models of production

The Solow Model is an economic model of production that incorporates the incorporates the idea of capital accumulation. Based on the Cobb-Douglas production function, the Solow Model describes production as follows: \[Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}\] With:

• \(\bar{A}\): total factor productivity (TFP)
• \(\alpha\): capital's share of output—usually \(1/3\) based on empirical data

In this simple model, the following statements describe the economy:

1. Output is either saved or consumed; in other words, savings equals investment
2. Capital accumulates according to investment \(I_t\) and depreciation \(\bar{d}\), beginning with \(K_0\) (often called the Law of Capital Motion)
3. Labor \(L_t\) is time-independent
4. A savings rate \(\bar{s}\) describes the invested portion of total output

Including the production function, these four ideas encapsulate the Solow Model:

Visualizing the model, namely output as a function of capital, provides helpful intuition before solving it.

Letting \((L_t,\alpha)=(\bar{L}, \frac{1}{3})\), it follows that \(Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}\). Utilizing this simplification and its graphical representation below, output is clearly characterized by the cube root of capital:

When investment is completely disincentivized by depreciation (in other words, \(sY_t=\bar{d}K_t\)), the economy equilibrates at a so-called "steady-state" with equilibrium \((K_t,Y_t)=(K_t^*,Y_t^*)\).

Using this equilibrium condition, it follows that: \[Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha} \rightarrow \bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}\] \[\rightarrow K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}\] \[\rightarrow Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}\]

Thus, the equilibrium intensive form (output per worker) of both capital and output are summarized as follows: \[(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}}) =((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}, \bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})\]

Using both mathematical intuition and manipulating the visualization above, we find that:

• \(\bar{A}\) has a positive relationship with steady-state output
• Capital is influenced by workforce size, TFP, and savings rate
• Capital output share's \(\alpha\) impact on output is twofold:
  1. Directly through capital quantity
  2. Indirectly through TFP
• Large deviations in capital from steady-state \(K^*\) induce net investments of larger magnitude, leading to an accelerated reversion to the steady-state
• Economies stagnate at the steady-state \((K^*,Y^*)\)—this model provides no avenues for long-run growth.

Lastly (and perhaps most importantly), exogenous parameters \(\bar{s}, \bar{d}\), and \(\bar{A}\) all have immense ramifications on economic status. For example, comparing the difference in country \(C_1\)'s output versus \(C_2\)'s using the Solow Model, we find that a difference in economic performance can only be explained by these factors: \[ \frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha} \]

We see that TFP is more important in explaining the differences in per capital output (\(\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)\)). Notably, the Solow Model does not give any insights into how to alter the most important predictor of output, TFP.

How, then, can we address these shortcomings?

The Romer Model provides an answer by both modeling ideas \(A_t\) (analagous to TFP in the Solow model) endogenously and utilizing them to provide a justification for sustained long-run growth.

The Model divides the world into two parts:

• Objects: finite resources, like capital and labor in the Solow Model
• Ideas: infinite, non-rivalrous items leveraged in production (note that ideas may be excludable, though)

The Romer Models' production function can be modelled as: \[Y_t=F(A_t,L_{yt})=A_tL_{yt}\] With:

• \(A_t\): the amount of ideas \(A\) in period \(t\)
• \(L_{yt}\): the population working on production-facing (output-driving) tasks

Assuming \(L_t=\bar{L}\) people work in the economy, a proportion \(\bar{l}\) of the population focuses on making ideas: \(L_{at}=\bar{l}\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\).

Further, this economy garners ideas with time at rate \(\bar{z}\): the "speed of ideas". Now, we can describe the quantity of ideas tomorrow as function of those of today: the Law of Ideal Motion (I made that up). \[A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta A_{t+1}=\bar{z}A_tL_{at}\]

Analagously to capital in the solow model, ideas begin in the economy with some \(\bar{A}_0\gt0\) and grow at an exponential rate. At its core, this is because ideas are non-rivalrous; more ideas bring about more ideas.

Finally, we have a model:

A visualization of the Romer Model shows that the economy grows exponentially—production knows no bounds (ceteris parbibus, of course). A graph of \(log_{10}(Y_t)\) can be seen below:

Playing with the sliders, this graph may seem underwhelming in comparison to the Solow Model. However, on a logarithmic scale, small changes in the parameters lead to massive changes in the growth rate of ideas and economices:

To find the output in terms of exogenous parameters, first note that \[L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}\]

Now, all that remains is to find ideas \(A_t\). It is assumed that ideas grow at some rate \(g_A\): \[A_t=A_0(1+g_A)^t\]

Using the growth rate formula, we find: \[g_A=\frac{\Delta A_{t+1}-A_t}{A_t}=\frac{A_t+\bar{z}A_tL_{at}-A_t}{A_t}=\bar{z}L_{at}=\bar{z}\bar{l}\bar{L}\]

Thus, ideas \(A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t\). Finally, output can be solved the production function: \[Y_t=A_t L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\]

We see the Romer model exhibits long-run growth because ideas have non-diminishing returns due to their nonrival nature. In this model, capital and income eventually slow but ideas continue to yield increasing, unrestricted returns.

