Financial deepening and economic growth

The core of Shapley–Shubik games and general equilibrium models with a Venn diagram is applied for a theory on the role of real finance in economic growth among advanced economies. Then the dynamic computable general equilibrium (DCGE) models for Germany, France, the UK, Japan and the USA are constructed to assess the validity of the over-financing hypothesis that has reappeared after the financial crisis of 2008. Actual financial deepening ratios observed in the nonconsolidated balance sheet of the OECD exceeded by factors of 3.5, 2.4, 5.1, 11.6 and 4.8 than the optimal financial deepening ratios implied by DCGE models, respectively, in these countries because of excessive leveraging and bubbles up to 19 times of GDP which were responsible for this great recession. Containing such massive fluctuations for macroeconomic stability and growth in these economies are not possible in conventional fiscal and monetary policy models and require a DCGE analysis like this along with adoption of separating equilibrium strategy in line of Miller–Stiglitz–Roth mechanisms to avoid problem of asymmetric information in the process of financial intermediation so that the gaps between actual and optimal ratios of financial deepening remain as small as possible.

the taxes despite growing risk of accumulation of public debts. Central banks have reduced the basic interest rate to record low rate since the beginning of central banking to expand the liquidity in the system; since January 2009, Federal fund rate has remained close to zero, Bank of England's basic rate is 0.5% and ECB's basic rate is now at 0.1%. Sources of credit levels of banks have been expanded under the quantitative easing. Why does a financial system collapse like this and how do they affect the long-run growth are questions of great interest.
Five major theories have been advanced to explain the role of financial sector in the economy in the literature. The first theory has its origin in the classical school of competitive and efficient markets. The fact that the process of capital accumulation and growth in modern economies is enhanced substantially by the financial markets that channel resources of millions of risk-averse savers to millions of riskneutral borrowers is well recognized. Schumpeter (1912) argued for financial development for economic growth, but Robinson (1953) viewed the financial development as a by-product of economic growth process. Importance of risk minimization and efficiency of portfolio allocation was noted by Markowitz (1959) and Merton (1973). Then Sidrauski (1967) and Tobin (1969) linked the balance sheet of the financial system to economic growth. The process of financial deepening (FD) and banking firms was discussed by Klein (1971) and Shaw (1973). These concepts were applied in the context of developing economies by McKinnon (1973) and Fry (1978). King and Levine (1993) and Levine (1997) tested these propositions empirically across countries. Hills et al. (2010) and Davies et al. (2010) studied recently the links of recessions and evolution of banking system in the context of the UK economy.
The second wave of literature in the FD and growth emphasizes the role of strategic modelling with Nash bargaining and signalling problems and coalition formation in line of Shapley (1953) and Shapley and Shubik (1969) and mechanism design of Rogerson (1985) and Roth (2008). Rasmusen (1987), Beaudry and Poitevin (1995), Cripps (1997), Dasgupta and Maskin (2000) and Roth (2008) further advanced strategic choices relating to investment. While the analysis of consequences of bank runs is found in Diamond et al. (1983), informal finance, stochastic factors and the financial structure and growth of economies are discussed in Townsend (1983), Boyd and Prescott (1986) and Bolnick (1987). Consequences of transaction cost in bilateral and multilateral negotiations (Balasko, 2003;Kiyotaki and Moore, 2006) and FD (Townsend and Ueda, 2006) were considered for developing models of coalition of intermediaries. Neoclassical and neo-Keynesian modelling paradigm of King et al. (1994) and Covas and Den Haan (2012) have refined linking of financial sectors to economic growth. Financial markets should be thick, less congested and safe for its participants as should the Kidney exchange centres be for the potential donors and receivers of Kidneys (Roth, 2008).
