ECO102 Lecture 07

• Potential GDP
  • That's at $y^{\ast }$
  • We have an exp growth curve.
  • It's AS, aggregate supply curve.
  • Developed countries have a potential GDP that grows at a steady rate.
  • Developing countries may double every 4 - 5 years or something like that: Rule of 72 or Rule of 70.
  • Robert Solow won a Nobel Prize for his work on growth. 4 of his students won the Nobel Prize for their work on growth.
• Basic Model
• Robert Solow's Neoclassical Growth Model
  • Output is determined by the amount of capital and labor.
    • $Y=K^{\alpha }{L}^{1-\alpha }$
    • Where $\alpha$ is the share of output that goes to capital.
    • Just a production function.
    • Most times we have $\alpha =\frac{1}{2}$ in ECO102
    • $Y=K^{\frac{1}{2}}{L}^{\frac{1}{2}}$
  • Capital grows according to the expression:
    • $K^{t=1}=K^{t=0}+I-\delta K^{t=0}$ where $\delta =0.1$
    • If $\delta =0.2$, then capital lasts $5$ years.
  • Saving is proportional to income:
    • $S=\sigma Y$
    • $\sigma$ is the $MPS$
    • $\sigma =0.4$ we save $40\mathrm{\%}$ of our income.
  • Investment is funded by saving:
    • $I=S$
  • Write the model in intensive form:
    • Take all the variables, divide by $L$.
    • $\frac{Y}{L}=\frac{\sqrt{K}\sqrt{L}}{L}$
      • $y=\frac{Y}{L}$
        • GDP per worker
      • $k=\sqrt{\frac{K}{L}}$
        • Capital per worker
      • $y=\sqrt{k}$
    • $k=\frac{K}{L}$
      • $k^{t=1}=k^{t=0}+s-\delta k$
      • $s=\delta y$
        • Income per worker is $y$, and we save a fraction of that, so $s$ is the saving per worker.
    • $k^{t=1}=\frac{K^{t=1}}{L}=\frac{K^{t=0}+I-\delta K^{t=0}}{L}$
    • #tk flesh this out.
  • Equilibrium is a little different here.
    • This is when the economy stops growing.
    • This means that $k^{t=1}=k^{t=0}$
      • Height at end of this year, is the same as beginning of this year. That's an analogy to the idea of a steady state.
    • $k^{t=1}-k^{t=0}=s-\delta k^{t=0}$
    • $s=\delta k$
    • Depreciation is just covered by savings.
  • Savings curve looks like $y=\sqrt{x}$
    • $s=\sigma \sqrt{k}$
  • You'd have a dashed line which is the depreciation line, which is $s=\delta k$.
  • Equilibrium capital is the intersection of $s=\delta k$ and $s=\sigma \sqrt{k}$
  • $y=\sqrt{k}$ is your gdp.
  • At $k^{\ast }$, you get $y^{\ast }$ capital.
  • #tk graph
  • The gap between income and savings is consumption.
  • $c^{\ast }=y^{\ast }-s^{\ast }$
  • If $\delta =0.1$ and $\sigma =0.4$
  • $0.1k=0.4\sqrt{k}$
  • $k=16.0$
  • $y=\sqrt{16}=4$
  • $s^{\ast }=1.6$
  • $c^{\ast }=2.4$
  • Catchup effect:
    • To illustrate.
    • If $k<k^{\ast }$ was low. Then $s$ would exceed depreciation, so $k$ would grow.
    • If $k>k^{\ast }$ then depreciation would exceed savings, so $k$ would shrink.
    • Either way we have a mechanism that pushes us towards $k^{\ast }$.
• Alpha has $k_A=1$ Bravo has $k_B=4$
  • Alpha is less developed, Bravo is more developed.
  • "Each worker has one shovel."
  • $y=\sqrt{k}$
  • If $k=1⟹y=1$
  • If $k=4⟹y=2$
  • $s=0.6y$ $\sigma =0.6$
  • $k^{t=1}=k^{t=0}+s-\delta k$
  • $d=0.1k$
  • Below we get that $k=1\to 1.5$ gets us a 22.5% increase in output, while $k=4\to 4.8$ gets us a 9.5% increase in output.
  • Less developed country with $1$ unit of capital will grow at 22.5% while the more developed country with $4$ units of capital will grow at 9.5%. This is the catchup effect.
  • Eventually the less developed country will catch up.
  • $$\begin{array}{cccccc}k& y& \sigma =0.6& \delta =0.1& k^{t=1}& \\ 1& 1& 0.6& 0.1& 1.5\\ 1.5& 1.225& 0.735& 0.15& 2.085\\ 4& 2& 1.2& 0.4& \\ 4.8& 2.19& 1.31& 0.48\end{array}$$
  • Proposition 2:
    • ${\delta }_0=0.1$
    • ${\delta }_1=0.2$
    • Some countries have higher depreciation rates than others.
    • Canada has a higher rate:
      • Salt on cars, rusty as hell.
    • If $\delta \uparrow$ what happens to $k$ and $y$?
      • Then the depreciation line gets steeper, so the equilibrium point gets lower, so $k$ and $y$ get lower.
      • This makes intuitive sense. If you have a higher depreciation rate, then you need to save more to maintain the same level of capital, so you end up with less capital and less output.
      • In previous example. We know at $k^{\ast }=16$ we have $y^{\ast }=4$
      • With a steeper line, higher $\delta$. $k<k^{\ast }$ so $y<y^{\ast }$
  • Proposition Savings:
    • ${\sigma }_0=0.4$
    • ${\sigma }_1=0.5$
    • $k^{\ast }=16$
    • $y^{\ast }=4$
    • If savings increase, we have a larger savings curve, so the equilibrium point gets higher, so $k$ and $y$ get higher.
