In this notebook we will also be going over the basics of the Solow model.
Below are the packages that we will be using for this tutorial.
import Pkg
Pkg.add("LaTeXStrings")
Pkg.add("Plots")
Pkg.add("Symbolics")
Pkg.add("Zygote")
Resolving package versions... No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Project.toml` No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Manifest.toml` Resolving package versions... No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Project.toml` No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Manifest.toml` Resolving package versions... No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Project.toml` No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Manifest.toml` Resolving package versions... No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Project.toml` No Changes to `C:\Users\Dawie\Dropbox\2022\318-macro\git\Macro-318\Manifest.toml`
using LaTeXStrings
using Plots
using Symbolics
using Zygote
In class we discussed the Solow model in detail. We started with some of the following assumption,
$$ N_{t+1} = (1 + n) N_{t} $$This means that the population grows at a constant rate $n$.
Notation: The notation has changed a bit here. We are now working with time subscripts. This is more traditional in the literature and also will help with our explanations down the line.
An important consideration for this model is the behaviour of consumers with respect to saving and investment.
Consumption and savings decisions are not microfounded in the Solow model. Decisions are made in a stylised way.
The savings rate $s \in (0, 1)$ is exogenous!
Saving is a portion of income $\rightarrow S_t = s Y_{t}$
Agents consume the rest of their income $ \rightarrow C_{t} = (1 - s)Y_{t}$
Saving is a key component since it will finance investment activities in the model.
Investment equals saving in this economy $\rightarrow I_t = sY_t = S_t$
In a closed economy with no government $\rightarrow C_t = Y_t - I_t$
Next we look at the firm side of the problem. We start with the production function.
One of the most important considerations in the Solow model is the production function. The production function is given by
$$ Y_{t}\left(K_{t}, N_{t}\right) = z \cdot F\left(K_{t}, N_{t}\right) $$For this example we can assume a Cobb-Douglas production function in discrete time $(t)$ with Hicks-neutral techonology $(z)$, and with constant returns to scale where $\alpha \in (0, 1)$ is the output elasticity of capital.
$$ \begin{equation*} Y_{t}\left(K_{t}, N_{t}\right) = z \cdot F\left(K_{t}, N_{t}\right) = z \cdot \left(K_{t} ^{\alpha} N_{t}^{1-\alpha}\right) \end{equation*} $$From our notes we have that there are decreasing and positive returns to scale in each factor individually. We can observe this from the partial derivatives. The first and second derivatives with respect to capital and labor are: $$ \begin{align*} \frac{\partial Y_{t}}{\partial K} = \alpha \cdot z\left(\frac{N_{t}}{K_{t}}\right)^{1-\alpha} = \alpha \cdot \frac{Y_{t}}{K_{t}} > 0 \; \text{and} \; \frac{\partial^2 Y_{t}}{\partial K^2} = -\alpha (1-\alpha) \cdot z\left(\frac{N_{t}}{K_{t}}\right)^{1-\alpha} < 0 \\ \frac{\partial Y_{t}}{\partial N} = (1-\alpha) \cdot z\left(\frac{K_{t}}{N_{t}}\right)^{\alpha} = (1-\alpha) \cdot \frac{Y_{t}}{N_{t}} > 0 \; \text{and} \; \frac{\partial^2 Y_{t}}{\partial N^2} = -\alpha (1-\alpha) \cdot z\left(\frac{K_{t}}{N_{t}}\right)^{1-\alpha} < 0 \end{align*} $$
We know from our experience with Julia so far that we can calculate these partial derivatives using the Symbolics package.
@variables K N z α
Y = z * (K ^ α) * (N ^ (1 - α))
∂Y_∂K = Differential(K)
∂Y_∂N = Differential(N)
∂2Y_∂K2 = Differential(K) ∘ Differential(K)
∂2Y_∂N2 = Differential(N) ∘ Differential(N);
expand_derivatives(∂Y_∂K(Y))
expand_derivatives(∂Y_∂N(Y))
expand_derivatives(∂2Y_∂K2(Y))
expand_derivatives(∂2Y_∂N2(Y))
Next, we write our production function in its intensive form. This means that we write the production function in per capita terms,
$$ \begin{align*} Y_{t} &= z \cdot \left(K_{t}^{\alpha}N_{t}^{1-\alpha} \right) \\ y_{t} &= \frac{Y_{t}}{N_{t}} = z \cdot \left[ \left(\frac{K_{t}}{N_{t}} \right)^{\alpha} \left(\frac{N_{t}}{N_{t}} \right)^{1-\alpha} \right] \\ y_{t} &= z \cdot f\left(k_{t}\right) = z \cdot k_{t}^{\alpha} \end{align*} $$Next we are going to plot the intensive form of the production function, as in the textbook.
We start by plotting the two dimensional version of the production function.
