### Archive

Archive for the ‘Solow Model’ Category

## Total factor productivity

Using a production function like $Y=K^{\alpha} L^{1-\alpha}$ is a nice and convenient way of modelling an economy, the problem is if you do that you are almost always going to find the model doesn’t work.

As an example, lets say you have a model economy with 1000 people (all workers) and 5000 units of capital, and a production function $Y=K^{0.3} L^{0.7}$. That means you will have output of $Y=5000^{0.3} 1000^{0.7}=1620.657$.

Now say over the next year the population grows by 4% and capital grows by 2%, so you have 1040 people and 5100 units of capital. The model would predict that you would have output of $Y=5100^{0.3} 1040^{0.7}=1675.693$. But the odds are when you actually measure output, you are going to get a higher figure than the model predicted. Say for example you find output is actually 1750. How can we explain this?

It is quite logical given the technological progress story. If there has been some form of technological progress which has augmented labour and made it more productive than labour was last year, so the model doesn’t hold any more.

We could take care of this by changing the model to this form: $Y=AK^{\alpha} L^{1-\alpha}$.

If you totally differentiate this to get an expression for the change in Y, you get $\Delta Y = \Delta A \frac{\partial Y}{\partial A} + \Delta K (\alpha) \frac{\partial Y}{\partial K} + \Delta L (1-\alpha) \frac{\partial Y}{\partial L}$.

Now we can sort out these partials:

If $Y=AK^{\alpha} L^{1-\alpha}$ then $\frac{\partial Y}{\partial A} = K^{\alpha} L^{1-\alpha}$. But this is just the same as $\frac{Y}{A}$.

Similarly $\frac{\partial Y}{\partial K} = \alpha A K^{\alpha - 1} L^{1-\alpha}$. This is the same as $\alpha \frac{Y}{K}$.

Finally $\frac{\partial Y}{\partial L} = (1-\alpha) A K^{\alpha} L^{-\alpha}$. This is the same as $(1-\alpha) \frac{Y}{L}$.

So this gives us a nice expression: $\Delta Y = \Delta A \frac{Y}{A}+ \Delta K \alpha \frac{Y}{K} + \Delta L (1-\alpha) \frac{Y}{ L}$.

Divide the whole thing through by Y and we get $\frac{\Delta Y}{Y} = \frac{\Delta A }{A}+ \alpha \frac{\Delta K }{K} + (1-\alpha) \frac{\Delta L }{ L}$.

This is now in terms of ‘proportionate change’ in A, K and L. We can think of it in terms of ‘growth of A’ and ‘growth of K’ and ‘growth of L’ by defining notation like this:

$g_A = \frac{\Delta A}{A}$, $g_K = \frac{\Delta K}{K}$, $g_L = \frac{\Delta L}{L}$.

So $g_Y = g_A + \alpha g_K + (1-\alpha) g_L$.

This tells us the growth in output is made up of three things. It is the growth in A, plus the share of capital in output times the growth in capital, plus the share of labour in output times the growth in labour. The growth in A then gives us a tool for capturing the growth that has developed not through increases in inputs (capital and labour) but through something else. That something else is technological progress.

A is usually described as Total Factor Productivity and the growth in A is called the Solow Residual.

You can relate the growth in A back to the concept of labour augmenting technological progress, because the Solow Residual is equivalent to (proof not shown here) $g_A = (1- \alpha)g_\pi$. The growth in TFP is the share of labour in output multiplied by the growth in labour augmenting technological progress.

The Solow Residual is a useful tool for growth accounting, in that it gives us a way of estimating how much technology is progressing. If we know how much capital and labour has increased, and we know how much output has increased, then we can work out the Solow Residual and get an estimate of technological progress from that.

Returning to the example from before, where capital had gone up from 5000 to 5100, labour had gone up from 1000 to 1040 and output had gone up from 1675.693 to 1750.

