This is an updated copy of the version on BadHistory. I plan to update it in accordance with the feedback I got.
I'd like to thank two people who will remain anonymous for helping me greatly with this post (you know who you are)
Three years ago a festschrift for Binay Bhushan Chaudhuri was published by Shubhra Chakrabarti, a history teacher at the University of Delhi and Utsa Patnaik, a Marxist economist who taught at JNU until 2010.
One of the essays in the festschirt by Utsa Patnaik was an attempt to quantify the "drain" undergone by India during British Rule. Her conclusion? Britain robbed India of $45 trillion (or £9.2 trillion) during their 200 or so years of rule. This figure was immensely popular, and got republished in several major news outlets (here, here, here, here (they get the number wrong) and more recently here), got a mention from the Minister of External Affairs & returns 29,100 results on Google. There's also plenty of references to it here on Reddit.
Patnaik is not the first to calculate such a figure. Angus Maddison thought it was £100 million, Simon Digby said £1 billion, Javier Estaban said £40 million see Roy (2019). The huge range of figures should set off some alarm bells.
So how did Patnaik calculate this (shockingly large) figure? Well, even though I don't have access to the festschrift, she conveniently has written an article detailing her methodology here. Let's have a look.

How exactly did the British manage to diddle us and drain our wealth’ ? was the question that Basudev Chatterjee (later editor of a volume in the Towards Freedom project) had posed to me 50 years ago when we were fellow-students abroad.

This is begging the question.

After decades of research I find that using India’s commodity export surplus as the measure and applying an interest rate of 5%, the total drain from 1765 to 1938, compounded up to 2016, comes to £9.2 trillion; since $4.86 exchanged for £1 those days, this sum equals about $45 trillion.

This is completely meaningless. To understand why it's meaningless consider India's annual coconut exports. These are almost certainly a surplus but the surplus in trade is countered by the other country buying the product (indeed, by definition, trade surpluses contribute to the GDP of a nation which hardly plays into intuitive conceptualisations of drain).
Furthermore, Dewey (2019) critiques the 5% interest rate.

She [Patnaik] consistently adopts statistical assumptions (such as compound interest at a rate of 5% per annum over centuries) that exaggerate the magnitude of the drain

Moving on:

The exact mechanism of drain, or transfers from India to Britain was quite simple.

Convenient.

Drain theory possessed the political merit of being easily grasped by a nation of peasants. [...] No other idea could arouse people than the thought that they were being taxed so that others in far off lands might live in comfort. [...] It was, therefore, inevitable that the drain theory became the main staple of nationalist political agitation during the Gandhian era.

- Chandra et al. (1989)

The key factor was Britain’s control over our taxation revenues combined with control over India’s financial gold and forex earnings from its booming commodity export surplus with the world. Simply put, Britain used locally raised rupee tax revenues to pay for its net import of goods, a highly abnormal use of budgetary funds not seen in any sovereign country.

The issue with figures like these is they all make certain methodological assumptions that are impossible to prove. From Roy in Frankema et al. (2019):

the "drain theory" of Indian poverty cannot be tested with evidence, for several reasons. First, it rests on the counterfactual that any money saved on account of factor payments abroad would translate into domestic investment, which can never be proved. Second, it rests on "the primitive notion that all payments to foreigners are "drain"", that is, on the assumption that these payments did not contribute to domestic national income to the equivalent extent (Kumar 1985, 384; see also Chaudhuri 1968). Again, this cannot be tested. [...] Fourth, while British officers serving India did receive salaries that were many times that of the average income in India, a paper using cross-country data shows that colonies with better paid officers were governed better (Jones 2013).

Indeed, drain theory rests on some very weak foundations. This, in of itself, should be enough to dismiss any of the other figures that get thrown out. Nonetheless, I felt it would be a useful exercise to continue exploring Patnaik's take on drain theory.

The East India Company from 1765 onwards allocated every year up to one-third of Indian budgetary revenues net of collection costs, to buy a large volume of goods for direct import into Britain, far in excess of that country’s own needs.

So what's going on here? Well Roy (2019) explains it better:

Colonial India ran an export surplus, which, together with foreign investment, was used to pay for services purchased from Britain. These payments included interest on public debt, salaries, and pensions paid to government offcers who had come from Britain, salaries of managers and engineers, guaranteed profts paid to railway companies, and repatriated business profts. How do we know that any of these payments involved paying too much? The answer is we do not.

