solowPy

This notebook demonstrates the functionality of the solowPy python package which is available both as a standalone package and as part of the QuantEcon library. The code, which was developed with funding from the Scottish Graduate Programme in Economics (SGPE) and the Scottish Institute for Research in Economics (SIRE), has found use in research and teaching at the Institute for New Economic Thinking (INET) at the Oxford Martin School it is hoped that others will find these notebooks useful for their own purposes.

N.B. If you are new to the Jupyter notebook environment, to evaluate one of the cells below press <shift>+enter

0. Motivation

The typical economics graduate student places great faith in the analytical mathematical tools that he or she was taught as an undergraduate. In particular this student is likely under the impression that virtually all economic models have closed-form solutions. At best the typical student believes that if he or she were to encounter an economic model without a closed-form solution, then simplifying assumptions could be made that would render the model analytically tractable without sacrificing important economic content.

The typical economics student is, of course, wrong about general existence of closed-form solutions to economic models. In fact the opposite is true: most economic models, particular dynamic, non-linear models with meaningful constraints (i.e., most any interesting model) will fail to have an analytic solution. I the objective of this notebook is to demonstrate this fact and thereby motivate the use of numerical methods in economics more generally using the Solow model of economic growth.

Economics graduate students are very familiar with the Solow growth model. For many students, the Solow model will have been one of the first macroeconomic models taught to them as undergraduates. Indeed, Greg Mankiw's Macroeconomics, the dominant macroeconomics textbook for first and second year undergraduates, devotes two full chapters to motivating and deriving the Solow model. The first few chapters of David Romer's Advanced Macroeconomics, one of the most widely used final year undergraduate and first-year graduate macroeconomics textbook, are also devoted to the Solow growth model and its descendants.

0.1 The basic Solow growth model

The Solow model of economic growth focuses on the behavior of four variables: output, $Y$, capital, $K$, labor, $L$, and knowledge (or technology or the "effectiveness of labor"), $A$. At each point in time the economy has some amounts of capital, labor, and knowledge that can be combined to produce output according to some production function, $F$.

$$ Y(t) = F(K(t), A(t)L(t)) $$

where $t$ denotes time.

The evolution of the inputs to production

The initial levels of capital, $K_0$, labor, $L_0$, and technology, $A_0$, are taken as given. Labor and technology are assumed to grow at constant rates:

\begin{align} \dot{A}(t) = gA(t) \dot{L}(t) = nL(t) \end{align}

where the rate of technological progrss, $g$, and the population growth rate, $n$, are exogenous parameters.

Output is divided between consumption and investment. The fraction of output devoted to investment, $0 < s < 1$, is exogenous and constant. One unit of output devoted to investment yields one unit of new capital. Capital is assumed to decpreciate at a rate $0\le \delta$. Thus aggregate capital stock evolves according to

$$ \dot{K}(t) = sY(t) - \delta K(t).$$

Although no restrictions are placed on the rates of technological progress and population growth, the sum of $g$, $n$, and $\delta$ is assumed to be positive.

The dynamics of the model

Because the economy is growing over time (due to exogenous technological progress and population growth) it is useful to focus on the behavior of capital stock per unit of effective labor

$$ k \equiv \frac{K}{AL}. $$

Applying the chain rule to the equation of motion for capital stock yields (after a bit of algebra!) an equation of motion for capital stock per unit of effective labor.

$$ \dot{k}(t) = s f(k) - (g + n + \delta)k(t) \tag {0.1.1} $$

That's it! The Solow model of economic growth reduced to a single non-linear ordinary differential equation describing the time evolution of capital stock (per unit effective labor), $k(t)$.