Third set of literature on finance and growth focuses on risk management and highlights the importance of the liquidity of the banking sector in theoretical or empirical settings in spirit of Epstein and Zin (1989), Fama (1980Fama ( , 2014, Spencer (1984Spencer ( , 2008, Bank of England (1999), Raghuram and Zingales (1998), Roubini and Sala-i-Martin (1992), Radelet et al. (1998), Cecchetti (2009), Brunnermeier (2009, Mendoza (2010) and Gai et al. (2008). Beck et al. (2000), Carlin and Mayer (2003) and Allena et al. (2014) survey the literature relating to the liberalization of financial sectors and associated problems including those of saving and loan associations in 1980 in the USA, bank runs and failures of giant banks in Japan in 1990s or the collapse of credit and housing markets in the USA and several EU economies recently including the credit crunch, bank failures, liquidity crises, stock market crash and bailouts in the UK, EU and the USA after the crisis in October 2008. Excellent intuition in these is found in Fama (2014) and Shiller (2014). New techniques on decomposing the impacts of shocks in the macroeconomy are developed further in Hansen (2012) and Brunnermeier and Sannikov (2014).
Finally, above propositions have been brought to empirical scrutiny as the data series on interest rates, deposits, stocks, bonds, foreign currency reserves and their prices are becoming increasingly available in recent years (see Taylor, 2010). Propositions of King and Levine (1993) and Levine (1997) have been tested for many economies in recent years (Beck et al., 2000;Arestis et al., 2001;Carlin and Mayer, 2003;Allena et al., 2014). Various studies exist on the evaluation of impacts of financial sector in the economy (Bank of England, 1999;Brunnermeier, 2009;and Cecchetti, 2009). How the asymmetry of information on depositors and savers results in volatilities of unimaginable proportions in these markets and how it affects the choices of economic agents and prospects of economies are analysed using testing theoretical models with empirical evidences. Financial markets often experience catastrophic failures whenever the expectations of lenders and borrowers do not match market realities.
Using four indicators of financial development for about 119 countries from 1960 to 1989, King and Levine (1993) had showed panel data analysis based empirical support for the Schumpeterian hypothesis that financial development leads to economic growth in contrast to the Robinsonian argument that growth rate of output had little connection to the financial development. The long-run growth is a function of real physical capital and not the financial leverages or derivatives that promote the artificial FD. Overfinancing, however, is a phenomenon that has become more serious in the last two decades. The results from the DCGE computations reveal that there are little differences between the optimal FD ratios (OFDRs) across countries, but there are large differences in actual financial ratios. Such gaps between these two measures are due to casino capitalism (Sinn, 2010) and asset bubbles or collective illusions as its consequence. It is pertinent here to consider Miller and Stiglitz's (2010) analytical model that weaves the financial intermediation with incentive distortions and information frictions to show how economy reacts during the time of fiscal shocks and financial instability while assessing implications of these bubbles.
None of the earlier studies have sufficiently addressed the issue of discrepancy between the optimal and actual financial ratios required for the growth as done in this article. Section II motivates the article with a short discussion of the underlying actual FD ratios (AFDR) from the OECD for five advanced economies. Section III presents concepts of an efficient competitive equilibrium mechanism theory contained in nonblocking core in Shapley-Shubik game and Pareto optimal core in a general equilibrium model that could be applied to analyse efficient allocations both in goods and asset markets. It illustrates the Schumpetarian view qualitatively that the growth of the financial sector is linked to the growth of the rest of the economy over time. Section IV illustrates how fluctuations in growth rates are caused by shocks in the financial sector with a simple endogenous growth model with financial intermediation in contrast to the Ramsey model in Bhattarai (2005) or cash in advance or money in utility function models in Bhattarai (2014). The article proceeds further in constructing multisectoral and multi-household DCGE models of Germany, France, UK, Japan and the USA in Section V to establish efficient and optimal paths of capital output ratios implied by underlying equilibrium mechanism illustrated in Figs 4 and 5. This article contributes to the literature by finding the degrees of the excess FDRs above the optimal ones required for smooth process of economic growth implied by the DCGE models of these economies. Conclusions, references andappendices supporting the study are in the final section.