$g_Y = \frac{1750-1675.693}{1675.693}=0.0443$, $g_K = \frac{5100-5000}{5000}=0.02$, $g_L = \frac{1040-1000}{1000}=0.04$,

So $0.0443 = g_A + 0.7 (0.02) + (0.3) (0.04) \Rightarrow g_A = 0.0183$. This means we have had TFP growth of 1.83%.

And from this we can say that ) $0.0183 = 0.7 g_\pi \Rightarrow g_\pi = 0.0261$ so we have had labour augmenting technological progress of 2.61%.

Of course the Solow Residual is just a way of estimating technological progress, it is very difficult to be totally accurate in practice, but it’s not a bad framework for thinking about it.

Categories: Macro, Solow Model

## The Solow model and the production function

I’ve presented the algebra of the Solow model in both the basic per capita case (without technological progress):

$k* = \frac{s}{(n +\delta)}y*$ and $y* = \frac{(n +\delta)}{s}k*$.

$\hat{k*} \approx \frac{s}{[\pi+n +\delta]}\hat{y*}$ and $\hat{y*} \approx \frac{[\pi+n +\delta]}{s}\hat{k*}$.

Now we can think of how this relates to a production function.

Typically the production function used to model output in the economy is of the form $Y=K^{\alpha}L^{1-\alpha}$. Here $\alpha$ is the share of capital in output and $1-\alpha$ is the share of labour in output. This particular form of production function is used to model the economy because it has constant returns to scale, which is fairly realistic (if you cloned an economy, it would probably produce the same amount of output, so by that logic if you double capital and labour in an economy, you would get double the amount of output).

If we want to use the small letter per-capita notation again, to show output and capital in ‘per worker’ terms, we can divide everything through by L: $\frac{Y}{L}=K^{\alpha}L^{-\alpha}$ so $y=\frac{K^{\alpha}}{L^{\alpha}} \Rightarrow y=k^{\alpha}$.

If $y=k^{\alpha}$ then $k=y^{\frac{1}{\alpha}}$

We can use this simple result, to express those earlier Solow equations in terms of the share of capital in output.

If in steady state, $k* = \frac{s}{(n +\delta)}y*$
then $k* = \frac{s}{(n +\delta)}k*^{\alpha} \Rightarrow k*^{1-\alpha} = \frac{s}{(n +\delta)} \Rightarrow k* = [\frac{s}{(n +\delta)}]^\frac{1}{1-\alpha}$.

And as $y=k^{\alpha}$
this means $y* = [\frac{s}{(n +\delta)}]^\frac{\alpha}{1-\alpha}$.

Similarly in the model with technological progress,

$k* \approx [\frac{s}{(\pi + n +\delta)}]^\frac{1}{1-\alpha}$
and
$y* \approx [\frac{s}{(\pi + n +\delta)}]^\frac{\alpha}{1-\alpha}$

Categories: Macro, Solow Model

## Labour-augmenting technical progress

The basic form of the Solow model gives us a bit of an unsatisfactory conclusion:

1. The economy will grow in terms of output per worker until it reaches a steady state level of output per worker. At steady state level of output per worker, the economy still grows, but it only grows at the rate of labour force growth (which we model as equal to the rate of population growth).

2. Raising the saving rate means you can lift yourself out of steady state and continue to grow for a while until you reach a new steady state. But you will still reach a new steady state point, just at a higher level of output per worker.

So this basically says that all economies will reach a point where they can no longer increase their living standards because output per worker will become constant. Is this realistic? Not really, and if Solow’s model had been left there, probably we wouldn’t be using it so widely.

If the Solow model is to really offer a good framework for thinking about growth, we need some way of explaining how countries can have continuously growing living standards. Solow does it by considering technological progress.

