5. Economic Growth

Key Learning Points

• Revisit our first encounter with economic dynamics—Solow-Swan Model
• Appreciate relevance of Solow-Swan in accounting for observed empirical regularities in long run economic data
• Use the Solow-Swan Model as a starting point for more modern extensions later
• Learn basic techniques of dynamic economics including scientific computing
• Ac, Ch. 2-3; See the growth chapters in Charles I. Jones’ undergraduate textbook for a refresher.

Let’s fast-forward a bit from Keynes’ world of (short-run) macroeconomics, for now, to the 1950s. We will revisit the prototype model of economic growth, due to Robert Solow and Trevor Swan. This framework later served as the backbone of modern business-cycle and macroeconomic policy modelling. In fact, if you looked at the flow of modern undergraduate textbooks, e.g., [Jo2010], and also the trend in modern macroeconomic and policy research, you’d see this chronologically nonlinear, but logically linear, pattern in the history of macroeconomic thinking and policy modelling:

• Solow model (and its other endogenous growth variants) as the basis for the long run growth path of economies.
• Aggregate shocks (New Classical) and policy-relevant short run market frictions (Keynesian, New Keynesian, Non-Walrasians or New Monetarists) as the source of temporary departures from the long run growth path—i.e., business cycle fluctuations.
• All of the above plus individual shocks to preferences, technologies, life-cycle problems, or incomplete market trading opportunities, as the source of agent heterogeneity, wealth and consumption inequality over the long run and/or over the business cycle. (This opens up new doors for the analysis of redistributive policies across individuals within a period and across time and generations.)

Note

The recent speech of Janet Yellen (US FRB President), entitled “Macroeconomic Research After the Crisis”, is a good synopsis of what’s important in macroeconomics today.

So let’s come back to the basics again: The long run growth model and its implied dynamic outcomes. Let’s motivate our study by recalling some empirical motivations.

5.1. Empirical Regularities

Around the time Trevor Swan (then at the ANU) and Robert Solow (MIT) were thinking about a theory of growth, Nicholas Kaldor [Ka1961] documented these facts on long-run data (for industrialized countries):

• The shares of labor and physical capital in GDP are nearly constant
• The ratio of physical capital to output is nearly constant
• The rate of return to capital (or the real interest rate) is nearly constant
• Physical capital per worker grows over time
• Per capita output grows over time, and its growth rate does not tend to diminish
• The growth rate of output per worker differs substantially across countries, but the rates tend to converge

[Ka1961]

Kaldor, Nicholas, “Capital Accumulation and Economic Growth,” in F.A. Lutz and D.C. Hague, eds., The Theory of Capital, St. Martins Press, 1961, pp. 177–222.

See also an updated version of Kaldor’s exercise by Chad Jones [Jo2010] which also deals with new facts on human capital, R&D, institutions and etc.

[Jo2010]

(1, 2) Charles I. Jones, “The New Kaldor Facts: Ideas, Institutions, Population, and Human Capital” (with Paul Romer) American Economic Journal: Macroeconomics, January 2010.

5.2. Population Growth

Let’s tame this creature. I’m pretty sure many of you have learned about this model ad nauseum. But let’s do this once more!

First, we list down the basic notations in a version of Solow-Swan:

• Time is denumerable. We index each date/period by \(t \in \mathbb{N} := \{ 0, 1, ... \}\).
• The stock of population/workers at date \(t \in \mathbb{N}\) is \(L_{t}\). Without loss, assume \(L_{0} = 1\). Population grows at constant rate \(n \geq 0\).
• The measure \(L_{t}\) of identical agents work, consume (\(C_{t}\)), saves/invests (\(I_{t}\)) and produces a single good (\(Y_{t}\)).
• The economy is closed.

5.2.1. Technology

There is a production technology that converts capital stock and labor into output:

(1)\[Y_{t} = F \left( K_{t}, L_{t} \right)\]

We assume available capital at the start of date \(t \in \mathbb{N}\) is used for producing output \(Y_{t}\) and in that process a fraction \(\delta\) of capital disappears or wears out. There is a technology that converts the remaining proportion \((1-\delta) \in (0,1)\) of date-\(t\) initial capital stock (\(K_{t}\)) and investment flow \(I_{t}\) into next period productive capital, \(K_{t+1}\):

(2)\[K_{t+1} = (1 - \delta) K_{t} + I_{t}\]

5.2.2. Saving

The agents consume according to a rule of thumb:

(3)\[C_{t} = (1 - s) Y_{t}\]

The parameter \((1-s) \in (0,1)\) is called the propensity to consume.

