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.