There are various ways that you can incorporate technological progress into the Solow model, but a simple and way of thinking about it is to think of technological progress as being labour augmenting, ie as the state of technology improves, it makes each worker more productive by augmenting their labour. Think about secretaries that used to type letters on typewriters. It may be that now with modern computers that can easily duplicate and edit documents, one secretary now can do the same amount of work as four secretaries could in the days of typewriters (I just made that up by the way). That means that the labour of the secretary has been augmented by the advance of technology, one secretary is now worth four secretaries in the past.

If you benchmark a particular year where the effective labour of one worker is 1, then you can model technological progress by seeing how technological progress makes effective labour grow. As technology improves, you may find that one worker becomes worth 1.4 effective workers (as benchmarked from the original year). This is a powerful concept, because if technology is improving then it means as your population and labour force grows, you can actually grow your number of ‘effective workers’ faster than the population grows. And going back to the concept of steady state, originally we said that when you are in steady state, your economy grows at the same rate as the labour force grows, so output per capita stays constant….if you are in steady state but with technological progress, then your economy would grow at the same rate as effective workers grow. If effective workers are growing faster than the population, then output per capita will rise and living standards will go up. For an economy, having more effective workers than there are people (because each person is worth more than 1 effective worker due to the technological advances) is a good thing because it means each worker only consumes the amount of one person, but produces the amount of more than one person.

We can look at the algebra of this, its quite similar to the algebra used before in the basic Solow model.

Start with $K_{t+1}= (1-\delta)K_t + sY_t$.

Now we can define the state of technology as being given by $E_t$, so we can divide throughout by effective workers, $\frac{E_t}{P_t}$:

$\frac{K_{t+1}}{E_t P_t}= \frac{(1-\delta)K_t}{E_t P_t} + \frac{sY_t}{E_t P_t}$.

In the basic model we used small letters to express output and capital per worker, we defined $y_t = \frac{Y_t}{P_t}$ and $k_t = \frac{K_t}{P_t}$.

Now we are going to think in terms of ‘per effective worker’ so we will put a hat on the small letters: $\hat{y}_t = \frac{Y_t}{E_t P_t}$ and $\hat{k}_t = \frac{K_t}{E_t P_t}$.

So we have $\frac{K_{t+1}}{E_t P_t}= (1-\delta)\hat{k}_t + s\hat{y}_t$

Now multiply the LHS by $\frac{E_{t+1} P_{t+1}}{E_{t+1} P_{t+1}}$ to get:

$\frac{K_{t+1}}{E_{t+1} P_{t+1}}\frac{E_{t+1} P_{t+1}}{E_t P_t}= (1-\delta)\hat{k}_t + s\hat{y}_t$ which means:

$\hat{k}_{t+1}\frac{E_{t+1} P_{t+1}}{E_t P_t}= (1-\delta)\hat{k}_t + s\hat{y}_t$.

Now we just need to create a definition for population growth: $\frac{P_{t+1}}{P_t}=(1+n)$,

and a definition for growth of technology: $\frac{E_{t+1}}{E_t}=(1+\pi)$,

and now we have this expression: $\hat{k}_{t+1}(1+n)(1+\pi)= (1-\delta)\hat{k}_t + s\hat{y}_t$

This is saying that the new per capita capital stock depends on the old per capita stock that is not depreciated, plus the newly accumulated capital stock due to investment. However, the LHS includes population growth and efficiency growth, which is a drag on the per capita stock – the faster population grows, the faster we need to increase capital to keep per efficiency unit of labour stock constant.

In the end we will tend to a steady state again, where $\hat{k}_{t+1}=\hat{k}_t = \hat{k*}$.

So $\hat{k*}(1+n)(1+\pi)= (1-\delta)\hat{k*} + s\hat{y*}$.

Rearranging this we get $\hat{k*}(1+n)(1+\pi) - (1-\delta)\hat{k*} = s\hat{y*} \Rightarrow \hat{k*}[(1+n)(1+\pi) - (1-\delta)] = s\hat{y*}$.

This lets us find an expression in terms of capital per effective worker:
$\hat{k*} = \frac{s\hat{y*}}{[\pi+n+\pi n +\delta]}$.

