class: center, middle, inverse, title-slide #
Solow Growth Model II
## Economic Development & Construction 0008 ###
Dr. Kumar Aniket
### Lecture 2 --- class:inverse, center, middle # Some Basic Math --- # UK's Economic Growth: Long view <img src="images/lec2/uk_longview.png" width="70%" style="display: block; margin: auto;" /> --- # UK's Economic Growth: Recent <img src="images/lec2/uk1.png" width="130%" style="display: block; margin: auto;" /> --- # UK's Economic Growth: Recent <img src="images/lec2/uk2.png" width="90%" style="display: block; margin: auto;" /> --- # UK and US's Economic Growth: Recent <img src="images/lec2/uk_us.png" width="120%" style="display: block; margin: auto;" /> --- # Growth Rates Primer I Change in the value of `\(x\)` between times period `\(t-1\)` and `\(t\)` is given by `$$\Delta{x}=x_{t}-x_{t-1}$$` Growth rate of `\(x\)` is given by `$$g_{x} = \frac{\Delta{x}}{x}$$` <table> <thead> <tr> <th style="text-align:right;"> x </th> <th style="text-align:left;"> Change </th> <th style="text-align:left;"> Growth </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 2 </td> <td style="text-align:left;"> </td> <td style="text-align:left;"> </td> </tr> <tr> <td style="text-align:right;"> 3 </td> <td style="text-align:left;"> 1 </td> <td style="text-align:left;"> 0.5 </td> </tr> <tr> <td style="text-align:right;"> 6 </td> <td style="text-align:left;"> 3 </td> <td style="text-align:left;"> 1 </td> </tr> <tr> <td style="text-align:right;"> 7 </td> <td style="text-align:left;"> 1 </td> <td style="text-align:left;"> 0.17 </td> </tr> <tr> <td style="text-align:right;"> 13 </td> <td style="text-align:left;"> 6 </td> <td style="text-align:left;"> 0.86 </td> </tr> <tr> <td style="text-align:right;"> 15 </td> <td style="text-align:left;"> 2 </td> <td style="text-align:left;"> 0.15 </td> </tr> </tbody> </table> --- # Growth Rates Primer II If variable are multiplied `$$z={x}\cdot{y}$$` the growth rates get added up `$$\frac{\Delta{z}}{z} = \frac{\Delta{x}}{x} + \frac{\Delta{y}}{y}$$` <table> <thead> <tr> <th style="text-align:center;"> x </th> <th style="text-align:center;"> y </th> <th style="text-align:center;"> z </th> <th style="text-align:center;"> growth (x) </th> <th style="text-align:center;"> growth (y) </th> <th style="text-align:center;"> growth (z) </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> 3 </td> <td style="text-align:center;"> 2 </td> <td style="text-align:center;"> 6 </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:center;"> 6 </td> <td style="text-align:center;"> 5 </td> <td style="text-align:center;"> 30 </td> <td style="text-align:center;"> 1 </td> <td style="text-align:center;"> 1.5 </td> <td style="text-align:center;"> 4 </td> </tr> <tr> <td style="text-align:center;"> 7 </td> <td style="text-align:center;"> 6 </td> <td style="text-align:center;"> 42 </td> <td style="text-align:center;"> 0.17 </td> <td style="text-align:center;"> 0.2 </td> <td style="text-align:center;"> 0.4 </td> </tr> <tr> <td style="text-align:center;"> 13 </td> <td style="text-align:center;"> 11 </td> <td style="text-align:center;"> 143 </td> <td style="text-align:center;"> 0.86 </td> <td style="text-align:center;"> 0.83 </td> <td style="text-align:center;"> 2.4 </td> </tr> <tr> <td style="text-align:center;"> 15 </td> <td style="text-align:center;"> 14 </td> <td style="text-align:center;"> 210 </td> <td style="text-align:center;"> 0.15 </td> <td style="text-align:center;"> 0.27 </td> <td style="text-align:center;"> 0.47 </td> </tr> </tbody> </table> --- # Effective Worker `\(K\)`: .ema[Physical capital stock] employed in the production process `\(L\)`: The quantity of .ema[labour] used (measured in person hours) for the production process `\(A\)`: The set of .ema[ideas] that empower the worker to use the capital stock <!