II. Actual Financial Deepening Ratios: Statistical Facts
In general, the size of the financial assets a country has is closely linked to the size of GDP as shown in Figs. 1 and 2 for five advanced countries. Contrast GDP of 15.5 trillion to financial assets of 156.5 trillion dollars for the US economy with GDP of 1.5 trillion and FA of 29 trillion pounds for the UK. Data for the financial assets were obtained from the OECD's nonconsolidated balance sheets in which the financial assets include currency and deposits, financial derivatives, securities, shares and equities for the years 2007-2011. GDP figures were obtained from the OECD as well.
The actual financial deepening ratio (FA/GDP is calculated by dividing the financial assets by GDP as shown in Fig. 3. UK had the highest FD ratios followed by Japan, France, the USA and Germany. Thus, UK's financial system has more excess Financial deepening and economic growth K. Bhattarai leveraging than other countries and is more vulnerable to financial crisis like those of 2008. In fact, all economies are vulnerable to good or bad financial sector policies, degree of over-financing and wide ranging inefficiencies, fluctuations in growth of output and other economic activities whenever the AFDRs deviate significantly from optimal ones. Financial assets are counterparts of physical capital in a well-balanced economy. Thus, in the classical system with saving investment identity, the rate of capital accumulation not only reflects the rate of economic growth but also the accumulation of financial wealth in the economy. A higher degree of FD through saving and investment activities promotes the level of income and raises the rate of economic growth. In real world, the level of economic advancement seems to have gone together with the level of FD until the deregulation of financial markets in mid-1980s. However, this tacit link seems to have broken in recent years.
From the OECD data summarized more precisely in Table 1, it is clear that the FD ratios are twice as large in UK than that in Germany. While Japan is close to the UK but France and the USA are closer to Germany in these ratios. Thus, the data make it clear that UK and Japan are more vulnerable to financial sector turbulences than France, the USA and Germany. It is important to show that financial and real sectors of the economy are mirror images of each other. Consider an asset A t ð Þ accumulation equation as: where C t is consumption, A t financial assets, W t endowment andr tþ1 return to asset net of tax and depreciation rate;r t ¼ ð1 À τ k Þðr À δÞ with r real interest rate, δ rate of depreciation and τ k capital income tax. When τ k ¼ 0, Equation 1 can be written one gets the macro balance proving the equivalence between the financial assets and physical capital stocks in Equation 2 as: Thus, the stocks of financial assets A t ð Þ must balance with the stocks of physical capital K t ð Þ in a smooth functioning economy with unrestricted borrowings and lending though their values remain sensitive to shocks in various market conditions as discussed in Sections IV and V.
Optimal and actual financial deepening ratios Þis the result of the growth process in the economy and varies across production sectors F i;t À Á according to variations in investment rates and levels of output among them. This happens as banks channel funds saved by households or enterprises for investment by firms at the real interest rate that matches cost and productivity of funds to the firms. Value of OFDR t is obtained by dividing the capital stocks by the GDP. AFDR AFDR t ð Þ is obtained by dividing the total of financial assets from the nonconsolidated balance sheet by the GDP.
OFDR t is the real measure of optimal FD, resulting from the optimization behaviour of consumers and firms in the economy. It should be equal to the ratio of financial assets to GDP in the financial market in an ideal world as shown in Equation 4. Such intertemporal equilibria are guaranteed by the flexibility of prices, wages and interest rates in the economy. Imbalances either due to the rigid or inflexible prices cause market imperfections or crises in the real world, giving a different value to the AFDR t . Gap between OFDR t and AFDR t is due to conditions in the financial markets. Good financial policies result in right set of accumulation process and higher growth rate of the economy over periods. Then these two measures are expected to be closer as I found in case of emerging economies illustrated in a related paper that I prepared for the Review of Development Economics. Wrong financial sector policies lead to mismatch between the volumes borrowed and lent, that often manifests in terms of bailouts or subsidies or preferential treatment of one sector against another, which distorts the accumulation process ultimately reducing the prospects of the economy in the long run. This causes a large gap between OFDR t and AFDR t as presented in Section V.
Financial deepening and economic growth

K. Bhattarai
What is the optimal ratio for a bubble-less smooth functioning of these economies? For each period t, OFDR t is aggregated from the sectoral optimal ratios, OFDR i;t ¼ K i;t Y i;t obtained from the solution of the DCGE model with many production sectors. This is discussed in Section V after explaining the meaning of the core allocations and stochastic growth underlying those DCGE calculations in Sections III and IV.

