in enumerate(grid) # for all possible R-values
		# loop over all possible action choices
		for (ia,achoice) in enumerate(grid)
			if r <= achoice   # check whether that choice is feasible
				w[ia] = -Inf
			else
				rlow = r - achoice + yvec[1] # tomorrow's R if y is low
				rhigh  = r - achoice + yvec[2] # tomorrow's R if y is high
				w[ia] = sqrt(achoice) + (1-π) * vitp(rlow) + (π) * vitp(rhigh)  # value of that achoice
			end
		end
		# find best action
		Vt[ir], ix = findmax(w) # stores Value und policy (index of optimal choice)
		at[ir] = grid[ix]  # record optimal action level
	end
	return (Vt, at)
end

# ╔═╡ 99b19ed4-92c8-11eb-2daa-2df3db8736bd
md"
#
"

# ╔═╡ 43539482-87e0-11eb-22c8-b9380ae1ebdb
function backwards2(grid, nperiods, π, yvec, β)
	points = length(grid)
	V = zeros(nperiods,points)
	a = zeros(nperiods,points)
	V[end,:] = sqrt.(grid)  # from before: final period
	a[end,:] = collect(grid)

	for it in (nperiods-1):-1:1
		x = Bellman2(grid, V[it+1,:], π, yvec, β)	
		V[it,:] = x[1]
		a[it,:] = x[2]
	end
	return (V,a)
end

# ╔═╡ a361963a-92c8-11eb-0543-7decc61720a8
md"
#
"

# ╔═╡ 5f764692-87df-11eb-3b9d-e5823eb4d1b3
let
	V,a = backwards2(Rspace,nperiods,π,yvec,β)
	cg = cgrad(:viridis)
    cols = cg[range(0.0,stop=1.0,length = nperiods)]
	pa = plot(Rspace, a[1,:], xlab = "R", ylab = "Action",label = L"a_1",leg = :topleft, color = cols[1])
	for it in 2:nperiods
		plot!(pa, Rspace, a[it,:], label = L"a_{%$(it)}", color = cols[it])
	end
	pv = plot(Rspace, V[1,:], xlab = "R", ylab = "Value",label = L"V_1",leg = :bottomright, color = cols[1])
	for it in 2:nperiods
		plot!(pv, Rspace, V[it,:], label = L"V_{%$(it)}", color = cols[it])
	end
	plot(pv,pa, layout = (1,2))
end

# ╔═╡ 8eb731f6-87e0-11eb-35dd-61ba070afc8b
md"

#

* Well at least a little bit better. 
* Still, what about those steps in there? What's going on?
"

# ╔═╡ 02d26410-87e2-11eb-0278-99ee0bbd2923
md"

## Continuous Choice

* In fact our action space is continuous here:
$$V_t(R_t) = \max_{0 \leq a_t \leq R_t} \left(u(a_t) + \beta \mathbb{E} \left[ V_{t+1}(R_{t+1}(R_t,a_t)) |R_t \right] \right)$$

* So let's treat it as such. We could direct optimization for the Bellman Operator!

#

"

# ╔═╡ 3b1f5120-87e2-11eb-1b10-85900b6fbeb6
function bellman3(grid,v0,β::Float64, π, yvec)
    n = length(v0)
    v1  = zeros(n)     # next guess
    pol = zeros(n)     # consumption policy function

    Interp = interpolate((collect(grid),), v0, Gridded(Linear()) ) 
    Interp = extrapolate(Interp,Interpolations.Flat())

    # loop over current states
    # of current resources
    for (i,r) in enumerate(grid)

        objective(c) = - (sqrt.(c) + β * (π * Interp(r - c + yvec[2]) + (1-π) * Interp(r - c + yvec[1])) )
        res = optimize(objective, 1e-6, r)  # search in [1e-6,r]
        pol[i] = res.minimizer
        v1[i] = -res.minimum
    end
    return (v1,pol)   # return both value and policy function
end