The intensive form of the production function $f$ is typically assumed to be to be strictly concave with

$$ f(0) = 0,\ lim_{k\rightarrow 0}\ f' = \infty,\ lim_{k\rightarrow \infty}\ f' = 0. \tag{0.1.2} $$

These additional restrictions on $f$, often referred to as Inada conditions, are actually stronger than required for the key results of the Solow model to hold. A common choice for the function $f$ which does satisfies the above conditions is known as the Cobb-Douglas production function

$$ f(k(t)) = k(t)^{\alpha}. $$

Assuming a Cobb-Douglas functional form for $f$ also makes the model analytically tractable (and thus contributes to the typical economics student's belief that all such models "must" have an analytic solution). Sato 1963 showed that the solution to the model under the assumption of Cobb-Douglas production is

$$ k(t) = \Bigg[\bigg(\frac{s}{n+g+\delta}\bigg)\bigg(1 - e^{-(n+g+\delta)(1-\alpha)t}\bigg)+ k_0^{1-\alpha}e^{-(n+g+\delta)(1-\alpha)t}\Bigg]^{\frac{1}{1-\alpha}}. \tag{0.1.3} $$

A notable property of the Solow model with Cobb-Douglas production is that the model predicts that the shares of real income going to capital and labor should be constant. Denoting capital's share of income as $\alpha_K(k)$, the model predicts that

$$ \alpha_K(k(t)) \equiv \frac{\partial \ln f(k(t))}{\partial \ln k(t)} = \alpha \tag{0.1.4} $$

Note that the prediction is that factor shares are constant along both the balanced growth path and during the disequilibrium transient (i.e., the period in which $k(t)$ is varying). We can test this implication of the model using data from the Penn World Tables (PWT).

From the above figure it is clear that the prediction of constant factor shares is strongly at odds with the empirical data for most countries. Labor's share of real GDP has been declining, on average, for much of the post-war period. For many countries, such as India, China, and South Korea, the fall in labor's share has been dramatic. Note also that the observed trends in factor shares are inconsistent with an economy being on its long-run balanced growth path.

0.2 A more general Solow growth model

While the data clearly reject the Solow model with Cobb-Douglas production, they are not inconsistent with the Solow model in general. A simple generalization of the Cobb-Douglas production function, known as the constant elasticity of substitution (CES) function:

$$ f(k(t)) = \bigg[\alpha k(t)^{\rho} + (1-\alpha)\bigg]^{\frac{1}{\rho}} $$

where $-\infty < \rho < 1$ is the elasticity of substitution between capital and effective labor in production is capable of generating the variable factor shares observed in the data. Note that

$$ \lim_{\rho\rightarrow 0} f(k(t)) = k(t)^{\alpha} $$

and thus the CES production function nests the Cobb-Douglas functional form as a special case.

To see that the CES production function also generates variable factor shares note that

$$ \alpha_K(k(t)) \equiv \frac{\partial \ln f(k(t))}{\partial \ln k(t)} = \frac{\alpha k(t)^{\rho}}{\alpha k(t)^{\rho} + (1 - \alpha)} $$

which varies with $k(t)$.

This seemingly simple generalization of the Cobb-Douglas production function, which is necessary in order for the Solow model generate variable factor shares, an economically important feature of the post-war growth experience in most countries, renders the Solow model analytically intractable. To make progress solving a Solow growth model with CES production one needs to resort to computational methods.

1. Creating an instance of the solow.Model class

We begin by creating an instance of the solow.Model class in the IPython notebook. As always, it is a good idea to read the docstrings...

From the docsting we see that in order to create an instance of the model we need to specify two primitives: the extensive production function, $F$, and a dictionary of model parameters.

More details on on how to create instances of the solow.Model class can be found in the Getting started notebook.

2. Finding the steady state

Traditionally, most analysis of the Solow model focuses almost excusively on the long run steady state of the model. Recall that the steady state of the Solow model is the value of capital stock (per unit effective labor) that solves

$$ 0 = sf(k^*) - (g + n + \delta)k^*. \tag{2.0.1} $$

In words: in the long-run, capital stock (per unit effective labor) converges to the value that balances actual investment,

$$sf(k),$$

with effective depreciation,

$$(g + n + \delta)k.$$

Given the assumption made about the aggregate production technology, $F$, and its intensive form, $f$, there is always a unique value $k^* >0$ satisfying equation 2.0.1.

