How much extra output do we get if we invest more?

 This expands on today's earlier post.

How much extra output do we get if we invest more?

How much extra output do we get if we invest more? Here’s some rules of thumb.

1.       From the RF growth mindset report, some useful data as background.

2.       Here’s our underperformance relative to the G7

 

3.       And here it is in bars:

 

4.       So we might regard our target is

a.       To get the US GDP per capita growth rates, we need another 1.55-1.03 = approx 0.5% growth per year.

b.       Notice that’s relative to the USA, relative to EU would be much less than that.

5.       How can we get to this via capital investment?  Here’s the tyranny of numbers.

6.       The extra output from 1 extra unit of capital sounds like an engineering problem.  But we can use some economic reasoning to get some measures.

7.       The average rate of return on capital is about 0.10 as measured by the ONS.  That says that the flow of profits from the existing capital stock in the UK, as a percentage of that capital stock, is just below 10%. 

8.       Let’s take a concrete example to check this.  In Feb2024, according to IBA Aero, the price of a new A320neo was $52m.  The annual rental cost was $400,000.  This ratio is 8%.  So that says that airlines are incurring a cost of renting a new aircraft that is 8% of the capital stock they would have to buy.

9.       If we further assuming that airlines are maximising profits, then they would rent more and more aircraft until the extra flow of output they can sell from renting another aircraft were equal to the extra costs.  But we have just worked out the extra costs are 8%. 

10.   That gives us part of the answer: for every increase in capital stock by one unit, output rises by 0.08 units (0.1 units for the economy as a whole)

11.   It’s more helpful to convert that into a percentage.  We can do this if we express the change in output following a change in capital, the 0.1, as a proportion of the baseline output/capital ratio.  That ratio, for most developed economies, is around 3.

12.   Thus we have the following:

a.       Average rates of return are around 0.1.  This tells us the extra output from one more unit of capital.

b.       Average output/capital is 3. 

c.       So the percentage extra output from a one percentage extra capital is 0.1/3 = approx. 0.3.

13.   What percentage rise in output do we get from a percentage rise in capital then?  For the economy as a whole the rule is

 

% rise in output from x% rise in capital asset A= 0.3 * share of capital asset A in the economy * % change in capital asset A.

14.   Let’s apply this rule to housing.  It is planned to raise housing input by 0.3%.  Housing is 40% of the total capital stock.  Thus the expected % rise is 0.3*0.4*0.3% =0.036%.  

Will building more houses boost growth?

An excellent meeting at the Resolution Foundation this morning discussed this point, following their new report, "The growth mindset: Sizing up the Government’s growth agenda" by Emily Fry & Gregory Thwaites.

1.  The new government is commited to building more houses.  It says this will boost growth, see their note 27.

2. Let's first be clear on levels and growth.  Will building more houses boost the level of GDP?  As Rupert Harrison on the panel said, this is what most people think. 

To most people, building more houses means allowed more people wearing hard hats producing some output. Surely that that must boost GDP?  Part of the job of economists, aside from quantifying things, is to point out unintended consequences. It is of course true that building more houses, with nothing else changing will raise measured GDP. Part of the components of GDP is investment come up and housing is an important share of investment.  So the question is will anything else change? As Rupert Harrison pointed out, All the estimates we have for the current economy is that it is running at full capacity. That means that any additional activity in housing simply transfer's activity away from other parts of the economy. In other words, what one might loosely call unintended consequences, turn out to be the key effect. Thus there is no effect on the level of GDP Via this mechanism.

This mechanism is a demand mechanism. That is to say, the mechanism most people have in mind, of more people wearing hard hats building buildings, is a mechanism whereby there is increased demand for resources in the economy, and that increased demand raises GDP common sense GDP measures the resources that the economy is producing. As is clear in this example, GDP will only rise if the increased demand is matched by supply (Or if there is surplus capacity in the economy set the increased demand does not displace any existing activity).

3. So what is the effect on supply? This is where the resolution foundation report very helpfully does the mathematics.

Their Figure 5 below sets out the data.  

The extra new housing that the report identifies turns out to be around 1.1% of the stock of housing. This in itself is an interesting number and shows the value of undertaking these calculations.  The announced target, of 1.5 million homes sounds like a large number. But this is an extra 60,000 homes per year over and above what we are already building. The key point is that there are around 30 million dwellings already existing. That's the additional building is around 0.3% of the stock of existing buildings. Thus the question is: what is the effect on the supply side of the economy of increasing the stock of existing buildings by 0.3%?

Is that we need to know how much extra output we get from a certain percentage change in the capital stock. Such extra output comes from the fact that increased capital stock raises the flow of capital services that are available for people to use in the economy.

The answer to that question sounds like an engineering answer. But this is where the economics of growth accounting comes in useful. If firms are behaving in any way rationally, they will equate the marginal product of capital to the real cost of capital. So for example if it costs British Airways $20 million to rent a Boeing 737 for a year they wouldn't bother to rent it unless they could make at least 20 million dollars per year in revenue.  But the real cost of capital is, in turn, the rate of return to capital which is something that statistical authorities calculate, for the UK, non-Continental Shelf firms, this is about 10%.

So there are two ways of getting to the percentage change in GDP: either the rate of return times the change in capital per unit of output, or the rate of return, expressed as a percentage of the baseline Y/K ratio, times the percentage change in capital.  For the latter, the Y/K ratio for the economy as a whole is around 3, with the housing share of the total economy capital stock of 40%.  So that the rate of return as a percentage of the baseline Y/K is 0.1*3*0.4 = 0.12.  If we then multiply that by the %change, of 0.3%, we get a % increase in growth of 0.03%. 

Update.

Another method is this (see Frontier economics, note 6 and the note to Table 3).  The elasticity is the rate of return times the K/Y ratio.  In the steady state, I=deltaK.  So K/Y is (I/Y)*(1/delta).  I/Y is about 0.2.  Delta is about 0.07.  This gives an elasticity of about 0.3