-- --> .emzb[Effective worker] `$$AL$$` .emzb[Capital stock per effective worker] `$$\frac{K}{AL}$$` --- # Growth Rates Primer III Capital per-effective labour ratio is given by `$$k=\frac{K}{AL}$$` where `\(A\)` is marginal productivity of labour. The growth rate of `\(k\)` is given by `$$\frac{\Delta{k}}{k} = \frac{\Delta{K}}{K} - \frac{\Delta{A}}{A} - \frac{\Delta{L}}{L}$$` <table> <thead> <tr> <th style="text-align:center;"> K </th> <th style="text-align:center;"> A </th> <th style="text-align:center;"> L </th> <th style="text-align:center;"> k </th> <th style="text-align:center;"> growth (K) </th> <th style="text-align:center;"> growth (A) </th> <th style="text-align:center;"> growth (L) </th> <th style="text-align:center;"> growth (k) </th> </tr> </thead> <tbody> <tr> <td style="text-align:center;"> 100.00 </td> <td style="text-align:center;"> 1.00 </td> <td style="text-align:center;"> 10.00 </td> <td style="text-align:center;"> 10.00 </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> <td style="text-align:center;"> </td> </tr> <tr> <td style="text-align:center;"> 120.00 </td> <td style="text-align:center;"> 1.05 </td> <td style="text-align:center;"> 10.30 </td> <td style="text-align:center;"> 11.10 </td> <td style="text-align:center;"> 0.2 </td> <td style="text-align:center;"> 0.05 </td> <td style="text-align:center;"> 0.03 </td> <td style="text-align:center;"> 0.10957 </td> </tr> <tr> <td style="text-align:center;"> 144.00 </td> <td style="text-align:center;"> 1.10 </td> <td style="text-align:center;"> 10.61 </td> <td style="text-align:center;"> 12.31 </td> <td style="text-align:center;"> 0.2 </td> <td style="text-align:center;"> 0.05 </td> <td style="text-align:center;"> 0.03 </td> <td style="text-align:center;"> 0.10957 </td> </tr> <tr> <td style="text-align:center;"> 172.80 </td> <td style="text-align:center;"> 1.16 </td> <td style="text-align:center;"> 10.93 </td> <td style="text-align:center;"> 13.66 </td> <td style="text-align:center;"> 0.2 </td> <td style="text-align:center;"> 0.05 </td> <td style="text-align:center;"> 0.03 </td> <td style="text-align:center;"> 0.10957 </td> </tr> <tr> <td style="text-align:center;"> 207.36 </td> <td style="text-align:center;"> 1.22 </td> <td style="text-align:center;"> 11.26 </td> <td style="text-align:center;"> 15.16 </td> <td style="text-align:center;"> 0.2 </td> <td style="text-align:center;"> 0.05 </td> <td style="text-align:center;"> 0.03 </td> <td style="text-align:center;"> 0.10957 </td> </tr> </tbody> </table> --- # Properties of Production Function Production function `$$Y = F(K,AL)$$` For the production functions, the inputs are capital stock `\(K\)` and the quantity of .emzb[effective worker] defined as `\(AL\)`. If the production function has .emob[diminishing returns to capital] output-capital ratio falls as more capital is employed, - i.e., `\(\frac{Y}{K}\)` falls as `\(K\)` increases - .emai[intuition:] each subsequent addition of capital produces a smaller increase in output at the margin --- # Constant Returns to Scale With constant returns to scale we can write the production function as `$$y = f(k)$$` where `\(k\)` is the .emzb[capital stock per effective worker] defined by `$$k=\frac{K}{AL}$$` and `\(y\)` is the is the .emzb[output per effective worker] defined by `$$y=\frac{Y}{AL}.