# ╔═╡ c6990bb0-92c8-11eb-2523-9f8a7ec1cd4a
md"
#
"

# ╔═╡ 0b83e920-87e3-11eb-0792-479eb843b429
function backwards3(grid, nperiods,β, π, yvec)
	points = length(grid)
	V = zeros(nperiods,points)
	a = zeros(nperiods,points)
	V[end,:] = sqrt.(grid)  # from before: final period
	a[end,:] = collect(grid)

	for it in (nperiods-1):-1:1
		x = bellman3(grid, V[it+1,:],β,  π, yvec)	
		V[it,:] = x[1]
		a[it,:] = x[2]
	end
	return (V,a)
end

# ╔═╡ d391139e-92c8-11eb-0888-b325a75fb10f
md"
#
"

# ╔═╡ 7003b4e8-87e3-11eb-28a9-f7e3668beac3
let
	V,a = backwards3(Rspace,nperiods,β,π,yvec)
	cg = cgrad(:viridis)
    cols = cg[range(0.0,stop=1.0,length = nperiods)]
	pa = plot(Rspace, a[1,:], xlab = "R", ylab = "Action",label = L"a_1",leg = :topleft, color = cols[1])
	for it in 2:nperiods
		plot!(pa, Rspace, a[it,:], label = L"a_{%$(it)}", color = cols[it])
	end
	pv = plot(Rspace, V[1,:], xlab = "R", ylab = "Value",label = L"V_1",leg = :bottomright, color = cols[1])
	for it in 2:nperiods
		plot!(pv, Rspace, V[it,:], label = L"V_{%$(it)}", color = cols[it])
	end
	plot(pv,pa, layout = (1,2))
end

# ╔═╡ 4504d988-87e4-11eb-05d5-c9f9f215f785
md"

#

* It's getting there!
* Much fewer wiggles in the consumption function.
* But we can do even better than that.
* Let's do *Policy Iteration* rather than *Value Iteration*!
"

# ╔═╡ 6828b538-87e4-11eb-3cdd-71c31dad5a6e
md"

## Policy Iteration

* In most economic contexts, there is more structure on a problem.
* In our consumption example, we know that the Euler Equation must hold.
* Sometimes there are other *optimality conditions* that one could leverage.
* In our case, we have this equation at an optimal policy $c^*(R_t)$:
$$\begin{align}
u'(c^*(R_t)) & = \beta \mathbb{E} \left[ u'(c(R_{t+1})) \right] \\
             & = \beta \mathbb{E} \left[ u'(R_t - c^*(R_t) + y)   \right]
\end{align}$$
* So: We *just* have to find the function $c^*(R_t)$!
"

# ╔═╡ ba19759c-87eb-11eb-1bec-51de9ac10e31
md"

## *Just*?

* There is a slight issue here. The Euler Equation in itself does not enforce the contraint that we cannot borrow any $R$. 
* In particular, $R_{t} \geq 0$ is missing from the above equation.
* look back above at the last plot of the consumption function. It has a kink!
* The Euler Equation applies only to the *right* of that kink!
* If tomorrow's consumption is low (we are saving a lot), today's consumption is relatively high, implying that we consume more than we have in terms of $R_t$.
* Satisfying the Euler Equation would require to set $R_t <0$ sometimes, which is ruled out.


"

# ╔═╡ a4018f28-87ec-11eb-07f0-af86523dd26e
md"
#

* So if we want to use the Euler Equation, we need to manually adjust for that.
* 😩
* Alright, let's do it. It's not too bad in the end.

#

* First, let's put down the marginal utility function and it's inverse:

"

# ╔═╡ 027fda74-87f1-11eb-1441-55d6e410bf4c
u_prime(c) = 0.5 .* c.^(-0.5)

# ╔═╡ 540d8814-87f1-11eb-0b8c-23357c46f93c
u_primeinv(u) = (2 .* u).^(-2)

# ╔═╡ d2788ffe-92c4-11eb-19e7-4b41d9f9ebdd
md"

#

* We need the _inverse_ of the marginal utility function?
* Yes. Here is why. For any candidate consumption level $c_t$
$$\begin{align}
u'(c_t) & = \beta \mathbb{E} \left[ u'(c(R_{t+1})) \right] \\
u'(c_t) & = \text{RHS}
\end{align}$$
* So, if we have computed the RHS, current period optimal consumption is just
$$c_t  = \left( u' \right)^{-1}\left(\text{RHS}\right)\hspace{1cm}(\text{EEresid})$$

where $\left( u' \right)^{-1}(z)$ denotes the _inverse_ of function $u'(x)$. 