So what was really happening is the government was paying its workers for services (as well as guaranteeing profits - to promote investment - something the GoI does today Dalal (2019), and promoting business in India), and those workers were remitting some of that money to Britain. This is hardly a drain (unless, of course, Indian diaspora around the world today are "draining" it). In some cases, the remittances would take the form of goods (as described) see Chaudhuri (1983):

It is obvious that these debit items were financed through the export surplus on merchandise account, and later, when railway construction started on a large scale in India, through capital import. Until 1833 the East India Company followed a cumbersome method in remitting the annual home charges. This was to purchase export commodities in India out of revenue, which were then shipped to London and the proceeds from their sale handed over to the home treasury.

While Roy's earlier point argues better paid officers governed better, it is honestly impossible to say what part of the repatriated export surplus was a drain, and what was not. However calling all of it a drain is definitely misguided.
It's worth noting that Patnaik seems to make no attempt to quantify the benefits of the Raj either, Dewey (2019)'s 2nd criticism:

she [Patnaik] consistently ignores research that would tend to cut the economic impact of the drain down to size, such as the work on the sources of investment during the industrial revolution (which shows that industrialisation was financed by the ploughed-back profits of industrialists) or the costs of empire school (which stresses the high price of imperial defence)

​

Since tropical goods were highly prized in other cold temperate countries which could never produce them, in effect these free goods represented international purchasing power for Britain which kept a part for its own use and re-exported the balance to other countries in Europe and North America against import of food grains, iron and other goods in which it was deficient.

Re-exports necessarily adds value to goods when the goods are processed and when the goods are transported. The country with the largest navy at the time would presumably be in very good stead to do the latter.

The British historians Phyllis Deane and WA Cole presented an incorrect estimate of Britain’s 18th-19th century trade volume, by leaving out re-exports completely. I found that by 1800 Britain’s total trade was 62% higher than their estimate, on applying the correct definition of trade including re-exports, that is used by the United Nations and by all other international organisations.

While interesting, and certainly expected for such an old book, re-exporting necessarily adds value to goods.

When the Crown took over from the Company, from 1861 a clever system was developed under which all of India’s financial gold and forex earnings from its fast-rising commodity export surplus with the world, was intercepted and appropriated by Britain. As before up to a third of India’s rising budgetary revenues was not spent domestically but was set aside as ‘expenditure abroad’.

So, what does this mean? Britain appropriated all of India's earnings, and then spent a third of it aboard? Not exactly. She is describing home charges see Roy (2019) again:

Some of the expenditures on defense and administration were made in sterling and went out of the country. This payment by the government was known as the Home Charges. For example, interest payment on loans raised to finance construction of railways and irrigation works, pensions paid to retired officers, and purchase of stores, were payments in sterling. [...] almost all money that the government paid abroad corresponded to the purchase of a service from abroad. [...] The balance of payments system that emerged after 1800 was based on standard business principles. India bought something and paid for it. State revenues were used to pay for wages of people hired abroad, pay for interest on loans raised abroad, and repatriation of profits on foreign investments coming into India. These were legitimate market transactions.

Indeed, if paying for what you buy is drain, then several billions of us are drained every day.

The Secretary of State for India in Council, based in London, invited foreign importers to deposit with him the payment (in gold, sterling and their own currencies) for their net imports from India, and these gold and forex payments disappeared into the yawning maw of the SoS’s account in the Bank of England.

It should be noted that India having two heads was beneficial, and encouraged investment per Roy (2019):

The fact that the India Office in London managed a part of the monetary system made India creditworthy, stabilized its currency, and encouraged foreign savers to put money into railways and private enterprise in India. Current research on the history of public debt shows that stable and large colonies found it easier to borrow abroad than independent economies because the investors trusted the guarantee of the colonist powers.

​

Against India’s net foreign earnings he issued bills, termed Council bills (CBs), to an equivalent rupee value. The rate (between gold-linked sterling and silver rupee) at which the bills were issued, was carefully adjusted to the last farthing, so that foreigners would never find it more profitable to ship financial gold as payment directly to Indians, compared to using the CB route. Foreign importers then sent the CBs by post or by telegraph to the export houses in India, that via the exchange banks were paid out of the budgeted provision of sums under ‘expenditure abroad’, and the exporters in turn paid the producers (peasants and artisans) from whom they sourced the goods.

Sunderland (2013) argues CBs had two main roles (and neither were part of a grand plot to keep gold out of India):

Council bills had two roles. They firstly promoted trade by handing the IO some control of the rate of exchange and allowing the exchange banks to remit funds to India and to hedge currency transaction risks. They also enabled the Indian government to transfer cash to England for the payment of its UK commitments.

​

The United Nations (1962) historical data for 1900 to 1960, show that for three decades up to 1928 (and very likely earlier too) India posted the second highest merchandise export surplus in the world, with USA in the first position. Not only were Indians deprived of every bit of the enormous international purchasing power they had earned over 175 years, even its rupee equivalent was not issued to them since not even the colonial government was credited with any part of India’s net gold and forex earnings against which it could issue rupees. The sleight-of-hand employed, namely ‘paying’ producers out of their own taxes, made India’s export surplus unrequited and constituted a tax-financed drain to the metropolis, as had been correctly pointed out by those highly insightful classical writers, Dadabhai Naoroji and RCDutt.