The agents are not wasteful; so what is not consumed must be saved:

(4)\[S_{t} = s Y_{t}\]

5.2.3. Characterizing Solution

Since the economy is closed, all savings must be invested, and the only investment opportunity is into capital formation. Recall the circular flow diagram in first-year macroeconomics? Here we have:

(5)\[S_{t} = I_{t}\]

So there we have it. Combine (5), (4) and (1) into (2), and we have a description of the Solow-Swan growth dynamics:

(6)\[K_{t+1} = (1 - \delta) K_{t} + s F( K_{t}, L_{t} ).\]

Don’t forget the description of population dynamics:

(7)\[L_{t+1} = (1+n)L_{t}\]

So together, given some initial states \((K_{0}, L_{0})\), the difference equations (6) and (7) characterize the trajectory of the Solow-Swan model economy, \(\tau (K_{0}, L_{0}) := \{ ( K_{t+1}, L_{t+1})(K_{0}, L_{0}) \}_{t \in \mathbb{N}}\).

5.2.4. Technology and equilibrium

It turns out that the solution (trajectory) of the Solow-Swan economy depends quite a lot on the assumption on the function \(F: \mathbb{R}_{+}^{2} \rightarrow \mathbb{R}_{+}\). So let’s put some more structure on it.

We assume that the function \(F\) has these properties:

1. \(F\) is twice-continuously differentiable on the set \(\mathbb{R}_{+}^{2}\).

2. \(F\) is homogeneous of degree one in its inputs. (What is another economics jargon for this mathematical property?)

3. The value of \(F\) is increasing in each input at a decreasing rate. How would you formalize these properties in terms of derivative functions of \(F\)?

4. \(F\) satisifies the Inada conditions:

   \[\begin{split}\qquad \lim_{a \searrow 0} \frac{\partial F(a,b)}{\partial a} = +\infty \\ \qquad \lim_{a \nearrow +\infty} \frac{\partial F(a,b)}{\partial a} = 0 \\ \qquad \lim_{b \searrow 0} \frac{\partial F(a,b)}{\partial b} = +\infty \\ \qquad \lim_{b \nearrow +\infty} \frac{\partial F(a,b)}{\partial b} = 0\end{split}\]

Exercise Technology and equilibrium (1)

1. An example of \(F\) that satisfies these assumptions is the Cobb-Douglas production function, where \(F(K,L) = K^{\alpha} L^{1-\alpha}\) and \(\alpha \in (0,1)\). If the real rate of return on renting capital (labor) is equal to the marginal product of capital (labor), show that in this instance, capital income as a share of total production is \(\alpha\). Likewise, show that labor’s share is \(1-\alpha\).
2. Show that these assumptions imply that if at least one input is zero then output is also zero. (“It takes two to Tango” as it were!)

Now given the assumption that \(F\) is homogeneous of degree one, we have the following property:

\[F(K, L) \times \frac{1}{L} = F \left( \frac{K}{L}, 1 \right) := f(k),\]

where \(k := K/L\). Using this fact, do the following exercise:

Exercise Technology and equilibrium (2)

1. Show that the two difference equations (with two state variables), (6) and (7), can be re-written in terms of a scalar state variable \(k_{t}\) as

(8)\[k_{t+1} = g(k_{t}) := \frac{(1-\delta) k_{t} + s f(k_{t})}{1+n}.\]

Observe that the function \(g : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) is a self-map. It takes points in the set of possible capital stocks back into the same set. Conventionally, we write \(g^{t}(k)\) to mean the value of a \(t\)-fold application of the map \(g\) on the point \(k\); or equivalently, this is written out as the value of the composite function \(g \circ g^{t-1} (\cdot)\) at the point \(k\), where \(t \geq 1\) and \(g^{0}(k) = k\).

We define a deterministic steady-state point, \(k^{\ast}\), to be an outcome such that

(9)\[k^{\ast} = g(k^{\ast}).\]

That is, it is a fixed point of the mapping \(g\). In other words, if the economy is at such a point \(k^{\ast}\), and repeated application of the mapping \(g^{t}(k^{\ast})\) for any \(t\geq 1\) results in again \(k^{\ast}\), then we say the economy is at a (deterministic) steady state. Where the context is understood below, we will use the term steady state in lieu of deterministic steady state.