#

### Example

* Suppose tomorrow's consumption function _does_ have the kink at $R_{t+1} = 1.5$
* I.e. $c_{t+1}^*(R_{t+1}) = R_{t+1},\forall R_{t+1} \leq 1.5$
* Let $y_{t+1} = 1.5, R_t = 1$. Now try out consumption choice $c_t = 1$!

$$\begin{align}
R_{t+1} &= R_t - c_t+ y_{t+1} \\
        &= 1 - 1 + 1.5 = 1.5 \\
\Rightarrow c_{t+1}^*(1.5) &= 1.5
\end{align}$$ 

because right at the kink! 
* Then, the Euler equation says: 
$$\begin{align}
u'(c_t) & = u'(c_{t+1}) \\
c_t & = (u')^{-1}\left(u'(c_{t+1})\right) \\
c_t & = c_{t+1} = 1.5
\end{align}$$
* Well, that in turn means that $R_t - c_t = 1 - 1.5 < 0$, i.e. we would have to borrow 0.5 units of $R$ in order to satisfy that Euler Equation!

#
"

# ╔═╡ c75eed74-87ee-11eb-3e9a-3b893294baec
function EEresid(ct::Number,        # candidate current (t) consumption choice 
		         Rt::Number,        # current level of resources
				 cfunplus::Vector,  # next period's consumption *function*
		         grid::Vector,      # grid values on which cfunplus is defined
				 yvec::Vector,      # vector of values for shock
		         β::Number,         # discount factor
				 π::Number)         # prob of high y
	
	# next period resources, given candidate choice ct
	Rplus = Rt - ct .+ yvec  # a (2,1) vector: Rplus for each income level

	# get implied next period consumption from cplus
	# we add point (0,0) here to make sure that this is part of the grid.
	citp = extrapolate(
				interpolate(([0.0, grid...],), [0.0,cfunplus...], Gridded(Linear())), Interpolations.Flat())
	cplus = citp.(Rplus)
	RHS = β * [1-π  π] * u_prime(cplus) # expecte marginal utility of tomorrows consumption
	
	# euler residual: expression EEresid from above
	r = ct .- u_primeinv(RHS)
	r[1]  # array of size 1
end    

# ╔═╡ ae56f4e2-87f9-11eb-10fc-e3eda66e8a1f
md"
#
"

# ╔═╡ 17ea7f6a-87e7-11eb-2d3e-b9a1e771b08e
function policy_iter(grid,c0,u_prime,β::Float64, π, yvec)
    
    c1  = zeros(length(grid))     # next guess
    # loop over current states
    # of current resources
    for (i,r) in enumerate(grid)
		# get euler residual if we consume all resources
		res = EEresid(r,r,c0, grid, yvec,β , π)
		if res < 0
			# we consume too little today, c_t is too small for the Euler Equation.
			# could only make it bigger by borrowing next period.
			# but we cant! so really we are consuming all we have:
			c1[i] = r
		else
			# no problem here: Euler Equation holds
			# just need to find that ct that makes it zero:
			c1[i] = fzero( x-> EEresid(x,r,c0, grid, yvec, β, π) , 1e-6, r)
		end
    end
    return c1
end

# ╔═╡ bcca454c-87f9-11eb-0670-0d85db6b6e37
md"
#
"

# ╔═╡ c1e65340-87f9-11eb-3cd7-05f14edf71d2
md"
#
"

# ╔═╡ 6fbaac56-87e0-11eb-3d09-39a4fd293e88
PlutoUI.Resource("https://s3.getstickerpack.com/storage/uploads/sticker-pack/meme-pack-1/sticker_19.png?363e7ee56d4d8ad53813dae0907ef4c0&d=200x200")