It doesn't appear that others appreciate their insight Roy (2019):

K. N. Chaudhuri rightly calls such practice ‘confused’ economics ‘coloured by political feelings’.

​

Surplus budgets to effect such heavy tax-financed transfers had a severe employment–reducing and income-deflating effect: mass consumption was squeezed in order to release export goods. Per capita annual foodgrains absorption in British India declined from 210 kg. during the period 1904-09, to 157 kg. during 1937-41, and to only 137 kg by 1946.

Dewey (1978) points out reliability issues with Indian agriculutural statistics, however this calorie decline persists to this day. Some of it is attributed to less food being consumed at home Smith (2015), a lower infectious disease burden Duh & Spears (2016) and diversified diets Vankatesh et al. (2016).

If even a part of its enormous foreign earnings had been credited to it and not entirely siphoned off, India could have imported modern technology to build up an industrial structure as Japan was doing.

This is, unfortunately, impossible to prove. Had the British not arrived in India, there is no clear indication that India would've united (this is arguably more plausible than the given counterfactual¹). Had the British not arrived in India, there is no clear indication India would not have been nuked in WW2, much like Japan. Had the British not arrived in India, there is no clear indication India would not have been invaded by lizard people, much like Japan. The list continues eternally.
Nevertheless, I will charitably examine the given counterfactual anyway. Did pre-colonial India have industrial potential? The answer is a resounding no.
From Gupta (1980):

This article starts from the premise that while economic categories - the extent of commodity production, wage labour, monetarisation of the economy, etc - should be the basis for any analysis of the production relations of pre-British India, it is the nature of class struggles arising out of particular class alignments that finally gives the decisive twist to social change. Arguing on this premise, and analysing the available evidence, this article concludes that there was little potential for industrial revolution before the British arrived in India because, whatever might have been the character of economic categories of that period, the class relations had not sufficiently matured to develop productive forces and the required class struggle for a 'revolution' to take place.

A view echoed in Raychaudhuri (1983):

Yet all of this did not amount to an economic situation comparable to that of western Europe on the eve of the industrial revolution. Her technology - in agriculture as well as manufacturers - had by and large been stagnant for centuries. [...] The weakness of the Indian economy in the mid-eighteenth century, as compared to pre-industrial Europe was not simply a matter of technology and commercial and industrial organization. No scientific or geographical revolution formed part of the eighteenth-century Indian's historical experience. [...] Spontaneous movement towards industrialisation is unlikely in such a situation.

So now we've established India did not have industrial potential, was India similar to Japan just before the Meiji era? The answer, yet again, unsurprisingly, is no. Japan's economic situation was not comparable to India's, which allowed for Japan to finance its revolution. From Yasuba (1986):

All in all, the Japanese standard of living may not have been much below the English standard of living before industrialization, and both of them may have been considerably higher than the Indian standard of living. We can no longer say that Japan started from a pathetically low economic level and achieved a rapid or even "miraculous" economic growth. Japan's per capita income was almost as high as in Western Europe before industrialization, and it was possible for Japan to produce surplus in the Meiji Period to finance private and public capital formation.

The circumstances that led to Meiji Japan were extremely unique. See Tomlinson (1985):

Most modern comparisons between India and Japan, written by either Indianists or Japanese specialists, stress instead that industrial growth in Meiji Japan was the product of unique features that were not reproducible elsewhere. [...] it is undoubtably true that Japan's progress to industrialization has been unique and unrepeatable

So there you have it. Unsubstantiated statistical assumptions, calling any number you can a drain & assuming a counterfactual for no good reason gets you this $45 trillion number. Hopefully that's enough to bury it in the ground.
^{1. Several authors have affirmed that Indian identity is a colonial artefact. For example see Rajan 1969:}

^{Perhaps the single greatest and most enduring impact of British rule over India is that it created an Indian nation, in the modern political sense. After centuries of rule by different dynasties overparts of the Indian sub-continent, and after about 100 years of British rule, Indians ceased to be merely Bengalis, Maharashtrians,or Tamils, linguistically and culturally.}

^{or see Bryant 2000:}

^{But then, it would be anachronistic to condemn eighteenth-century Indians, who served the British, as collaborators, when the notion of 'democratic' nationalism or of an Indian 'nation' did not then exist. [...] Indians who fought for them, differed from the Europeans in having a primary attachment to a non-belligerent religion, family and local chief, which was stronger than any identity they might have with a more remote prince or 'nation'.}

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