Does History confirm that prosperity, capitalism, and some form "democracy" or representative republican form of government always go hand in hand?

The trivial thing to do is to look at the list of countries sorted by GDP per capita.

At the top positions of the list one can find several good old oil-rich monarchies. Surely the source of the wealth for those countries is.... democracy?

One can find EU member states deep down in the middle of the list. Can an EU member state be non-democratic?

Now let's look at the very bottom of the list. How many countries there are not trying to be democracies? Not a single monarchy, princedom or a satrapy. They all are Republics.

If you're comparing states with significant amounts of representative action (historially very few) vs those ruled by oligarchies or monarchies, the answer is clearly that prosperity is nearly guaranteed during and after the periods of higher representation.

In comparison, oligarchies and monarchies almost always lead to long periods of stagnation. At best, these arrangements generate short periods of growth before heavy stagnation. I wrote a couple blog entries on ancient representative governments and the rise of modern democracy that give a summary of all the oldest representative governments. They all, without fail, have become prosperous.

Modern "democracies" are murkier because the amount of representation they actually have is often hidden - many places that call themselves democracies or republics are really oligarchies or dictatorships in disguise.

To complement Mark's answer, from the Marxist perspective where "prosperity" could only be viewed as the maximisation of value under bourgeois control.

I was reading an abstract today on the commodity form's [use in human communities, resulting in reflected legal and political ideology as people consider their practice in the light of commodity society, and these relation's production of] conflicting construction of rights doctrines from property. The conflicting constructions require individuals to be able to own property and the state to be able to regulate this ownership in the interests of the general condition of individuals owning property.

Such a society has tended to

• Recognise individuals before the law
• Promote free (dispossessed) labour as the general form of labour
• Deal with the social problems from above by democratisation of the political sphere combined with the dedemocratisation of the economic sphere (chiefly through promoting employer controlled labour markets)

There are strong countervailing tendencies such as the requirement to preserve the commodity form (reaction, fascism, bastard capitalisms like the Soviet Union); or, the opportunities of primary accumulation through enslavement (Federici's Caliban and the Witch); or, the requirement to preserve the commodity form (industrial democracy, works councils).

These effects don't occur at the national level either, but in terms of transnational structures of capital ("imperialism," "the metropole/the periphery"); though obviously capital has been provincial or national at different times.

Any answer to this question will necessarily import a theoretical perspective.

No. It does not always go hand in hand. In fact, if history is any guide, it tended to be the opposite. There are a lot more successful, stable empires than democracy. Many empires saw long period of peace and prosperity that they were called "Pax something", like Pax Romana, Pax Mongolica and Pax Ottomana. Other than those we also have the Imperial China, the Persian (Achaemenid) Empire, and the Islamic Caliphates. If this list does not convince you, take a look at this wikipedia article and you will find most of them are non-democratic.

In fact, maybe it is harder to find democracies successful enough to rival the above examples. Probably the British Empire (though I'm not sure how democratic it was during its peak) and the contemporary United States and Europe. Many successful modern democracies (e.g. post-war Japan, that you mentioned in the comments) have only been around for a couple of decades, so in historical perspective they have only existed for a small period.

That said, in my opinion there are many problems in using history in comparing democracies and empires. The main one - as you mentioned in the comments is that the modern kind of democracies have not been around for long. Nationwide (as opposed to city-state) elections, universal vote and the secret ballot have all been around only relatively recently. We need more time to fairly judge whether it is more correlated to prosperity than non-democratic empires.

Heavily revised based on comments

This question is NOT simple; I would argue that it cannot be answered, but I can outline what I think the parameters of the answer might be.

In "The Origins of Political Order", Fukayama makes a throwaway comment that the key is that the government be accountable to the people; democracy is only one of the ways that accountability can be created. He doesn't develop the concept (at least not to the extent that I would have liked), but I suspect it is the key to the answer to this question. Let’s begin by setting some boundaries to make the question reasonable.