# ╔═╡ a510ff40-9302-11eb-091e-6929367f6783
md"

# Infinite Time Problems

* Let's now introduce the infinite time version of this.
* For that purpose, let's rely on the well known Optimal Growth model:

$$\begin{align}
   V(k) &= \max_{0<k'<f(k)} u(f(k) - k') + \beta V(k')\\
  f(k)  & = k^\alpha\\
  k_0   & \text{ given} 
\end{align}$$

* The solution in finite time was simple by going backwards.
* Here, we need to rely on results from Functional Analysis: Remember the Contraction Mapping Theorem?
* The Contraction Mapping Theorem or the [Banache Fixed Point Theoreom](https://en.wikipedia.org/wiki/Banach_fixed-point_theorem) says there is a unique fixed point in an appropriately chosen function space. We arrive at it from an *arbitrary* starting point by just iterating on the operator $T(V)$ until $V = T(V)$
"


# ╔═╡ 06ff3932-9304-11eb-07cc-d798d56c0931
md"

# Implementing Value Function Iteration

### Checklist

1. Set parameter values
1. define a grid for state variable $k \in [0,2]$
1. initialize value function $V$
1. start iteration, repeatedly computing a new version of $V$.
1. stop if $d(V^{r},V^{r-1}) < \text{tol}$.
1. plot value and policy function 
1. report the maximum error of both wrt to analytic solution

#
"

# ╔═╡ 2e121c60-9304-11eb-29c4-afd0a289343f
begin
	alpha     = 0.65
	beta      = 0.95
	grid_max  = 2  # upper bound of capital grid
	n         = 150  # number of grid points
	N_iter    = 3000  # number of iterations
	kgrid     = 1e-2:(grid_max-1e-2)/(n-1):grid_max  # equispaced grid
	f(x) = x^alpha  # defines the production function f(k)
	tol = 1e-9
end

# ╔═╡ 3c8e560a-9304-11eb-2c45-7df67768f75b
md"

## Analytic Solution

* If we choose $u(x)=\ln(x)$, the problem has a closed form solution.
* We can use this to check accuracy of our solution.
"

# ╔═╡ 4849479a-9304-11eb-1c06-4b53a4163f85
begin 
	ab        = alpha * beta
	c1        = (log(1 - ab) + log(ab) * ab / (1 - ab)) / (1 - beta)
	c2        = alpha / (1 - ab)
	# optimal analytical values
	v_star(k) = c1 .+ c2 .* log.(k)  
	k_star(k) = ab * k.^alpha   
	c_star(k) = (1-ab) * k.^alpha  
	ufun(x) = log.(x)
end

# ╔═╡ 5984cd7a-9304-11eb-0bd4-ebac2da4f300
md"
#
"

# ╔═╡ 5d642bca-9304-11eb-2b23-dfa47b04bb22
# Bellman Operator
# inputs
# `grid`: grid of values of state variable
# `v0`: current guess of value function

# output
# `v1`: next guess of value function
# `pol`: corresponding policy function 

#takes a grid of state variables and computes the next iterate of the value function.
function bellman_operator(grid,v0)
    
    v1  = zeros(n)     # next guess
    pol = zeros(Int,n)     # policy function
    w   = zeros(n)   # temporary vector 

    # loop over current states
    # current capital
    for (i,k) in enumerate(grid)

        # loop over all possible kprime choices
        for (iprime,kprime) in enumerate(grid)
            if f(k) - kprime < 0   #check for negative consumption
                w[iprime] = -Inf
            else
                w[iprime] = ufun(f(k) - kprime) + beta * v0[iprime]
            end
        end
        # find maximal choice
        v1[i], pol[i] = findmax(w)     # stores Value und policy (index of optimal choice)
    end
    return (v1,grid[pol])   # return both value and policy function
end

# ╔═╡ 67d367c4-9304-11eb-0bfc-21b9bfec2fb9
md"
#
"

# ╔═╡ 6b33c6ca-9304-11eb-3766-fdcc42b64f2d
# VFI iterator
#
## input
# `n`: number of grid points
# output
# `v_next`: tuple with value and policy functions after `n` iterations.
function VFI(op::Function)
    v_init = zeros(n)     # initial guess
    for iter in 1:N_iter
        v_next = op(kgrid,v_init)  # returns a tuple: (v1,pol)
        # check convergence
        if maximum(abs,v_init.-v_next[1]) < tol
            verrors = maximum(abs,v_next[1].-v_star(kgrid))
            perrors = maximum(abs,v_next[2].-k_star(kgrid))
            return (v = v_next[1], p =v_next[2], errv = verrors, errp = perrors, iter = iter)
        elseif iter==N_iter
            @warn "No solution found after $iter iterations"
            return (v = v_next[1], p =v_next[2], errv = verrors, errp = perrors, iter = iter)
        end
        v_init = v_next[1]  # update guess 
    end
end