K = 100
α = 0.5
k = 0:0.5:K
z = 1
function production_per_capita(k, z)
return y = z .* k .^ α
end
plot(production_per_capita(k, z), label = "z = 1", legend = :topleft, lw = 2)
┌ Info: Precompiling GR_jll [d2c73de3-f751-5644-a686-071e5b155ba9] └ @ Base loading.jl:1423
Now we will generate the same plot, but this time we will include a tangent line that provides the value for the marginal product of capital at a point. This is similar to Figure 7.12 in Chapter 7 of the textbook.
g(k; z = 1) = z .* k .^ α
a = round(g'(20), digits = 2)
b = round(g'(65), digits = 2)
gradient_line(f, x_0) = (x -> f(x_0) + f'(x_0) * (x - x_0))
plot(g, 0:1:100, legend = :bottomright, label = L"y = zf(k)", xlabel = L"$k = $capital per worker", ylabel = "\$y = \$ output per worker", lw = 2.5)
plot!(gradient_line(g, 20), 5:0.1:40, label = L"MP_K = %$a", lw = 3, alpha = 0.6, color = :black)
scatter!([20], [g(20)], label = "", markersize = 6, color = :black)
plot!(gradient_line(g, 65), 45:0.1:85, label = L"MP_K = %$b", lw = 3, alpha = 0.6, color = :red)
scatter!([65], [g(65)], label = "", markersize = 6, color = :red)
From above one can observe that the marginal product of capital (the slope of the graph) is decreasing with an increase in capital per worker. This illustrates the concavity of the function with respect to capital.
We make another assumption regarding capital. It wears out over time. This is referred to as depreciation $d$. Capital stock then changes over time according to the following equation,
$$ K_{t+1} = (1 - d)K_t + I_{t} $$Now that we have the behaviour of firms and consumers we can put everything together to retrieve the competitive equilibrium result.
One of the equilibrium conditions from the consumer side of the economy is that,
$$ Y_t = C_t + I_t $$This means that current output is equal to aggregate consumption plus aggregate investment. From our equation for the change in capital over time we can substitute the investment component as follows,
$$ K_{t+1} = (1 - d)K_{t} + Y_t - C_t $$We can then rearrange this equation to write in terms of $Y_t$,
$$ Y_{t} = C_{t} + K_{t+1} - (1 - d)K_{t} $$Next, we use $C_{t} = (1-s)Y_{t}$ from the consumer's saving and investment decisions,
$$ Y_{t} = (1-s)Y_{t} + K_{t+1} - (1 - d)K_{t} $$Finally, we rearrange and simplify,
$$ \begin{align*} Y_{t} - (1 - s)Y_{t} &= K_{t+1} - (1 - d)K_{t} \\ Y_{t}(1-1+s) & = K_{t+1} - (1 - d)K_{t} \\ K_{t+1} &= sY_{t} + (1 - d)K_{t} \end{align*} $$This means that capital stock in the future period is the quantity of aggregate savings in current period plus the capital stock left over from the current period that has not depreciated.
We can now substitute $Y$ using our definition from of the production function,
$$ K_{t+1} = sz \cdot F\left(K_{t}, N_{t}\right) + (1 - d)K_{t} $$We would like to express this equation of motion in per capita terms. We can do this by dividing through by the number of workers,
$$ \frac{K_{t+1}}{N_t} = sz \cdot \frac{F\left(K_{t}, N_{t}\right)}{N_t} + (1 - d)\frac{K_{t}}{N_t} $$Next, we multiply the left hand side by $1 = \frac{N_{t+1}}{N_{t+1}}$, which then gives us,
$$ \begin{align*} \frac{K_{t+1}}{N_t}\frac{N_{t+1}}{N_{t+1}} & = sz \cdot \frac{F\left(K_{t}, N_{t}\right)}{N_t} + (1 - d)\frac{K_{t}}{N_t} \\ \frac{K_{t+1}}{N_{t+1}}\frac{N_{t+1}}{N_{t}} & = sz \cdot F\left(\frac{K_{t}}{N_t}, \frac{N_{t}}{N_t}\right) + (1 - d)\frac{K_{t}}{N_t} \\ k_{t+1}(1 + n) & = sz \cdot f\left(k_t, 1\right) + (1 - d)k_t \\ k_{t+1} & = \frac{sz \cdot f\left(k_t\right)}{1+n} + \frac{(1 - d)k_t}{1+n} \end{align*} $$The key equation that we want to use in this tutorial is then,
$$ k_{t+1} = \frac{sz \cdot f\left(k_t\right)}{1+n} + \frac{(1 - d)k_t}{1+n} $$This equation summarises what we need to know about competitive equilibrium in the Solow growth model.
In order to find the steady state of the Solow model, we can utilise the equation of motion for capital. The steady state in this model is going to be the point where there is no change in capital over time. In other words, it is the value that maintains a constant capital stock per capita over time. We will first determine this steady state algebraically, but then in our section on difference equations we will come back to a more accurate depiction of the notion of steady state.