As $\pi n$ is likely to be small we can express this as an approximation:

$\hat{k*} \approx \frac{s}{[\pi+n +\delta]}\hat{y*}$.

Alternatively, in terms of output per effective worker,

$\hat{y*} \approx \frac{[\pi+n +\delta]}{s}\hat{k*}$.

Categories: Macro, Solow Model

## The Golden Rule level of capital

An increase in the saving rate allows you to reach a higher steady state level of capital per worker and output per worker.

However, if you just keep on increasing the saving rate, you start to defeat the point of growth, you want to have more output available for consumption now, as that is what determines living standards. If you have high output but everybody is saving all their income for the future then nobody has anything to spend on consumption now.

So there is a point where you can maximise the benefit for the population in terms of consumption. If you save too little, and consume too much now, you don’t increase your capital stock by enough to get much growth. If you save too much, you might increase your capital stock but you aren’t leaving enough for consumption now. Given that the economy is going to head to steady state in the long run, if you choose just the right saving rate, you can strike the right balance and optimise your living standards when you get there.

How do you do this?

First think about the basic concept that all national income is either saved or consumed.

$y=c+i$ means income per worker is equal to consumption per worker plus investment per worker (remember investment per worker depends on saving per worker). So $c = y - i$.

When we are in steady state, $c*=y*-i*$. In steady state, investment per worker is equal to depreciation per worker, so we can rewrite this as $c{*}= y{*}- \delta k{*}$.

This gives us an important idea: in steady state, consumption per worker is equal to the difference between output per worker and depreciation per worker, it is what is left of national income per worker once depreciation per worker has been taken care of. So in the Solow diagram, it is the difference between the output per worker curve, and the depreciation line.

The brown arrow shows the difference between output per worker and depreciation at different points. The difference will be at its widest when the slope of the production function (output per worker curve) is parallel to the depreciation line.

On this diagram the saving rate has been chosen so that it intersects the depreciation line at the point where we have the golden rule level of capital per worker. This is the saving rate that would get the economy into steady state at a point that would maximise consumption per worker in steady state.

The slope of the production function is the marginal productivity of capital, it tells you the amount of extra output you get from adding another unit of capital – here of course we are thinking in terms of capital and output ‘per worker’.

We can say then that the condition for the golden rule is that $MP_K = \delta$, marginal productivity of capital equals the depreciation rate, as $\delta$ is the slope of the depreciation line, $\delta k$.

An important note here is that the depreciation line giving ‘break even’ investment is actually a line of slope $\delta + g_n$ when you take into consideration population growth. As the population grows, then the amount of capital you need to ‘break even’ increases over and above depreciation, because you have to not only replace the capital that has depreciated, but you have to add some new capital for the new workers to use, if you are going to keep the same amount of capital per worker to break even. This is called capital widening, as your population grows you need to add more capital just to keep your capital per worker ratio constant. Only once you have added enough to equip the new workers with the same level of capital that the existing workers had, are you starting to get capital deepening which is increasing the overall level of capital per worker.

So in this case, the condition for the golden rule would be that $MP_K = \delta + g_n$

In the version of the Solow model that uses labour augmenting technological progress, the condition becomes $MP_K = \delta + g_n + g_{\pi}$.

Categories: Macro, Solow Model

## Why the saving rate is important

The saving rate shows the proportion of national income which is saved, and therefore available in the form of loanable funds for firms to use for investment, which is what adds new capital stock to the economy and increases its productive capacity.

We saw here how the economy reaches a point of steady state, where the amount of investment just equals the amount of old capital that depreciated, so the capital stock (in per capita terms) stays constant:

At point labelled C on this diagram, the economy has reached steady state, and it cannot grow its output per worker past the level of yC as it is impossible to increase the amount of capital per worker beyond point C. Any increase in capital will just mean that next year there is more depreciation than investment, so the capital stock will slip back to the steady state level. You can express a relationship between steady state capital and steady state output algebraically like this: $k* = \frac{s}{(n +\delta)}y*$.