# ╔═╡ 72cc3cdc-9304-11eb-3548-95962c3513ec
md"
#
"

# ╔═╡ 776c4712-9304-11eb-1824-a14cde26b895
function plotVFI(v::NamedTuple)
    
    p = Any[]
	    # errors of both
	if eltype(v.p) == Int
		policy = kgrid[v.p]
	else
		policy = v.p
	end
    
    # value and policy functions
	if !isnan(v.v[1])
		push!(p,plot(kgrid,v.v,
            leg = false,title = "Vfun iterations: $(v.iter)",
            ylim=(-50,-30)))
		push!(p,plot(kgrid,v.v .- v_star(kgrid),
        title = "max Vfun error: $(round(v.errv,digits=3))"))
		perrors = policy .- k_star(kgrid)
	else
		push!(p,plot(title = "Vfun iterations: $(v.iter)"), plot())
		perrors = policy .- c_star(kgrid)
		
	end
    push!(p,plot(kgrid,policy,
            title = "policy function"))
    push!(p,plot(kgrid,perrors,
        title = "max policy error: $(round(v.errp,digits=3))"))

    plot(p...,layout=grid(2,2) , leg = false)
    
end

# ╔═╡ 7999a272-9304-11eb-254c-af591bae0620
plotVFI(VFI(bellman_operator))

# ╔═╡ a4ba5a96-ad59-4319-bd94-636f525853c6
function VFI_converge(op::Function,steps)
    v_init = zeros(n)     # initial guess
	pl = plot(kgrid, v_star(kgrid), color = :red, label = "true")
    for iter in 1:steps
		plot!(pl, kgrid, v_init, label = "", color = :grey)
        v_next = op(kgrid,v_init)  # returns a tuple: (v1,pol)
        # check convergence
        if maximum(abs,v_init.-v_next[1]) < tol
            verrors = maximum(abs,v_next[1].-v_star(kgrid))
            perrors = maximum(abs,v_next[2].-k_star(kgrid))
            return (v = v_next[1], p =v_next[2], errv = verrors, errp = perrors, iter = iter)
        elseif iter==N_iter
            @warn "No solution found after $iter iterations"
            return (v = v_next[1], p =v_next[2], errv = verrors, errp = perrors, iter = iter)
        end
        v_init = v_next[1]  # update guess 
    end
	pl
end

# ╔═╡ d52d3d31-e447-4120-92ae-3d4da7a02219
md"""
steps = $(@bind stps Slider(1:100, show_value = true, default = 1))
"""

# ╔═╡ c881819e-ca81-40b0-9e0f-93abcb43a2b4
VFI_converge(bellman_operator, stps)

# ╔═╡ 84e23d2a-9388-11eb-2483-3342d1683129
md"

## continous choice

* like before, now let's treat the choice dimension as a continuous variable
* This is almost identical to before with finite time.

#
"

# ╔═╡ 7e0a9d82-9304-11eb-1d3d-bb47fc55d033
function bellman_operator2(grid,v0)
    
    v1  = zeros(n)     # next guess
    pol = zeros(n)     # consumption policy function

    Interp = interpolate((collect(grid),), v0, Gridded(Linear()) ) 
    Interp = extrapolate(Interp,Interpolations.Flat())

    # loop over current states
    # of current capital
    for (i,k) in enumerate(grid)

        objective(c) = - (log.(c) + beta * Interp(f(k) - c))
        # find max of ojbective between [0,k^alpha]
        res = optimize(objective, 1e-6, f(k))  # Optim.jl
        pol[i] = f(k) - res.minimizer   # k'
        v1[i] = -res.minimum
    end
    return (v1,pol)   # return both value and policy function
end

# ╔═╡ aa891724-9388-11eb-32bb-8f8d15dc3761
md"
#
"

# ╔═╡ 597080ba-9307-11eb-2780-87474bfc5cff
plotVFI(VFI(bellman_operator2))

# ╔═╡ 6f9b2b8a-9389-11eb-2731-398e1342538a
md"
#

* Finally, policy function iteration!
* Again, remember the euler Equation here.