III. Classical Theory: Core of Finance, Growth and Efficiency
The dynamic economy implied by models mentioned in Section I is better explained by diagrams in Figs. 4 and 5. Figure 4 shows the distinct possibilities of excess or shallow financing in comparison to the normal equilibrium path in the middle. Then the fluctuations around the steady state are shown in Fig. 5 where the E-E is the allocation representing the core equilibrium path, LL the market valuations of lenders and BB the market valuation of borrowers. The gap between LL and BB reflects the subjective differences in the assessment of prospects of financial assets and these gaps are the reason for trades among lenders and borrowers. Wide fluctuations in these are not only the sources of cycles but also the sources of crises.
Market equilibrium path E-E represents a no friction complete information world of lenders and borrowers. It ignores the asymmetry of information in financial markets, which is the underlying cause of deviation of asset accumulation path of borrowers (BB) and lenders (LL) of the equilibrium path (EE). The main intuitive points from classical theory of finance and growth thus are as follows: (1) Assets are results of consumption saving behaviour resulting from the inter-temporal optimization of households or firms.
(2) There is an equilibrium allocation EE for each time period of the economy that is at the core of the equilibrium. (3) Lenders and borrowers start with different amounts of endowments and bargain continuously in order to gain more from the transaction. (4) Underlying productivity and preferences cause differentiation in valuation by the buyers and sellers in the asset markets. Therefore, the valuation can be generalized in n number of cases. (5) Corrective measures are taken by individuals or the policy makers when these valuations significantly deviate away from the underlying equilibrium destabilizing the whole financial system. Miller-Stiglitz mechanism of bubbles and crashes is helpful in advancing above thoughts by designing incentive compatible contracts contained in Maskin and Tirole (1990) and Roth (2008) to separate normal borrower and lenders from risky ones under asymmetric information to solve moral hazard or adverse selection problems required to ensure efficient equilibrium path EE by minimizing gaps in their evaluation as shown above by LL and BB lines in Fig. 5. Arbitrage in the financial market should be set in such a way that it guarantees the efficient and Pareto optimal core of the economy in coalition games and growth with dynamic general equilibrium in the economy.
Arbitrage and core in games and general equilibrium models Arbitrage conditions set at the core of the economy lead to efficient decisions in the financial markets and promote growth. Game theory and general equilibrium models show how optimal choices made by consumers and producers facing the resource constraint are efficient when these set of points belong to the core of an economy. Arriving to these unique set points in the core involves continuous bargaining over the gains from the intra-and inter-temporal trade on goods, services and financial assets. Technically, the Shapley value of a bargaining game is given by the payoff from a nonblocking coalition in a Shapley-Shubik game, and it is a set of Pareto efficient points. Similarly, the core of a general equilibrium lies in the contract curve where it is difficult to make one economic agent better off without making another worse off. The core of the coalition in the game and that in a general equilibrium model basically represents the same efficient point and relative price as proven in Equation 4. These are consistent to the efficient arbitrage conditions in an efficient financial market. As the optimal allocation of resources to economic agents possible with given endowments confirms to the first and the second theorems of welfare economics, solutions either of game or DCGE models characterize the optimal allocation of resources after more complex bid and offer interactions among economic agents. This also happens to be the key process in the financial markets. This set of efficient points is illustrated by the intersections of three circles at the Economic agents in the financial markets tend to play a zero sum and noncooperative game when they are outside this core set. The benefits of coalition and cooperation far exceed from noncooperation (Gale, 1986). Even when agreements are made for cooperation, there are questions on whether such coalitions are stable. There are always temptations for at least some players to cheat and break the cooperative agreements in anticipation of raising their own share from the total gains against other players. However, such process sets a motion of negative externality and retaliations, resulting in mistrusts and eventually a low value of the game. No player can fool other players for long as they will discover the cheaters and penalize them more than what they could gain by cheating, thus giving the noncooperative Nash outcome of the game.