We are going to replicate Figure 7.13 from the textbook and indicate the steady state point on the graph. We are looking for the point where $k_{t} = k_{t+1} = k^{*}$. The value of $k^{*}$ is known as the steady state value of capital. This value will occur where the transition equation for capital intersects with the $45$ degree line (since this line reflects all the points where $k_{t+1} = k_{t}$).
We would like to know what the steady state value is going to be, so we first calculate it algebraically. Solving for $k^{*}$ from the equilibrium condition entails the following,
$$ \begin{align*} k^{*} &= \frac{sz \cdot f\left(k^{*}\right)}{1+n} + \frac{(1 - d)k^{*}}{1+n} \\ k^{*} &= \frac{sz \cdot (k^{*})^{\alpha}}{1+n} + \frac{(1 - d)k^{*}}{1+n} \\ (1 + n)k^{*} &= sz \cdot (k^{*})^{\alpha} + (1 - d)k^{*} \\ (d + n)k^{*} &= sz \cdot (k^{*})^{\alpha} \\ {k^*}^{1-\alpha} &= z \cdot \frac{s}{d + n} \\ k^* &= \left[ z \cdot \frac{s}{d + n} \right]^{\frac{1}{1-\alpha}} \end{align*} $$# parameter values (selected arbitrarily, can you think of values that make sense economically?)
s = 0.4 # savings rate at 90%! This seems quite high.
z = 1 # total factor productivity
d = 0.15 # depreciation rate
n = 0.5 # population growth rate
α = 0.4 # output elasticity of capital
# calculate k_star from the formula above
k_star = round((z .* s ./ (d .+ n)) .^ (1 / (1 - α)), digits = 2)
# equation of motion for capital
h(k; s = s, z = z, d = d, n = n) = (s .* z .* g(k)) ./ (1 .+ n) .+ ((1 .- d) .* k) ./ (1 .+ n)
# code for plotting the function -- try to read one line at a time.
capital_plot = plot(h, 0:0.001:0.75, legend = :topleft, label = L"k_{t+1} = \frac{sz \cdot f\left(k_t\right)}{1+n} + \frac{(1 - d)k_t}{1+n}", xlabel = L"k_t", ylabel = L"k_{t+1}", lw = 2.5, color = :blue, xticks = ([k_star], [L"k^{*} = %$k_star"]), y_ticks = ([k_star], [L"k^{*} = %$k_star"]))
plot!(capital_plot, x -> x, lw = 2, color =:black, lab = L"k_{t+1} = k_{t}") # 45 degree line plot added
scatter!([k_star], [k_star], label = "", markersize = 6, color = :black) # point of intersection
We see from the graph above that the two lines intersect at the point where $k^{*} = 0.45$. The value will change if we select a different parameterisation of the model.
Now we can move on to plotting Figure 7.14 from the textbook. From this graph we can easily determine where the steady state value is. This graph basically has two components. First, there is an investment curve $szf(k^{*})$. Second, we have a curve that captures the reasons why capital might decrease. There are two reasons in this model why capital might decrease, depreciation and dilution. Depreciation accurs via the depreciation rate and dilution through the population growth rate. Capital is diluted among the population if the population grows faster than the capital growth rate. In the literature, the line $(n + d)k^{*}$ is known as the break-even line. It shows the level of investment needed to compensate for depreciation and population growth.
From our steady state equations above we had that,
$$ (d + n)k^{*} = sz \cdot f(k^{*}) $$This solves for the steady state capital stock per worker. Next we replicate Figure 7.14 using this equation.
y_star = z .* (k_star .^ α)
i_star = s .* y_star
c_star = y_star - i_star
break_even(k; n = n, d = d) = (n .+ d) .* k
inv_curve(k; s = s, z = z) = s .* z .* g(k)
inv = plot(inv_curve, 0:0.01:1.2, leg = :topleft, lw = 2, lab = L"szf(k^{*})", color = :blue, alpha = 0.7)
break_crv = plot!(inv, break_even, lw = 2, lab = L"(n + d)k^{*}", color = :black, alpha = 0.7)
plot!(break_crv, g, lw = 2, lab = L"zf(k^{*})", color = :red, alpha = 0.5)
vline!([k_star], lab = "", color = :black, alpha = 0.5, ls = :dash, xticks = ([k_star], [L"k^{*}"]))
hline!([y_star], lab = "", color = :black, alpha = 0.5, ls = :dash, yticks = ([round(y_star, digits = 2), round(i_star, digits = 2)], [L"y^{*}", L"i^{*}"]))
hline!([i_star], lab = "", color = :black, alpha = 0.5, ls = :dash)
scatter!([k_star], [i_star], label = "", markersize = 6, color = :black)