#
"

# ╔═╡ 710a0212-9307-11eb-2cf3-e75b9c5eddab
begin
	uprime(x) = 1.0 ./ x
	fprime(x) = alpha * x.^(alpha-1)
end

# ╔═╡ 67c4bbb8-9307-11eb-3ecb-13d7af9d899a
function policy_iter(grid,c0,u_prime,f_prime)

	c1  = zeros(length(grid))     # next guess
	pol_fun = extrapolate(interpolate((collect(grid),), c0, Gridded(Linear()) ) , Interpolations.Flat())

	# loop over current states
	# of current capital
	for (i,k) in enumerate(grid)
		objective(c) = u_prime(c) - beta * u_prime(pol_fun(f(k)-c)) * f_prime(f(k)-c)
		c1[i] = fzero(objective, 1e-10, f(k)-1e-10) 
	end
	return c1
end



# ╔═╡ d18511ea-87e7-11eb-08ce-a9176a32fbd1
function backwards_pol(grid, nperiods,β, π, yvec)
	u_prime(c) = 0.5 * c^(-0.5)
	points = length(grid)
	V = zeros(nperiods,points)
	c = zeros(nperiods,points)
	c[end,:] = collect(grid)
	V[end,:] = sqrt.(c[end,:] )  

	for it in (nperiods-1):-1:1
		c[it,:] = policy_iter(grid, c[it+1,:], u_prime, β,  π, yvec)
		# now calulate implied value function
		v_fun = extrapolate(interpolate((collect(grid),), V[it+1,:], Gridded(Linear()) ) , Interpolations.Flat())
		V[it,:] = sqrt.(c[it,:]) + β * ((1-π) * v_fun(grid .- c[it,:] .+ yvec[1]) + (π) * v_fun(grid .- c[it,:] .+ yvec[2]))
	end
	return (V,c)
end

# ╔═╡ c339ad8a-87e7-11eb-2802-0991af7b2a78
let
	V,a = backwards_pol(Rspace,nperiods,β,π,yvec)
	cg = cgrad(:viridis)
    cols = cg[range(0.0,stop=1.0,length = nperiods)]
	pa = plot(Rspace, a[1,:], xlab = "R", ylab = "Action",label = L"a_1",leg = :topleft, color = cols[1])
	for it in 2:nperiods
		plot!(pa, Rspace, a[it,:], label = L"a_{%$(it)}", color = cols[it])
	end
	pv = plot(Rspace, V[1,:], xlab = "R", ylab = "Value",label = L"V_1",leg = :bottomright, color = cols[1])
	for it in 2:nperiods
		plot!(pv, Rspace, V[it,:], label = L"V_{%$(it)}", color = cols[it])
	end
	plot(pv,pa, layout = (1,2))
end

# ╔═╡ 96a55b70-9389-11eb-13a1-3146098e9cb9
md"

#

"

# ╔═╡ 7b137330-9307-11eb-2644-390cd9c1e569
function PFI()
    c_init = kgrid
    for iter in 1:N_iter
        c_next = policy_iter(kgrid,c_init,uprime,fprime)  
        # check convergence
        if maximum(abs,c_init.-c_next) < tol
            perrors =  maximum(abs,c_next.-c_star(kgrid))
            return (v = fill(NaN,n), p =c_next, errv = 0.0, errp = perrors, iter = iter)

        elseif iter==N_iter
            warn("No solution found after $iter iterations")
            return (v = fill(NaN,n), p =c_next, errv = 0.0, errp = perrors, iter = iter)
        end
        c_init = c_next  # update guess 
    end
end

# ╔═╡ 9d4211e6-9389-11eb-03a4-559460e28c44
md"
#
"

# ╔═╡ 3b50f766-9386-11eb-2101-97a38644fda8
plotVFI(PFI())

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version = "0.34.3"