A financial coalition among players should be consistent to the individual rationality, group rationality and coalition rationality because of the super-additivity property. This implies that the value of the game in a coalition is greater than the sum of the value of the game of playing alone noncooperatively by those individual members. In case of three players, this means: v Þ , for each of its member than when playing alone with payoffs v 1 1 ð Þ; v 2 ð Þ; and v 3 ð Þ. Cooperation and team spirit generate extra benefits. Considering three sets, 1, 2 and 3, of possible allocations in a market, there is only a tiny set of core equilibrium as illustrated by the intersection of 1, 2 and 3 in the Venn diagram in Fig. 6. Financial arbitrage made at this core is efficient and optimal and bring  Debreu and Scarf (1963) had proven the equivalence of a competitive equilibrium to the core of the game for economies with and without production by contradiction when preferences are nonsatiable, strictly convex and continuous. Scarf (1967) theorem states that a balanced n person game has a nonempty core. Financial markets open each time, bid-offer process sets the prices of assets and exchange takes place in the core. This process continues forever. Thus, the competitive equilibrium is equivalent to the allocation at the core, 'An exchange economy with convex preferences always gives rise to a balanced n person game and such will always have a nonempty core (Scarf, 1967).' Financial deepening and economic growth smooth growth in the economy. Thus, efficient allocations in the economy are only a small subset of all possible allocations. Proliferations of financial assets as observed in the OECD data in this section are the union of sets rather than their interactions at the core.
On the other hand, a general equilibrium is given by the relative prices that clear all markets in the economy. It is derived using a sequence of correspondence and optimizing relations by which consumers and producers make prudent choices subject to resource or technology constraints and public policies. Consumers' optimal choice set is complete, transitive, continuous, monotonous and convex; u : R n ! Rjx 2 X f g . These contain quantities of n commodities (x ¼ x 1; x 2; . . . : :x n ) in nonnegative orthant of X 2 R n : These maximize utility uðxÞ subject to budget constraints p:x y. Given the input and output prices, w ! 0 and p ! 0 producers choose output levels y 2 R n þ to maximize their profits, π p; w ð Þ ¼ p:y À w:x;; y : R m ! Rjy 2 Y f g : Here y is produced using labour and capital. The general equilibrium system results in the Pareto optimal allocation when it is not possible to improve the level of welfare of one person without lowering the level of welfare of another person. Financial allocations emerging from this core given by the relative prices that guarantee equilibrium in the system lead to the most efficient outcome in terms of welfare and growth though these are often distorted by the tax, transfer, spending policies of government as well as tariffs and trade system in the global economy. The wide-ranging backward and forward linkage effects of the financial markets run on arbitrage principles are consistent to the feasibility and optimality of inter-temporal plans of consumers and producers at the core. This optimal core itself is, however, subject to shocks of financial frictions and technologies of production from time to time and can cause significant fluctuations in economic growth. How it happens is briefly illustrated in the next section in one sector growth model with financial intermediation to provide a background for the DCGE model in Section V.

IV. Model of Financial Intermediation and Endogenous Growth
Let a dynamic economy be expressed with a simple stochastic technology Y t ¼ z t K t where z t ,N 0; σ 2 ð Þ. Capital stock accumulates from investment, I t ¼ K tþ1 À 1 À δ ð ÞK t . Amount of investment deviates from saving depending on the efficiency of financial markets 0 < ϕ < 1 ð Þ , I t ¼ ϕ S t and I Y ¼ ϕ S Y as in (Bhattarai, 2005). Assuming market clearing Y t ¼ C t þ S t and a steady growth rate of the economy K tþ1 ¼ ð1 þ gÞ K t and the parameters z; φ; s and δ in Table 2 determine the growth rate of the economy as shown in Equation 5 2 and in Fig. 7. Kiyotaki and Moore (2006) illustrate importance of the bilateral and multilateral commitment in maintaining the efficiency of the financial system ϕ ð Þ like this. 3  (2014) numerically shows how financial crises of 2008 could be explained due to the shocks to these real sides of the financial system with standard dynamics contained in simple cash in advance in Sargent (1987) and money in utility theories Sidrauski (1967) in small prototype models.