Although it is trivial to derive an analytic expression for the long-run equilibrium of the Solow model for most intensive production functions, the Solow model serves as a good illustrative case for various numerical methods for solving non-linear equations.

The solowpy.Model.find_steady_state method provides a simple interface to the various 1D root finding routines available in scipy.optimize and uses them to solve the non-linear equation 2.0.1. To see the list of currently supported methods, check out the docstring for the Model.find_steady_state method...

More details on on how to the various methods of the solow.Model class for finding the model's steady state can be found in the accompanying Finding the steady state notebook.

3. Graphical analysis using Matplotlib and IPython widgets

Graphical analysis is an important pedagogic and research technique for understanding the behavior of the Solow (or really any!) model and as such the solow.Model class has a number of useful, built-in plotting methods.

Static example: the classic Solow diagram

Interactive example: the classic Solow diagram

There are number of additional plotting methods available (all of which can be turned into interactive plots using IPython widgets). See the Graphical analysis notebook for more details and examples.

4. Solving the Solow model

Solving the Solow model requires efficiently and accurately approximating the solution to a non-linear ordinary differential equation (ODE) with a given initial condition (i.e., an non-linear initial value problem).

4.1 Solow model as an initial value problem

The Solow model can be formulated as an initial value problem (IVP) as follows.

$$ \dot{k}(t) = sf(k(t)) - (g + n + \delta)k(t),\ t\ge t_0,\ k(t_0) = k_0 \tag{4.1.0} $$

The quantecon library has its own module quantecon.ivp for solving initial value problems of this form using finite difference methods. Upon creation of our instance of the solow.Model class, an instance of the quantecon.ivp.IVP class was created and stored as an attribute of our model...

...meaning that we can solve this initial value problem by applying the solve method of the ivp attribute!

We can plot the finite-difference approximation of the solution as follows...

Finite-difference methods only return a discrete approximation to the continuous function $k(t)$. To get a continuous approximation of the solution we can combined finite-difference methods with B-spline interpolation using the interpolate method of the ivp attribute.

We can graphically compare the discrete, finite-difference approximation with the continuous, B-spline approximation as follows.

Accuracy of our numerical methods

When doing numerical work it is important to understand the accuracy of the methods that you are using to approximate the solution to your model. Typically one assesses the accuracy of a solution method by computing and evaluating some residual function:

$$ R(k; \theta) = sf(\hat{k}(\theta)) - (g + n + \delta)\hat{k}(\theta) - \dot{\hat{k}}(\theta) $$

where $\hat{k}(\theta)$ is our computed solution to the original differential equation. We can assess the accuracy of our finite-difference methods by plotting the residual function for our approximate solution using the compute_residual method of the ivp attribute.

We can then plot the residual as follows. Our approximation is accurate so long as the residual is everywhere "close" to zero.

For more details behind the numerical methods used in this section see the the Solving the Solow model notebook.

5. Impulse response functions

Impulse response functions (IRFs) are a standard tool for analyzing the short run dynamics of dynamic macroeconomic models, such as the Solow growth model, in response to an exogenous shock. The solow.impulse_response.ImpulseResponse class has several attributes and methods for generating and analyzing impulse response functions.

Example: Impact of a change in the savings rate

One can analyze the impact of a doubling of the savings rate on model variables as follows.

Example: Interactive impulse reponse functions

Using IPython widgets makes it extremely easy to analyze the various impulse response functions.

For more details and examples see the accompanying Impulse response function notebook.

6. The Solow model, declining labor's share, and secular stagnation

Recently there has been much discussion about the reasons for the declining labor share in the U.S. (and in other developing countries). Elsby (2013) has a nice paper that looks at the decline of labor share in the U.S.; as well as much debate about whether or not developed economes are experiencing some sort of secular stagnation (i.e., negative trend in per capita income/output) more generally. In this section I see to what extent we can simultaneously capture a decline in labor's share of income as well as a decline in the growth rate of per capita income in a Solow growth model.

See how well you can get a Solow model to account for trends in factor shares and growth rates of per capita output for your favorite country by replacing the ISO3 country code!