So this tells us that the economy has got stuck at steady-state. But there is a way of breaking out.

If you increase the saving rate, then it shifts the saving (or investment) function on the diagram upwards:

Here we have a new saving rate, which means there is a new investment function, the purple line. So the economy can move past the old capital per worker (k*) which was at point C and continue to grow until it reaches a new steady state k* at point D. This means that there will be a new output per worker (y*), which has gone up (not by much on the diagram, because the production function was diminishing quite sharply at this stage) from yC to yD.

However this will still mean we are at a new steady state, just at a higher level of k* and y*.

So the moral of the story is in this form of the Solow model, which does not include technological progress, an increase in the saving rate can get you to a higher steady state level of output per worker, but you will still reach a new steady state, and you will get stuck again.

Categories: Macro, Solow Model

## The algebra of the Solow model

When I looked at the Harrod-Domar model on this blog I basically presented two forms,
a basic version: $g =\frac{s}{\theta} -\delta$,

and a version in per capita terms: $(1+g*)=\frac{(1-\delta)}{(1+n)} +\frac{s}{\theta (1+n)}$.

Because the Solow model is based around the concept of two factors of production (capital and labour), the presence of labour is central to the model because this is why you get diminishing returns to capital as you add more capital, so it makes sense to think about the Solow as a sort of ‘per capita’ model, by thinking in terms of output and capital per worker.

To form a Solow equation you go back to the basic equation of capital accumulation: $K_{t+1}=K_t+I_t-\delta K_t$ which can be rewritten as $K_{t+1}= (1-\delta)K_t + I_t$.

Because investment is equal to the saving rate multiplied by income, you can substitute that value here to get $K_{t+1}= (1-\delta)K_t + sY_t$.

Now we are going to use the same small letter notation for ‘per worker’ that we did in the per-capita Harrod-Domar model. Again we are treating ‘population’ and ‘workers’ as the same thing here, which is not an exactly correct assumption, but that is something we can deal with in the later versions of the Solow model.

So we can define our small letters in terms of population, ie $k_t = \frac{K_t}{P_t}$ and $y_t = \frac{Y_t}{P_t}$.

Now if you divide the equation above throughout by population you get $\frac{K_{t+1}}{P_t}= \frac{(1-\delta)K_t}{P_t} + \frac{sY_t}{P_t}$.

Now we can start using small letters, $\frac{K_{t+1}}{P_t}= (1-\delta)k_t + sy_t$.

This is tidier but the left hand side (LHS) is a bit of a nuisance because we have K in terms of year t+1 and P in terms of t, so we can’t just use the small letter to express it in per capita terms. So there is a trick to deal with this LHS, multiply top and bottom by the same thing, ie multiply it by $\frac{P_{t+1}}{P_{t+1}}$.

This gives us $\frac{K_{t+1}}{P_t+1} \frac{P_{t+1}}{P_t}= (1-\delta)k_t + sy_t$ ,
which means we can use a small k on the LHS, $k_{t+1} \frac{P_{t+1}}{P_t}= (1-\delta)k_t + sy_t$.

We can define population growth as $\frac {P_{t+1}}{P_t} = (1+n)$,

so $k_{t+1} (1+n) = (1-\delta)k_t + sy_t$.

This is a tidier equation. It is saying that next year’s capital stock (per worker) depends on the amount of this year’s capital stock that is not depreciated, plus the extra capital stock we have added through investment. However, the LHS includes population growth, which is a drag on the per capita stock – the faster population grows, the faster we need to increase capital to keep per capita stock constant. These models consider population growth to be constant.

Eventually we will reach a point where the amount of new capital accumulated in a year will be just enough to keep the per capita capital stock constant, when you have taken depreciation and population growth into account. This will be the steady state level of capital per capita, which we can call k*, which is associated with a steady state level of output per capita, which we can call y*.

