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Since Piketty offered a new view of capital/income ratio, numerous attempts have been made to examine the relationship between return on capital, economic growth and the capital/income ratio. This paper attempts to shed new light on this field. More precisely, following recent literatures that pay attention to dynamics of external balance sheets of countries, we examine if Piketty’s results for large countries are robust for a country that takes the world rate of return on capital as given and whose savings rate increases gradually from negative value. It is revealed that for such a country, (1) Kuznets curve is drawn and (2) capital/income ratio decreases in accordance with a rise in savings rate and return on capital.

Researches on national accounting system have entered a new phase since Piketty and his coauthors offered a sweeping new view of capital/income ratio, that is, the capital/income ratio increases if rate of return on capital (r) is greater than economic growth rate (g) (Atkinson, Piketty and Saez (2011) [

Rowthorn (2014) [

Mankiw (2015) [

The present paper, in the spirit of these studies, attempts to shed new light on the aspects that are missed in the research on the capital/income ratio. More precisely, following the recent literatures that pay attention to the dynamics of the external balance sheets of countries (Lane and Milesi-Ferretti (2007) [

Analysis of the present paper demonstrates that Kuznets curve (a hump-shaped trajectory ontime-capital/ income ratiospace, shown by Kuznets (1955) [

Let us consider a small open country that takes the world rate of return on capital as given. Following Piketty and Zucman (2014) [

whose accumulation dynamics is

where S(t) is the saving in period t.

With reference to national income also, as in Piketty and Zucman (2014) [_{D}(t), and income from the net foreign assets in period t, rF(t). That is to say,

by making use of (1). Y_{D}(t) is assumed to be produced by a Cobb-Douglas function

If we normalize the labor to be unity for the simplicity of analysis, and assume the capital receives its marginal product (i.e.,

Note that in the setting of the present paper, the optimal amounts of capital and domestic output remain constant over time. Thus, we can say that results of the present paper are virtually the same as those derived by assuming that capital and domestic output are already at their optimal levels.

By substituting (4) and (5) into (3), we have the national income in period t, Y(t), as

If we assume that a fraction s(t) of the national income is saved in period t and make use of (2), we obtain the wealth accumulation dynamics as

which reduces to the following derivative of W with respect to time t

assuming continuous time horizon in order to simplify the analysis.

Differentiating (6) and substituting (8) into it, we have

Since s(t) < 1, it follows that r is greater than g (i.e., r > g), the inequality Piketty (2014) [

Now, if we let β(t) denote the capital/income ratio in period t (i.e.,

Since

In the following, in order to obtain a concrete solution, we assume s(t) to be a linear function of t, i.e.,

If we insert

which is solved as

by noting that θ, η and r are exogenous variables, setting

Now, let us specify C = 1, so that we have

by assuming the initial conditions to be

Graph of (14) is depicted ont-β space as a hump-shaped trajectory as in

Proposition:

Capital/income ratio β(t) first increases and then decreases as time goes by in a country that takes the constant world rate of return on capital as given and whose savings rate increases gradually from negative value.

This proposition is what Kuznets curve implies. So that, we can say from this proposition that Kuznets curve, which Piketty (2014) [

By making use of Equation (14), we can also show the effect of the change in exogenous variables to the shape of Kuznets curve. That is to say, the curve shifts inwards as in

Thus, we have the following corollary.

Corollary:

Capital/income ratio β decreases if (1) the savings rate increases or (2) the return on capital increases, in a country that takes the constant world rate of return on capital as given and whose savings rate increases gradually from negative value.

Corollary (1) is in sharp contrast to Piketty’s Second Law of Capitalismin Piketty (2014),

means that a rise in the savings rate s increases β. Corollary (2) also contrasts to Piketty (2014) [

In the present paper, we investigate if Piketty’s results for large countries are robust for a country that takes the world rate of return on capital as given and whose savings rate increases gradually from negative value. The main findings of this extended Piketty’s model are: for such a country, (1) Kuznets curve is drawn and (2) capital/income ratio decreases in accordance with a rise in the savings rate and the return on capital, all of which are in sharp contrast to Piketty’s results for large countries, although the framework of this paper is basically the same as that of Piketty’s. From these results, we can say that what applications for large countries don’t apply for small countries.

In the setting of the present paper, capital and domestic output are constant over time, and hence, the engine of the economic growth is the foreign direct investment. This kind of modeling may be justified for a developing country where capital and domestic output reach their optimal levels quickly since each level is not so large. However, it remains to be seen if the results of the present paper are robust when it takes time for them to reach their optimal levels so as to necessitate formulating the process of capital accumulation. We take up such analysis next.

YasunoriFujita, (2016) Piketty’s Capital-Income Theory Reconsidered for a Small Open Economy with Increasing Savings Rate. Open Journal of Statistics,06,25-30. doi: 10.4236/ojs.2016.61004