[[deps.StyledStrings]]
uuid = "f489334b-da3d-4c2e-b8f0-e476e12c162b"
version = "1.11.0"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "7.7.0+0"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
version = "1.11.0"

[[deps.TranscodingStreams]]
git-tree-sha1 = "0c45878dcfdcfa8480052b6ab162cdd138781742"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.11.3"

[[deps.Tricks]]
git-tree-sha1 = "7822b97e99a1672bfb1b49b668a6d46d58d8cbcb"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.9"

[[deps.URIs]]
git-tree-sha1 = "67db6cc7b3821e19ebe75791a9dd19c9b1188f2b"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
version = "1.11.0"

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
version = "1.11.0"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "d95fe458f26209c66a187b1114df96fd70839efd"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.21.0"
weakdeps = ["ConstructionBase", "InverseFunctions"]

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    InverseFunctionsUnitfulExt = "InverseFunctions"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "975c354fcd5f7e1ddcc1f1a23e6e091d99e99bc8"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.4"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Vulkan_Loader_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Wayland_jll", "Xorg_libX11_jll", "Xorg_libXrandr_jll", "xkbcommon_jll"]
git-tree-sha1 = "2f0486047a07670caad3a81a075d2e518acc5c59"
uuid = "a44049a8-05dd-5a78-86c9-5fde0876e88c"
version = "1.3.243+0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "7558e29847e99bc3f04d6569e82d0f5c54460703"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+1"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "93f43ab61b16ddfb2fd3bb13b3ce241cafb0e6c9"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.31.0+0"

[[deps.WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "c1a7aa6219628fcd757dede0ca95e245c5cd9511"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "1.0.0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "6a451c6f33a176150f315726eba8b92fbfdb9ae7"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.13.4+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "a54ee957f4c86b526460a720dbc882fa5edcbefc"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.41+0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "15e637a697345f6743674f1322beefbc5dcd5cfc"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.6.3+0"

[[deps.Xorg_libICE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "326b4fea307b0b39892b3e85fa451692eda8d46c"
uuid = "f67eecfb-183a-506d-b269-f58e52b52d7c"
version = "1.1.1+0"

[[deps.Xorg_libSM_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libICE_jll"]
git-tree-sha1 = "3796722887072218eabafb494a13c963209754ce"
uuid = "c834827a-8449-5923-a945-d239c165b7dd"
version = "1.2.4+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "d2d1a5c49fae4ba39983f63de6afcbea47194e85"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.6+0"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "47e45cd78224c53109495b3e324df0c37bb61fbe"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+0"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "bcd466676fef0878338c61e655629fa7bbc69d8e"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "555d1076590a6cc2fdee2ef1469451f872d8b41b"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "936081b536ae4aa65415d869287d43ef3cb576b2"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.53.0+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b70c870239dc3d7bc094eb2d6be9b73d27bef280"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.44+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

# ╔═╡ Cell order:
# ╟─d367efe3-f2e1-40b2-8bd1-47bb827424d0
# ╟─36610294-87dc-11eb-21e8-5b1dd06c6020
# ╟─5fab5e80-87ce-11eb-111a-d5288227b97c
# ╟─705e96f2-87ce-11eb-3f5a-eb6cdb8c49d4
# ╟─34403942-87d2-11eb-1172-7bd285bf7d75
# ╟─b9ed265e-87d2-11eb-1028-05b4c4ccbc74
# ╟─07992abe-87d4-11eb-0fd7-1bef7fc2ada8
# ╟─93148b06-87d2-11eb-2356-fb460acf8e66
# ╟─3d97787a-87d2-11eb-3713-ab3dad6e5e8b
# ╟─a41aff3a-87d8-11eb-00fe-e16c14870ba2
# ╟─c6ed3824-87d9-11eb-3e0a-b5a473fdb952
# ╟─51983542-87f9-11eb-226a-81fe383f0763
# ╠═6ddfc91c-87da-11eb-3b5c-dfe006576f29
# ╟─aeb2b692-87db-11eb-3c9a-114acdc3d159
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