• I’m going to ignore historical examples that are pre-capitalist, because they undermine the assumptions of the question (prosperity, capitalism and some form of democracy) I’m going to arbitrarily set a boundary of 1600CE ; the boundary is arbitrary because justifying the boundary would extend the answer even closer to book length, and I’m going to try to be brief. This also neglects the transitions between mercantilist capitalism, consumer capitalism, managed capitalism (Breton Woods) and post-modern capitalism. I simply can’t explain that briefly.
• I’m also going to constrain the question to countries where policy choices affect prosperity, capitalism and democracy (this eliminate countries with the Dutch Disease; they are interesting to study, but I believe they’re distinct from the question. Countries cannot fiat the existence or value of natural resources.)
• I’m going to exclude nations that restrict ownership of property or participation in economic transactions. Nations that permit slavery, or exclude minorities from normal economic activities are exercising policy choices, but I find the choices repugnant. This is a problematic assumption because of the problem of Gastarbeiters in Europe, immigrants in the US, etc. Those are fascinating questions, but I’m already failing at my goal of being brief.

Definition of terms

We need to define democracy, capitalism and prosperity before we can proceed.

Democracy (I’m going to ignore the distinction between democracy, republic, and a couple of others.) is difficult to define because North Korea, East Germany and the United States all claim to be democratic, yet all fall short of the ideal in different ways. For simplicity of argument, let us suppose a bipolar scale with some democratic ideal at one end, and some form of authoritarian regime at the other. I’ll use Linz’s criteria to distinguish – countries on the democratic end have (1) stronger/more effective constraints on political institutions, (2) legitimacy based on something other than emotion, (3) minimal constraints on political mobilization by the inhabitants and (4) formally defined and stable definitions of executive power. Countries on the authoritarian end of the axis have minimal constraints on political institutions, legitimacy based on emotional appeal, strong constraints on political mobilization and informal or unstable definitions of executive power. This is actually the easiest of the terms to define.

Capitalism – I’m going to make the question far harder by including both pure capitalism and welfare state capitalism. This significantly weakens the utility of the answer, but if we restrict welfare state capitalism from the analysis there are a dearth of examples. For simplicity let us ignore the problem of sovereign wealth funds (which significantly weakens the utility of China as an example) and assume that capitalism is correlated with the % of GDP which is in private control vs public control.

Prosperity Nobody is happy with the established definitions of prosperity; they are only distinguished by the fact that they are the least bad of the known alternatives. (I’m going to ignore the Nepalese efforts to measure “Gross National Happiness”, and Marxist exchange entitlements and a hundred other metrics).
We could measure prosperity by long term positive trend in GDP or in long term positive trend in per-capita GDP. I don’t have the numbers in front of me, but I strongly suspect that this choice will define the answer. If we measure by per-capita GDP then I strongly suspect that we’re measuring a participatory measure of economic power against a participatory measure of political power. I am willing to bet my lunch that there is a positive correlation between those two factors. If on the other hand we measure against total GDP, then we have a fascinating question. In fact, I think that is the question that OP is really seeking.

If we proceed on these assumptions (I don’t argue that they’re sound, merely that they provide reasonable boundaries for initial discussion), then I would argue that there is a positive relationship between economic participation, political participation and prosperity. The question is transformed to the Principal-Agent Problem.

If we restrict the discussion to modern consumer capitalism within the context of welfare state representative government, then I submit the correlation will be even stronger. Hayek, Gorbachov and Deng Xioping all agree that prosperity is correlated with the ability of the general population to benefit from prosperity.

It appears that authoritarian governments can employ strong economic policies to kickstart capitalism and raise prosperity. Both Japan and China have done this successfully. Of course the same policy can fail as has happened repeatedly. There are ideological arguments whether that is sustainable – some would argue that the current Japanese economic problems arise from governmental vetoes on normal economic corrective activities. I think there are more developmental economists studying this problem than there are countries in the dataset.