A while back, Dietrich Vollrath wrote a famous post called "You can't reform your way to rapid growth." We should add to this: you can't invest your way to rapid growth either. The government's "number one mission" of boosting economic growth is therefore a challenging one.

To see the issue, consider an aggregate production function:

Y = TK^aL^b

where Y is output; K is capital; L is labour; T is total factor productivity or, if you prefer, technology; a is the elasticity of output with respect to capital; and b the elasticity of output with respect to labour.

This implies that growth in labour productivity is equal to the growth in capital stock per worker multiplied by a (the elasticity of output with respect to capital) plus growth in technology.

So, how big is a? Or, in other words, how much extra output do we get if we increase the capital stock?

The answer is: not much.

One common approach here - well, more of a first pass really - is to assume perfect competition and constant returns. If so, it can be shown that a is equal to capital's share of income. Which is not much. Last year, corporate profits were 21.7% of GDP. If we allow for some of the income of the self-employed also being capital income, then a is around 0.3.

Which implies that a 10% increase in the capital stock, all else equal, will give us a 3% rise in output.

But 10% of the capital stock is a huge sum. The ONS estimates that the net capital stock excluding housing is £3.5 trillion. So a 10% rise in it is equivalent to 15% of GDP. It would take many years (and much else!) to achieve that.

If this sounds modest, remember that investment doesn't merely to add to output. It can reduce it. When Lidl invests in a new store, it reduces Tesco's sales, for example. And investment in wind turbines is intended to reduce the output of gas-fired power stations.

We've lots of evidence from different times and places that the elasticity of output with respect to capital is indeed small. In his famous paper which kickstarted this approach to thinking about economic growth Robert Solow estimated (pdf) that only one-eighth of the increase in US GDP per worker between 1900 and 1949 was due to increases in the capital stock. The rest, he said, was due to technical progress. In Fully Grown, Dietrich Vollrath estimated that from 1950 to 2000 a rise in the US's physical capital stock per person accounted for only 0.64 percentage points of the 2.2 per cent annual rise in real GDP per capita. Nick Crafts has estimated (pdf) that less than one-third of growth in advanced economies between 1913 and 1973 was due to a bigger capital stock. And Gavan Conlon and colleagues have estimated (pdf) that a fall in capital growth accounted for less than half of the slowdown in the UK's hourly productivity growth between 2001-07 and 2014-19.

Now, we shouldn't be fixated on precise numbers here. Aggregate production functions and measures of both the capital stock and depreciation have massive theoretical and practical problems: Solow himself said that such measures "will really drive a purist mad." But the picture here is clear: it takes a lot of capital spending to deliver only moderate increases in GDP.

Intuition, I suspect, confirms this. Imagine if the capital stock were, say, twice as large as it is. How much higher would productivity be?

Almost certainly not twice as high. Having twice as much office space won't make us twice as productive. Having twice as many lorries won't allow drivers to make twice as many deliveries in a week. Having more trains will get you to work in a less frazzled state, but not so much so as to double your productivity. More software would not necessarily eliminate bugs but would perhaps have more features which we just rarely use. And so on.

Let's put this another way. If the elasticity of output with respect to capital were large, then we might presume that capital investment was very profitable. But if this were the case, then we would already be seeing lots of it. Which we are not. Granted, this is partly because economic instability and a lack of finance have held back capital spending. But you have to argue that these are very powerful constraints indeed to solve this basic paradox.

Instead, elasticity optimists have a better potential response. It's that investment does not raise output merely by increasing the amount of kit we have. It's that new capital goods embody better technologies, so investment raises total factor productivity.

Yes, but by how much? The newest lorry, or latest version of any software package, isn't very much better than five-year old ones.

But what about investment in brand-new industries? Here, we run into two problems. One is brute maths. New industries, by definition, are small and so even if they expand tremendously their macroeconomic impact is small; an industry that accounts for 1% of GDP and trebles over ten years will add only 0.2 percentage points a year to GDP growth - and less insofar as it detracts demand from older businesses. The other is that new techs are often initially unreliable (do a Google image search for London Bridge) and it's hard to find workers and managers who understand them. That constrains both their expansion and efficiency. For these reasons, Nick Crafts has said (pdf) that "the early years of a general purpose technology will see little or no impact on aggregate productivity growth." Between 1760 and 1830, UK GDP per head grew by only 0.2% a year even in the face of technical changes that transformed lives.

In fact, there are two other reasons why increased capital spending won't increase growth very much.
One is that many investments won't yield the expected gains. This has little to do with the state making bad decisions. It's simply because of what Keynes called "the dark forces of time and ignorance which envelope our future". Nobody can reliably foretell which projects will pay off and whch won't and so some investment spending will inevitably be wasted. Indeed, because decisions are driven by sentiment ("animal spirits") Salman Arif and Charles Lee have found that increases in capital spending lead to lower, not higher, macroeconomic growth in the short-run.

There's a second problem. Devoting more of our national income to investment requires that we devote less to private or public consumption. That would entail uncomfortable changes in our standard of living. It would also require hundreds of thousands of people to change jobs, away from consumer sectors and towards producing capital goods; where else can the builders, turbine and solar panel installers come from? That would take many years simply because people are slow to move and retrain. There's little sign that the government is ready to face such changes in our economic priorities.

Obviously, none of this is to decry increased investment. It will raise growth - just not by much.

Instead, what I'm saying is that raising long-term trend GDP growth significantly is damnably hard, and it cannot be achieved by a single magic bullet or reliance upon a few new industries. Instead, what's needed is a broad spectrum approach using many policies. Not least of these is to adopt the Obama doctrine: "don't do stupid shit" such as preventing growth by planning controls, trade barriers or political instability. On this, Labour has at least made a start. But it is the start of a long and difficult journey.