$$` --- # Output-capital ratio <img src="images/lec2/worksheet2a.png" width="85%" style="display: block; margin: auto;" /> --- class:inverse, center, middle # Solow Growth Model --- # Goods Market Equilibrium People consume `\((1-s)\)` proportion of their income `$$C = (1-s)Y$$` And save the rest `$$S = Y-C= s Y$$` Economy's output is either consumed or invested by its denizens $$Y {~\equiv~} C + I $$ It implies that **saving equals investment** in the economy `$$S {~\equiv~}I$$` .right[.emb[non-consumed output becomes investment]<sup>1</sup>] .footnote[ 1. Output cannot easily be transformed into investment good. This implies that resources that produce output can be redirected towards investment goods. ] --- # Investment ### Capital Goods Market Equilibrium Investment `\(I\)` gets divided up between .ema[depreciation] and adding to .emb[capital stock] in the economy `$$I = \Delta{K} + \delta{K}$$` .pull-right[ where `\(\delta\)` proportion of current capital stock `\(K\)` needs to replaced every year ] Change in capital stock is given by $$ \Delta{K} = I - \delta{K}$$ --- # Output-capital ratio <img src="images/lec2/iscurve1.png" width="85%" style="display: block; margin: auto;" /> --- # Simplifying Assumptions There is no population growth: `$$\frac{\Delta {L}}{L} = 0$$` Marginal Product of labour is constant: `$$\frac{\Delta {A}}{A} = 0$$` --- # Fundamental equation Capital accumulation is saving `\(sY\)` that is in excess of depreciation `\(\delta{K}\)` `$$\Delta{K} = sY - \delta{K}$$` dividing through by `\(K\)` $$ \frac{\Delta{}K}{K} = s \left(\frac{Y}{K}\right) - \delta $$ .pull-right[ the closer the value of `\(s\frac{Y}{K}\)` gets to `\(\delta\)`, the smaller the growth rate of capital `\(g_{K}=\frac{\Delta{}K}{K}\)` ] If the economy's production function has diminishing returns to capital, the economy heads to convergence --- # Convergence to Steady State <img src="images/lec2/worksheet2b.png" width="85%" style="display: block; margin: auto;" /> --- # Steady State Capital accumulation stops when `\(\Delta{K}=0\)`. This implies that $$sY = \delta{K} $$ **Steady-state condition**: the .ema[output-capita ratio] equals a constant (depreciation saving rate ratio) $$\frac{Y}{K} = \frac{\delta}{s} $$ .pull-right[ where `\(\frac{Y}{K}=\frac{Y}{L}/\frac{K}{L}\)` ] - .ema[Higher] the saving rate `\(s\)`, the .emb[richer] the country - .ema[Lower] the depreciation rate `\(\delta\)`, the .emb[richer] the country --- # New Assumptions Population grows at the rate `\(n>0\)` `$$\frac{\Delta {L}}{L} = n$$` Marginal Product of labour grows at the rate `\(g>0\)` `$$\frac{\Delta {A}}{A} = g$$` Since `\(k=\frac{K}{AL}\)`, we can write this as `\(K=k\cdot{A}\cdot{L}\)` $$\frac{\Delta {K}}{K} = \frac{\Delta {k}}{k} + g + n $$ --- # Revisiting the Fundamental equation .pull-left[ Fundamental equation $$ \frac{\Delta{}K}{K} = s \left(\frac{Y}{K}\right) - \delta $$ ] .pull-right[ Growth rate of capital `\(K\)` $$\frac{\Delta {K}}{K} = \frac{\Delta {k}}{k} + g + n $$ ] Growth rate of capital per-effective worker `\(k=\frac{K}{AL}\)` is $$ \frac{\Delta {k}}{k} = s \left(\frac{Y}{K}\right) - (\delta+n+g) $$ Further the economy from the steady state, the faster it grows. The closer its gets, the smaller the growth rate. --- # Convergence to Steady State <img src="images/lec2/worksheet5a.png" width="85%" style="display: block; margin: auto;" /> --- # Steady State Growth rate In steady state $$ \frac{\Delta {k}}{k}=0$$ This implies growth rate of .emb[capital-labour ratio] `\(\frac{K}{L}\)` in steady state is equal to the growth rate of .emc[marginal productivity of labour]. .pull-right[ `$$k = \frac{\left(\frac{K}{L}\right)}{A}$$` For `\(k\)` to be constant, growth rate of numerator should be equal growth rate of the denominator ] The faster .emc[marginal productivity of labour] grows, the more capital each worker has, the richer the worker becomes. --- # Taking Stock Convergence driven by .emc[diminishing returns to capital] What if returns to capital did not diminish? - No convergence to steady state - Perpetual growth .pull-right[ .embb[Infrastructure] has complementarities with capital and thus plays a crucial role in ensuring that returns to capital are .emc[non-diminishing] [*Erie canal*](https://en.wikipedia.org/wiki/Erie_Canal) *Railway Construction* by the British Raj in India ] --- # Erie canal .pull-left[ Opened on October 26, 1825 and ran 584 km from Hudson River to Lake Erie ] .pull-right[ It was faster than carts pulled by draft animals and cut transport costs by about 95% ] <img src="https://upload.wikimedia.org/wikipedia/commons/b/b8/ErieCanalMap.jpg" width="100%" style="display: block; margin: auto;" /> --- # Non-diminishing returns to Capital <img src="images/lec2/worksheet7b.png" width="85%" style="display: block; margin: auto;" /> --- ### Output per-worker versus capital per-worker (1951-2017) <img src="figs/unnamed-chunk-20-1.svg" style="display: block; margin: auto;" /> --- # US: Production Function (1840 - 1990) <img src="images/lec2/grlong1.png" width="70%" style="display: block; margin: auto;" /> --- # US: Production Function (1840 - 1990) <img src="images/lec2/grlong2.png" width="70%" style="display: block; margin: auto;" /> --- # UK: Production Function (1760-1990) <img src="images/lec2/grlong3.png" width="70%" style="display: block; margin: auto;" /> --- # UK: Production Function (1760-1990) <img src="images/lec2/grlong4.png" width="70%" style="display: block; margin: auto;" /> --- # Production Functions (1900-1990) <img src="images/lec2/grlong5.png" width="70%" style="display: block; margin: auto;" /> --- # Economic Growth (1760-1990) <img src="images/lec2/4a.jpg" width="100%" style="display: block; margin: auto;" /> --- # Economic Growth (1760-1990) <img src="images/lec2/4b.jpg" width="100%" style="display: block; margin: auto;" /> --- # Economic Growth (1760-1990) <img src="images/lec2/4c.png" width="100%" style="display: block; margin: auto;" /> --- # Conclusions Capital accumulation crucial for economic development - The more capital workers have, the more they can produce and consume Diminishing returns to capital constrains the growth - Factors that makes returns to capital non-diminishing crucial for growth .pull-right[ .embb[Infrastructure] like plays a crucial role in ensuring returns to capital are non-diminishing e.g., roads, railways, canals, electricity, education and health infrastructure ] <style> h1, h2, h3 { color: #EC5800; } p { line-height: 1.5em; } rr { color: #002E63; } a { color: #002E63; } .inverse { background-color: #D2691E; } .tab { display:inline-block; margin-left: 15px; } .ema {color: rgb(43,106,108);} .emb {color: rgb(184,13,72); } .emc {color: #1034A6; } .emd {color: rgb(64,64,64); } .eme {color: #614051; } .emg {color: #6D351A; } .emgr {color: #696969; } .emo {color: rgb(229,65,6); } .emob {color: rgb(229,65,6); font-weight: bold;} .emoi {color: rgb(229,65,6); font-style: italic;} .emz {color: #004225; } .emzb {color: #004225; font-weight: bold;} .emzi {color: #004225; font-style: italic;} .emab {color: rgb(43,106,108);font-weight: bold;} .embb {color: rgb(184,13,72); font-weight: bold;} .emcb {color: #1034A6; font-weight: bold;} .emdb {color: rgb(64,64,64); font-weight: bold;} .emai {color: rgb(43,106,108);font-style: italic;} .embi {color: rgb(184,13,72); font-style: italic;} .emci {color: #1034A6; font-style: italic;} .emdi {color: rgb(64,64,64); font-style: italic;} .emdb {color: rgb(64,64,64); font-weight: bold;} .footnote {color: gray;} .red { color: red; } .mybox { color:#3D2B1F; background-color:#3D2B1F10; margin:1em; padding: 1em; border-radius: 10px; } .thebox { color:#704214; background-color: #F5DEB320; margin: 0.1em 0.5em 0.1em 0.5em; padding: 0.1em 1em 0.1em 1em; border-radius: 10px; border-color: #704214; border-style: solid; border-width: 2px; } </style> <!-- rgb(184,13,72) #plum rgb(242,151,36) #orange rgb(43,106,108) #dark-teal rgb(64,64,64) #dark-grey --> <!-- #386890 #A40000 #FFB347 -->