The next exercise requires a bit of high-school mathematical analysis on the real line.

Exercise Technology and equilibrium (3)

1. Prove that given the restrictions on \(F\) above, there exists a unique steady state equilibrium in terms of the sufficient state variable \(k\) that is nontrivial (i.e. there is positive production).

2. We say that the (scalar) dynamical system (8) has a fixed point \(k^{\ast} \in (0, \infty)\) that is stable if for an \(\varepsilon > 0\) there is some \(\delta \in (0,\varepsilon)\) such that for all \(T \geq t \in \mathbb{N}\), \(| g^{t}(k_{0}) - k^{\ast}| < \delta\) implies that \(| g^{T}(k_{0}) - k^{\ast}| < \varepsilon\). A more restricted notion of stability is as follows: We say that the state \(k^{\ast}\) is an asymptotically stable fixed point of the (scalar) dynamical system (8) if it is stable, and, there is a \(\delta > 0\) such that if \(| g^{t}(k_{0}) - k^{\ast} | < \delta\) for any \(t \in \mathbb{N}\), then \(| g^{T}(k_{0}) - k^{\ast} | \rightarrow 0\) as \(T \rightarrow \infty\). Can you try verbalizing these definitions, using words your sixteen-year-old self would have understood?

3. Argue that the self-map in (8) is increasing. Using the definitions of stability above, prove that the nontrivial steady state equilibrium is asymptotically stable and that it is unique.

4. Now verify that the unique nontrivial steady state \(k^{\ast} \in (0,+\infty)\) solves

   \[( n + \delta ) k^{\ast} = s f \left( k^{\ast} \right).\]

5. Can you explain in words what the last condition says? (Be mindful of the economic interpretation here.) What about another \(k^{\ast \prime} = 0\)? Is this also a steady state of the economy?

6. Can you construct a counterexample of \(F\) where the steady-state equilibrium is not unique?

Now let’s make use of the property that \(F\) is homogeneous of degree one in the following warm-up exercises. The purpose of these exercises is for developing the computational side of our skillset, so you are strongly encouraged to work through them carefully.

Exercise Technology and equilibrium (4)

1. Outline an algorithm for solving for the trajectory \(\tau (K_{0}, L_{0})\).

2. You can actually solve for \(\tau (K_{0}, L_{0})\) given parameter values by hand (maybe with the help of a calculator or spreadsheet). Try “coding” this up in a spreadsheet software. All we need is to instantiate the production function \((K,L) \mapsto F(K,L)\). Let’s pick a Cobb-Douglas example: \(F(K,L) = K^{\alpha}L^{1-\alpha}\). Pick some arbitrary numbers for the parameters for now: \(\alpha = 1/3\), \(\delta = 0.10\), \(n = 0.015\), and \(s=0.15\).

3. Now, let’s do this instead using a programming language like Python.

4. Once you get the Python example running, try different parameter values. Provide economic explanations disciplined by the Solow-Swan model structure for:

   • What happens if \(s\) is increased?
   • What happens if \(\alpha\) is bigger?
   • What happens if \(\delta\) is bigger?

5.3. Decentralized Economy

Now let’s return to the Solow-Swan model. Notice how we had a story as if there were a single agent/planner for the whole economy? Or equivalently, economists call this a centralized problem. In most countries, the economy is decentralized and allocations of resources are somehow coordinated through some market pricing mechanism(s). Let’s see how that looks like in this artificial economy.

Now consider the same model environment but with a different structure of ownership:

• There are three markets: final good, capital rental and labor rental markets
• Households own the initial capital stock and supply all their labor endowment to firms
• A continuum of identical and perfectly competitive firms (in both output and input markets) rent capital and labor services from households
• The representative firm maximizes profits—i.e. the difference between its total revenue and total cost

Without loss, let’s normalize the absolute price of the final good to unity. So then, we can let \(r\) and \(w\), respectively, denote the relative prices of renting capital and labor determined in the competitive input markets. These relative prices are sometimes called the real return on capital and the real wage, respectively.