Further, all economy continually and perpetually grow along a constant "Balanced Growth Path" as previously defined by \(Y_t\) as a function of the endogenous variables. This directly contrasts the Solow model, in which an economy converges to a steady-state via transition dynamics.

Changes in the growth rate of ideas, then, alter the growth rate of output itself—in this case, parameters \(\bar{l}, \bar{z}\), and \(\bar{L}\). This is best exemplified by comparing the growth rate before and and after a parameter changes. In the below example, a larger \(\bar{l}\) initially drops output due to less workers being allocated to production. Soon after, though, output recovers along a "higher" Balanced Growth Path.

Notably, while both the Romer and Solow Models help to analyze growth across countries, they both are unable to resolve one question: why can and do investment rates and TFP differ across contries? This is a more fundamental economic question involving culture, institutions, and social dynamics—one day I hope we'll have an answer.

While the Romer Model provides an avenue for long-run economic growth, it is anything but realistic—surely economies due not grow at an ever-increasing blistering rate into perpetuity. A model in which:

• Economies grow faster the further below they are from their balanced growth path
• Economies grow slower the further above they are from their balanced growth path

would certainly be more pragmatic. The Solow Model's capital dynamics do, in some sense, mirror part of this behavior with respect to the steady-state (output converges quicker/slower to the steady state the further/closer it is from equilibrium).

Combining the dynamics of the Romer Model's ideas and the Solow Model's capital stock could yield the desired result. Intuitively, incorporating capital into output via the Solow Model's production function, as well as including the Law of Capital Motion seems like one way to legitimately create this so-called "Romer-Solow" model:

Based on the the motivations for creating this model, it is more useful to first analyze the growth rates of equilibrium long run output \(Y_t^*\).

According to the production function, \[g_Y=g_A+\alpha g_K+(1-\alpha)g_{L_y}\]

From previous analysis it was found that \(g_A=\bar{z}\bar{l}\bar{L}\).

Based on the Law of Capital Motion, \[g_K=\frac{\Delta K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}\]

Because growth rates are constant on the Balanced Growth Path, \(g_K\) must be constant as well. Thus, so is \(\bar{s}\frac{Y_t}{K_t}-\bar{d}\); it must be that \(g_K^*=g_Y^*\).

The model assumes population is constant, so \(g_{\bar{L}}=0\rightarrow g_{\bar{L}_yt}=0\) as well.

Combining these terms, we find: \[g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\] \[\rightarrow g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\]

Solving for \(Y_t^*\) is trivial after discovering \(g_K=g_Y\) must hold on a balanced growth path.

Invoking the Law of Capital Motion with magic chants, \[g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}\]

Isolating \(Y_t^*\), \[Y_t^*=A_t^* (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha ({(1-\bar{l})\bar{L}})^{1-\alpha}\] \[\rightarrow {Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}\]

Plugging in the known expressions for \(A_t^*\) and \(g_Y^*\), a final expression for the Balanced Growth Path output as a function of the endogenous parameters and time is obtained: \[ Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\]

First looking at the growth rate of output, \(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), idea-driving factors and an increased allocation of labor to output increase the equilibrium Balanced Growth Path—the level of long-run growth. Thus, this model captures the influences of both capital and ideas on economic growth.

Looking at \(Y_t^*\), ideas have both a direct and indirect effect on output. Firstly, ideas raise output because they increase productivity (directly); second, with the introduction of capital stock, ideas also increase capital, in turn increasing output further (indirectly). Mathematically, this is evident in both instances of \(g_A^*\) in the formula for output \(Y_t^*\)—note that \(\frac{1}{1-\alpha},\frac{\alpha}{1-\alpha}>0\) for any \(\alpha\in(0,1)\), so \(\frac{d}{dg_A^*}Y_t^*>0\).

Expectedly, output has a positive relationship with the savings rate and a negative relationship with the depreciation rate.

Using the visualization below, we see a growth pattern similar to that of the Romer Model. However, the Romer-Solow economy indeed grows at a faster rate than the Romer model—I had to cap \(\bar{L}\) at \(400\) and \(\alpha\) at \(0.4\) because output would be too large for JavaScript to contain in a number (the graph would disappear).

Playing with the parameters, the previous mathematical findings are validated. For example, because \(g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\), only changes in parameters \(\alpha,\bar{z},\bar{l}\), and \(\bar{L}\) affect the growth rate of output, manifesting as the y-axis scaling up/down on a ratio scale.

However, do economics grow faster/slower the further below/above they are from their Balanced Growth Path, as initially desired? While this can be mathematically proven (of course), sometimes a visualization helps.

The graph below illustrates the transition dynamics of Romer-Solow Model. Namely, \((\bar{z}, \bar{l}, \bar{L}, \alpha)=(0.5, 0.5, 100, 0.33)\forall t<t_0\), then update to the slider values when \(t>t_0\).

Finally, it is clear that economies converge to their Balanced Growth Path as desired—something slightly more convoluted to prove from the complex expression for \(Y^*\) derived earlier. For example, with an increase in \(\alpha_0\), output grows at an increasing rate after the change, then increases at a decreasing rate as it converges to the new higher Balanced Growth Path. Increasing parameters \(\bar{z},\bar{l},\bar{L}\) yield similar results, although the changes are visually less obvious.