diff --git a/posts/economics/models-of-production.html b/posts/economics/models-of-production.html index 47896bc..3c5a6e2 100644 --- a/posts/economics/models-of-production.html +++ b/posts/economics/models-of-production.html @@ -53,477 +53,542 @@ in Intermediate Macroeconomics (ECON 3020) during the Spring semester of 2024 at the University of Virginia.

-

solow

-

introduction

+

solow

-

- 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: -

- -

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

-
    -
  1. - Output is either saved or consumed; in other words, savings - equals investment -
    2. -
  2. - Capital accumulates according to investment \(I_t\) and - depreciation \(\bar{d}\), beginning with \(K_0\) (often called - the - Law of Capital Motion) -
    3. -
  3. Labor \(L_t\) is time-independent
    4. -
  4. - A savings rate \(\bar{s}\) describes the invested portion of - total output -
    5. -
-

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

-
-
-
    -
  1. \(C_t + I_t = Y_t\)
    2. -
  2. \(\Delta K_{t+1} = I_t - \bar{d} K_t\)
    3. -
-
-
-
    -
  1. \(L_t = \bar{L}\)
    2. -
  2. \(I_t = \bar{s} Y_t\)
    3. -
-
-
-
-
-

solving the 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: -

-
-
-
-
-
-
    -
  • -
    - - 1.00 - -
    -
    • -
  • -
    - - 0.50 - -
    -
    • -
-
-
-
    -
  • -
    - - 0.50 - -
    -
    • -
  • -
    - - 0.33 - -
    -
    • -
-
-
-

- 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: - \[(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}}) - =((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}, - \bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})\] -

-
-
-

analysis

-
-
-

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

- -

- 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. -

-
- - - -

romer

-

introduction

-
-

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:

- -

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

- -

- 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:

-
-
-
    -
  1. \(Y_t=A_tL_{yt}\)
    2. -
  2. \(\Delta A_{t+1} = \bar{z}A_tL_{at}\)
    3. -
-
-
-
    -
  1. \(L_{yt}+L_{at}=\bar{L}\)
    2. -
  2. \(L_{at}=\bar{l}\bar{L}\)
    3. -
+ href="https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function" + >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. +
  2. + Capital accumulates according to investment \(I_t\) and + depreciation \(\bar{d}\), beginning with \(K_0\) (often called + the + Law of Capital Motion) +
    3. +
  3. Labor \(L_t\) is time-independent
    4. +
  4. + A savings rate \(\bar{s}\) describes the invested portion of + total output +
    5. +
+

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

+
+
+
    +
  1. \(C_t + I_t = Y_t\)
    2. +
  2. \(\Delta K_{t+1} = I_t - \bar{d} K_t\)
    3. +
+
+
+
    +
  1. \(L_t = \bar{L}\)
    2. +
  2. \(I_t = \bar{s} Y_t\)
    3. +
+
-

- 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: -

-
-
+
+

solving the model

-
-
-
    -
  • -
    - - 0.50 - -
    -
    • -
  • -
    - - 505 - -
    -
    • -
+
+

+ 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: +

+
+
-
-
    -
  • -
    - - 0.50 - -
    -
    • -
  • -
    - - 5000 - -
    -
    • -
+
+
+
    +
  • +
    + + 1.00 + +
    +
    • +
  • +
    + + 0.50 + +
    +
    • +
+
+
+
    +
  • +
    + + 0.50 + +
    +
    • +
  • +
    + + 0.33 + +
    +
    • +
+
+

+ 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: + \[(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}}) + =((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}, + \bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})\] +

-

- 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: -

-
- - - - - - - - -
+
+

analysis

+
+
+

+ 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. +
    2. Indirectly through TFP
      3. +
    +
    • +
  • + 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. +

-

solving the model

+

romer

-

- 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: \[Y_t=A_t - L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\] -

- - - - -
-

analysis

-
-

- 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 - "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 with - 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. -

-
-
-
-
-
-
    -
  • -
    - - 0.50 - -
    -
    • -
+

introduction

+
+

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:

+
+
+
    +
  1. \(Y_t=A_tL_{yt}\)
    2. +
  2. \(\Delta A_{t+1} = \bar{z}A_tL_{at}\)
    3. +
+
+
+
    +
  1. \(L_{yt}+L_{at}=\bar{L}\)
    2. +
  2. \(L_{at}=\bar{l}\bar{L}\)
    3. +
+
+
+