At the steady state point, the amount of capital per worker next year will be the same as the capital per worker this year, so $k_{t+1} = k_t = k*$ and the amount of output per worker next year will be the same as the output per worker this year, $y_{t+1} = y_t = y*$.

So at steady state, $k*(1+n) = (1-\delta)k* + sy*$.

This allows us to find an expression which links k* to y*, ie $k*(1+n) - (1-\delta)k* = sy* \Rightarrow (n +\delta) k* = sy* \Rightarrow k* = \frac{s}{(n +\delta)}y*$.

Alternatively, $y* = \frac{(n +\delta)}{s}k*$.

Categories: Macro, Solow Model

## The concept of steady state

The idea of an economy reaching steady state is central to the Solow growth model. This means a point where the diminishing returns to factor have kicked in to an extent that the economy can’t become any more productive in per capita terms by simply adding more capital, instead it reaches a maximum limit where output per capita will stay constant.

The reason this happens in the Solow model is because of the concept of depreciation in capital accumulation. The rate at which capital depreciates is usually modelled as being constant. So the more capital you have in any one year, the more depreciation you are going to get, and the more investment (new capital) you will need just to sustain the amount of capital you had last year – break even investment.

Steady state is often illustrated on a diagram, so you need to get familiar with diagrams looking like this:

The black curve is the production function, it gives output per worker. I am using small letters here to illustrate ‘per worker’ (not all the population are workers so this isn’t exactly ‘per capita’, but we can deal with that aspect of the model later). So the production function is written as y=f(k), ie output per worker is a function of capital per worker, which is saying that the amount of output each worker produces is a function of the amount of capital each worker has.

The blue curve is investment per worker. This is basically the production function multiplied by the saving rate, it is saying that at a given level of output per worker, this amount will be saved, so it tells us the amount of saving per worker. As we equate saving with investment (savings provide loanable funds which firms use to invest) this gives us the amount of investment per worker going on in the economy. This is important as investment shows us the amount of new capital we are adding.

The red line is depreciation per worker, it is a straight line because the depreciation rate is modelled as being constant. The more capital per worker we have, the more wears out every year, so this line basically tells us the amount of capital we need to add to break even (replace the worn out capital).

At any value of capital per worker where investment per worker is above depreciation per worker (ie the blue curve is above the red line) then you are continuing to add to your capital stock.

Here’s an example:

At point A here in terms of capital per worker, we can see the level of output per worker that corresponds to it (yA). Here the blue curve is above the red line, the brown arrow shows how much new capital we are adding to the economy when we are at this point.

Of course as we are adding new capital to the economy, we will move to a new point where there is more capital per worker available.

Now we are at point B, with more capital per worker than we had at point A and more output per worker as well. But notice how the brown arrow is smaller this time, it is showing a smaller gap between the blue curve and the red line. This is because although we are adding more investment at point B than we did at point A, we are also facing much more depreciation because we have more capital to maintain, so a lot more of the investment will be taken up in just replacing worn out old stock, and less is left over for adding new capital.

Eventually we get to point C. Here we have reached a point where the amount of investment is equal to the break even point of investment, ie we are just investing enough to cover depreciation and aren’t adding any new stock. This is the steady state. We can’t break ahead of this because attempting to add new capital would mean that our investment fell below depreciation, so we would have more depreciation than investment next year and would fall back to the steady state point. So the level of output per worker, yC, that corresponds to the steady state level of capital per worker that we have reached, is the maximum level of output per worker that we are going to get.

The economy, under this simple version of the model, is stuck at this steady state point.

Note that when you get to the steady state of output per worker, it means that your economy will still grow, just that it will only grow at the rate of growth of the labour force (which we are modelling here as being equal to the rate of population growth). If you are in steady state and have population growth of 3% then you will have output growth of 3%, but that won’t raise living standards, all it will mean is you are growing enough output to maintain the living standards for the population as it gets bigger.

Categories: Macro, Solow Model