Note

A Matter of Accounting

If the rental rate \(r_{t}\) is what the competitive firm faces in the capital rental market, then from the household/capital-owner’s accounting perspective, each additional unit of capital rented out should return a rate \(r^{h}_{t}\) over and above the depreciation rate of capital (\(\delta\)), the latter being a consequence of use in production over date \(t\). So we have:

\[r_{t} = r_{t}^{h} + \delta.\]

Exercise Decentralized Economy (1)

1. Write down the profit function of the firm in terms of claims to units of the final good.

2. Show that the first order conditions characterizing the firm’s optimal demand for capital and labor services are, respectively:

   \[\begin{split}r_{t} = \frac{\partial F(K_{t}, L_{t})}{\partial K_{t}}, \\ w_{t} = \frac{\partial F(K_{t}, L_{t})}{\partial L_{t}}.\end{split}\]

3. Define a competitive equilibrium for this economy. This will be an infinite sequence of allocations and relative prices that must satisfy certain restrictions. Spell them all out clearly.

4. National Accounting done Three Ways: Show that in a competitive equilibrium the household’s total income must be equal to the total production of the economy. In turn these must all be equal to the total expenditure demand in the economy.

5. Verify that in competitive equilibrium the firm earns zero economic profits.

6. Can you modify your Python programs earlier to now incorporate this more general version of the Solow-Swan model?

5.4. Exogenous Growth

Consider a variation now where the production technology is represented by

(10)\[F(K_{t}, A_{t}L_{t}).\]

The new variable, \(A_{t}\) is assumed exogenous and is often called Harrod-neutral or labor augmenting technical progresss.

Assume

(11)\[A_{t+1} = (1 + g) A_{t};\]

and \(A_{0}\) is known.

For there to exist a steady state fixed point in this model’s competitive equilibrium, we will have to work with a change of variables. Define original level \(X_{t}\) of a variable to be in efficiency units by \(\tilde{x} := X_{t}/A_{t}L_{t}\), where \(X \in \{ C, I, Y, K \}\).

Exercise Exogenous Growth (1)

Assume again \(F\) is Cobb-Douglas with \(\alpha \in (0,1)\).

1. Show that the competitive equilibrium you defined just now can be distilled into a single difference equation in terms of capital per efficiency units of labor \(\tilde{k}\):

   (12)\[\tilde{k}_{t+1} = \frac{ (1-\delta) \tilde{k}_{t} + s f\left(\tilde{k}_{t}\right) }{(1+n)(1+g)}; \qquad f(\tilde{k}) := \tilde{k}^{\alpha}.\]

2. Show that there is a unique non-trivial fixed point representing the transformed model’s steady state equilibrium indexed by:

   (13)\[\tilde{k}^{\ast} = \left( \frac{s}{(1+n)(1+g)-(1-\delta)}\right)^{\frac{1}{1-\alpha}}.\]

3. Verify that this model, along the steady state equilibrium path, can address all six of the stylized facts of growth in industrialized nations, listed at the beginning of this chapter. (This is a longer exercise worth doing!)

5.5. Golden Rule

Notice that in a steady state, sometimes called a balanced growth path, we have capital in efficiency units in equation (13) being constant. But it is a function of the parameter \(s\), inter alia. So let’s make explicit this dependency for the purposes of our study here and write \(\tilde{k}^{\ast} \equiv \tilde{k}^{\ast}(s)\).

Here’s a normative economics question: Suppose a planner outside of this model competitive equilibrium were able to manipulate household’s saving behavior—by changing the parameter \(s\). Since we are talking about normative issues, you might wonder, in public finance or welfare economics one typically thinks in terms of a measure of agents’ welfare or utility representations. In the Solow-Swan model, these objects are not defined. So our notion of “what ought to be better or best” will be measured in terms of consumption. In particular, we ask: What is the steady-state consumption-maximizing level of the savings rate, \(s_{gold}\)?

Exercise Golden Rule (1)

1. Assume our friendly Cobb-Douglas function again for production. Show that the golden-rule savings rate is \(s_{gold} = \alpha\) in this example.
2. Dynamic (In)efficiency: Explain what happens to the balanced-growth path consumption if \(s > s_{gold}\)? Explain what happens to the balanced-growth path consumption if \(s < s_{gold}\)?

Note

Later, when we deal with the topic of optimal growth, we will revisit this exercise again to compare with the implications from the optimal growth model.