    • $s_0=0.4y$
    • $s_1=0.5y$
    • $\sigma \uparrow ⟹k\uparrow \wedge y\uparrow$
    • What policies can we do to $\sigma \uparrow$?
      • Preconditions:
        • Stable economy (do you need to make a run for it?)
          • Best would've been to borrow a lot from Iran before the revolution, then pay nothing.
          • But you don't want to be on the wrong side, invest everything and get nothing back.
        • Trust in the government (do you think the government will waste your money?)
      • Tax incentives for saving.
      • Registered Education Savings Plan (RESP)
      • Registered Retirement Savings Plan (RRSP)
      • Tax Free Savings Account (TFSA)
      • Government bonds.
  • Consider different $\alpha$
    • $Y=K^{\alpha }{L}^{1-\alpha }$
    • ${\alpha }_0=\frac{1}{2}$
      • $Y=\sqrt{K}\sqrt{L}$
    • ${\alpha }_1=\frac{2}{3}$
      • $Y=K^{\frac{2}{3}}{L}^{\frac{1}{3}}$
      • Causes an increase in capital per worker, so $k$ and $y$ get higher.
      • As well as increase in GDP per worker.
    • Policies to effect this:
      • Policies which create stability.
        • Some countries have a lot of political instability, so they have a hard time attracting investment, so they have a hard time growing.
        • Nationalization of someone's investments could mean their money is gone, so they won't invest in that country.
      • Compare GDP of North and South Korea. North Korea has a lot of political instability, so they have a hard time attracting investment, so they have a hard time growing.
• Labour Force Growth
  • Suppose Labour force $\eta =0.1$, grows at $10\mathrm{\%}$ per year.
  • The end of year labour force: $L^{t=1}=L^{t=0}(1+\eta )$
  • The break even line, $d=(\eta +\delta )k$.
  • If $\eta +\delta$ is higher then the break even line gets steeper, so the equilibrium point gets lower, so $k$ and $y$ get lower.
  • Vice versa.
• Prop 5:
  • Growing population, results in lower GDP per worker.
  • Toy example:
    • If we have $Z=\frac{ab}{c}$
    • $\mathrm{\Delta }Z\simeq \mathrm{\Delta }a+\mathrm{\Delta }b-\mathrm{\Delta }c$
  • Since $y=\frac{Y}{L}$
  • $\mathrm{\Delta }y\simeq \mathrm{\Delta }Y-\mathrm{\Delta }L$
  • In equilibrium we have $\mathrm{\Delta }y=0$
  • That means that $\mathrm{\Delta }Y=\mathrm{\Delta }L=\eta =0.1$
  • A growing population will have a lower level of gdp. But the level of gdp will grow at the same rate as the population, so the gdp per worker will be constant.
  • 4 dollars per year for each year for the rest of your life?
  • Or 2 dollars per year for each year for the rest of your life, but it grows at 10% per year?
  • $(2)(1+0.1{)}^{55}=378.118284942558$ is the amount you would have after 55 years if you took the second option. So the second option is better.
  • $\mathrm{\Delta }GDP=0.1$
    • #tk exam question
  • $\frac{\mathrm{\Delta }GDP}{\text{Worker}}=?$
• Prop 6
  • Is a growing population bad for economy?
  • Not really, there's a good correlation between population growth and economic growth. But it can be bad if the population growth is too high, because then you have a lot of people competing for a limited amount of resources, which can lead to poverty and other problems.
• Tech Progress
  • Labour Augmenting Tech Progress
    • $Y=A\sqrt{K}\sqrt{L}$ is neutral tech progress.
    • Neutral $A=1,2,3,4$
    • If we have $Y=\sqrt{K}\sqrt{AL}$, the index affects productivity of workers.
    • Example:
      • Double length bus, so you can fit twice as many people in the bus, so you can get twice as much output from the same amount of labour.
  • If $\gamma =0.2$ and $\eta =0.1$ and $\delta =0.1$ and $\sigma =0.4$
  • The break even line. $d=(\gamma +\eta +\delta )k$
  • $k^{\ast }=16$
  • $y=0.1k$
  • $y^{\ast }=4$
  • Now slope increases to $0.4$
  • $y=0.4k$
  • $k^{\ast }=1$
    • Huge decline.
  • $y^{\ast }=1$
  • Is tech progress a good thing? Originally gdp was 4 per worker, now it's 1 per. Seems like a bad thing.
  • It's not because:
    • $y=\frac{Y}{AL}$
    • $\mathrm{\Delta }y=\mathrm{\Delta }Y-\mathrm{\Delta }A-\mathrm{\Delta }L$
    • $\mathrm{\Delta }Y=\mathrm{\Delta }A+\mathrm{\Delta }L+\mathrm{\Delta }y$
    • In equilibrium $\mathrm{\Delta }y=0$
    • If we have $\mathrm{\Delta }A=0.2$
    • We get $\mathrm{\Delta }Y=0.3$
    • So the gdp grows at 30% per year, which is a good thing.
    • 4 dollars per year for each year for the rest of your life?
      • Some fuckass bmw
    • Or 1 dollar per year for each year for the rest of your life, but it grows at 30% per year?
    • $(1)(1+0.3{)}^{55}=1848776.34987589$
      • Honda after vtec
    • The second option is better.
    • $\frac{\mathrm{\Delta }Y}{\mathrm{\Delta }L}=\mathrm{\Delta }y+\mathrm{\Delta }A$
      • $\mathrm{\Delta }A=0.2$
    • Income per capita is growing at 20% per year, which is a good thing.
  • What's percent change in gdp per effective worker?
    • $0$
  • GDP per worker is $0.2$
  • $\mathrm{\Delta }GDP=0.3$
• Prop 8
  • Is tech progress bad? Nah.