Dynamic Efficiency and Asset Bubbles
under Financial Market Friction #Masaya Sakuragawa
masaya@econ.keio.ac.jp

December 31, 2007
Abstract
This paper investigates whether the criterion for the real-interest-rate versus
the-economic-growth-rate is effective in evaluating dynamic efficiency/inefficiency andunderstanding rational bubbles in a dynamic economy with frictions of financial markets. In theclosed economy, this criterion is effective for the sustainable bubbles, but not in terms ofdynamic efficiency/inefficiency. Asset bubbles do not necessarily restore efficiency, but rather
asset bubbles typically coexist with capital under-accumulation. The effectiveness of thatcriterion is restored in the small-open economy if the output produced by borrowers isinternational collateral.

#We are gratefully thankful to Koichi Hamada, Masashi Manabe, and participants at seminar held inKeio.
11. Introduction
The criterion for the real interest rate versus the economic growth rate is a problem oftheoretical and empirical importance in evaluating the efficiency of the economy andunderstanding the sustainability of asset bubbles. Diamond (1965), Ihori (1978), and Tirole(1985) demonstrate that when the return to capital is equal to the real interest rate, this criterion
becomes a benchmark for understanding dynamic efficiency and sustainability of bubbles.
However, various kinds of financial frictions, including uncertainty, transaction costs,
asymmetric information, and other incentive problems will deter this equality to continue tohold in the actual economy so that the effectiveness of this criterion is an open question. Abel etal (1989) and Bohn (1995) provide examples in which the safe interest rate is smaller than theeconomic growth rate due to risk premium but the economy is dynamically efficient.
This problem is stringent also from the empirical prospect because the current speculative
bubbles will be associated with the historically low real interest rate. Figure 1 illustrates averageinterest rates and the average economic growth rate of G7 countries (U.S., U.K., Japan,Germany, France, Canada, and Italy).1 Remarkably, before the end of 1990s, all the interest
rates were greater than the economic growth rate except for the deposit rate, but after that, manyof the interest rates began to be smaller than the economic growth rates. Since 2000 and around,several kinds of bubbles recurrently emerged, including IT bubbles, housing market booming in
US, the global booming in the stock markets, and appreciations in gold and oil prices.2

1Each of interest rates and the economic growth rate is a simple average of G-7 countries (source:IFS). We exclude the 1991 data of German in calculating averages. Money market rate, treasury bills,
treasury bills: 3years or longer, deposit rate, and max overdraft reflect call rate, short-term rate of thegovernment bond, long-term rate of the government bond, short-term deposit interest rate, and loaninterest, respectively.
2

Additionally, behind asset bubbles in China that have been sustained for more than twodecades is the far higher economic growth rate than the interest rates.

2

Tirole (1985) demonstrates that asset bubbles arise when capital is overly accumulated in
the bubbleless economy. His finding is, however, inconsistent with the current world economy
in that Tirole (1985) can not explain the coexistence of “global bubbles” and weak incentive forinvestment relative to great global savings, in other words, the “asset shortage” following theterminology of Caballero (2006). Caballero and Krishnamurthy (2006) develop an insightfulmodel of a small-open economy in which capital under-accumulation and asset bubbles cancoexist. In theirs, asset bubbles are used for collateral for financing productive investment. Their
contribution is to show the complementary role of bubbles with capital accumulation, incontrast with Tirole where bubbles and capital accumulation are substitutable.
The purpose of this paper is to investigate the effectiveness of the criterion for thereal-interest-rate versus the-economic-growth-rate in evaluating dynamic efficiency/inefficiency
and understanding rational bubbles in the presence of financial market frictions. In doing so, weconstruct a simple model which is as close as possible to Diamond (1965), Ihori (1978), andTirole (1985) except for that there is a friction in the financial market. As the extensive literatureargues, borrowing constraints or credit rationing that arise in response to the friction of financial
markets leads to the breakdown of the equality between the return to capital and the real interest
rate (e.g., Stiglitz and Weiss (1981), Gale and Hellwig (1985), Williamson (1986), Bernanke andGertler (1989), and others). The discrepancy between the two rates has different results on
dynamic efficiency and sustainability of asset bubbles, and the criterion relative to the one in thestandard economy.
In the closed economy, the criterion for the real interest rate versus economic growth rate iseffective for the sustainable bubbles, but not in terms of dynamic efficiency/inefficiency. Assetbubbles move the capital stock down to the smaller level than the Golden Rule so that asset
bubbles and capital under-accumulation coexist. Asset bubbles do not in general restore3

dynamic efficiency, but do only if capital is too overly accumulated in the bubbleless economy.
The theoretical finding explains asset bubbles, asset shortage, and the low real interest rate
simultaneously, all of which features the current world economy. The effectiveness of thatcriterion is restored in the small-open economy with the real interest rate being constant if thecapital good becomes international collateral. In the small-open economy, asset bubbles nevercrowd out capital accumulation, but has the role of raising the real interest rate faced bydomestic investors.
This paper is organized as follows. Section 2 sets up the model and studies the benchmark
economy. Section 3 analyzes the closed economy when there is the friction of financial market,
and Section 4 the small-open economy. Section 5 concludes.

2. Basic Model
Let us consider an economy of overlapping generations that lasts for infinity. At each periodt = 0,1,2,..., ∞, the economy is populated by a continuum of ex ante identical agents that live
for two periods. Letting N t denote the number of young people at t, the population grows at
rate n > 0 , satisfying N t = (1 + n)t N 0 = (1 + n)t . At each period the final good is produced byfirms that use labor and capital as inputs according to the constant-returns-to-scale technology
described as Yt = F ( K t , N t ) , where K t and N t are aggregate supplies of capital and labor,and Yt is the output of the final good. That technology is described as a per-capita formby yt ≡ Yt N t = F ( K t N t ,1) ≡ f (kt ) , where k t is the capital-labor ratio and yt is theper-capita output of the final good. f (.) is thrice continuously differentiable, increasing,concave, satisfying f (0) = 0 , and lim f '( k t ) = +∞ . Since the production technology isk t →0homogeneous of degree one, output of the final good can be described in terms of the action of a4