+ 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: +

+
+
+
+
+
+
    +
  • +
    + + 0.50 + +
    +
    • +
  • +
    + + 505 + +
    +
    • +
+
+
+
    +
  • +
    + + 0.50 + +
    +
    • +
  • +
    + + 5000 + +
    +
    • +
+
+
+

+ 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: +

+
+ + + + + + + + +
+

solving the model

+
+

+ 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: \[Y_t=A_t + L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\] +

+ + + + +
+

analysis

+
+

+ 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 + "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 + with 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. +

+
+
+
+
+
+
    +
  • +
    + + 0.50 + +
    +
    • +
+
+
+

+ 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. +

+
-

romer-solow

-

introduction

+

romer-solow

-

hi

-

hello

+

introduction

+
+

+ 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 the so-called 'Romer-Solow' model: +

+
+
+
    +
  1. \(Y_t=K_t^\alpha L_{yt}^{1-\alpha}\)
    2. +
  2. \(\Delta K_{t+1}=\bar{s}Y_t-\bar{d}K_t\)
    3. +
  3. \(\Delta A_{t+1} = \bar{z}A_tL_{at}\)
    4. +
+
+
+
    +
  1. \(L_{yt}+L_{at}=\bar{L}\)
    2. +
  2. \(L_{at}=\bar{l}\bar{L}\)
    3. +
+
+
+
+

solving the model

+
content
diff --git a/scripts/post.js b/scripts/post.js index bd408af..1c6c1c5 100644 --- a/scripts/post.js +++ b/scripts/post.js @@ -3,6 +3,38 @@ document.documentElement.style.setProperty( getTopicColor(urlToTopic()), ); +const tagToHeader = new Map([ + ["H2", "#"], + ["H3", "##"], +]); + +const makeFold = (h, i) => { + const toggle = document.createElement("span"); + toggle.style.fontStyle = "normal"; + toggle.textContent = "v"; + + // only unfold first algorithm problem + if (urlToTopic() === "algorithms" && i === 0) toggle.textContent = "v"; + + h.parentElement.nextElementSibling.style.display = + toggle.textContent === ">" ? "none" : "block"; + h.parentE; + toggle.classList.add("fold-toggle"); + toggle.addEventListener("click", () => { + const content = h.parentElement.nextElementSibling; + toggle.textContent = toggle.textContent === ">" ? "v" : ">"; + content.style.display = toggle.textContent === ">" ? "none" : "block"; + }); + + const mdHeading = document.createElement("span"); + const header = tagToHeader.has(h.tagName) ? tagToHeader.get(h.tagName) : ""; + mdHeading.textContent = `${header} `; + mdHeading.style.color = getTopicColor(urlToTopic()); + + h.prepend(mdHeading); + h.prepend(toggle); +}; + document.addEventListener("DOMContentLoaded", () => { document.querySelectorAll("article h2").forEach((h2) => { const mdHeading = document.createElement("span"); @@ -11,27 +43,6 @@ document.addEventListener("DOMContentLoaded", () => { h2.prepend(mdHeading); }); - document.querySelectorAll(".fold h3").forEach((h3, i) => { - const toggle = document.createElement("span"); - toggle.textContent = "v"; - - // only unfold first algorithm problem - if (urlToTopic() === "algorithms" && i === 0) toggle.textContent = "v"; - - h3.parentElement.nextElementSibling.style.display = - toggle.textContent === ">" ? "none" : "block"; - toggle.classList.add("fold-toggle"); - toggle.addEventListener("click", () => { - const content = h3.parentElement.nextElementSibling; - toggle.textContent = toggle.textContent === ">" ? "v" : ">"; - content.style.display = toggle.textContent === ">" ? "none" : "block"; - }); - - const mdHeading = document.createElement("span"); - mdHeading.textContent = "## "; - mdHeading.style.color = getTopicColor(urlToTopic()); - - h3.prepend(mdHeading); - h3.prepend(toggle); - }); + document.querySelectorAll(".fold h2").forEach(makeFold); + document.querySelectorAll(".fold h3").forEach(makeFold); });