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.