1142
K. Bhattarai Such excess volatility in economic growth causes further bubbles or crash in contagious fashion as shown by Miller and Stiglitz (2010) resulting in panic runs to the banks or exuberances as shown in Figs 4 and 5 above with a wide gaps between OFDR t and AFDR t . Policy analyses and prescriptions that are not based on the structural features of the economy and heterogeneity in consumption, production and trade can hardly come up with a concrete solutions required to resolve the problem. Despite a large body of theoretical and empirical literature on finance and growth mentioned in Section I, very little work has taken place in analysing the financial markets with a dynamic general equilibrium model. This lacuna in the literature motives us for this effort on constructing a DCGE model to explain the implications of FD on efficiency, growth and redistribution refining Bhattarai (1997). The main ingredients of these DCGE models are presented in the next section.

V. Finance in a Dynamic CGE Model
A DCGE model quantifies how the size of financial sector relates and contributes to the economic growth at the core over time. There are mainly two different theories relevant to a DCGE. One is the classical approach which takes finance as a by-product of investment and saving activities among economic agents. The size of the financial sector is basically determined by the rate of saving and investment, marginal productivity of capital and size of the economy in it. This structure resembles a competitive market economy with the neoclassical or Ramsey process of economic growth. In more recent theories, the size of financial sector and the economy are endogenously linked to each other and determined by the risk-taking behaviour or risk-pooling arrangements of economic agents. Risky projects usually have higher rate of return, but investors are willing to take risky projects only when the risk is pooled among borrowers and lenders by an insurance mechanism. An economy with greater degree of risk pulling will have higher rate of investment and growth and larger financial sector because of implementation of more productive investment plans in general. Higher level of income in turn allows more amounts to be saved and invested. Greater the degree of capital accumulation, the bigger are the coalitions of intermediaries and larger the size of the financial sector (Townsend, 1983;Greenwood and Jovanovic, 1990;and Acemoglu and Zilibotti, 1997;Balasko, 2003;Brunnermeier and Sannikov, 2014). This theory is supportive of the deregulation and liberalization of financial sector after 1980s. It is, however, irony that the risk-taking behaviour can reach out of proportions and create bubbles and lead to collapse of the financial system as it happened unprecedented in financial crisis of 2008 (and in several episodes of them that preceded it). Such devastating experience has made economists think about structural theories of bubbles originating in the financial sector that spreads adverse consequences not only on asset prices but also on investment, growth, employment and welfare of the households in the economy. The DCGE model proposed here takes main points of above theories and properly accounts for the inter-temporal preferences of households between the current and future consumption (and saving), long-run decision of investors in accumulating capital and the policies of the government that often affects either positively or negatively on choices of these heterogeneous firms and households. It is pertinent to present the generic structure of a dynamic general equilibrium model here and to apply it to the five advanced economies selected for this study with a focus on the OFDRs emerging from the optimizing behaviour of consumers and producers in these economies.

Consumers
Consumers are looking forward in the DCGE model. They are interested in smoothing out their lifetime consumption in order to guarantee a certain level of utility or standard of life for each period in their life, given the subjective discount factors 0 < β h < 1. This requires inter-temporal optimization over the lifetime, maximizing lifetime utility U h 0 À Á given the present value of the lifetime income Equation 8 and budget constraints Equation 9.
Financial deepening and economic growth Each consumer starting from initial endowment of physical capital K h 0 À Á and labour time L h 0 À Á makes decision to consume C h i À Á and work LS h t ¼ L h t À L h t and save from its full income I h t À Á in each period leaving it to the banking system to channel those savings to the potential investors.
Households supply factors of production, K h t and LS h t , to firms. They receive net of tax wage income in return to labour supply w h t 1 À t l ð ÞL h t h i and capital income r t 1 À t k ð ÞK h t Â Ã in return to their investment. They pay taxes on their capital and labour incomes and may receive transfer payments (R h t ) from the government on the mean tested basis.

Firms
Firms are central to the supply of goods and services. Given the production technology, optimal choices of inputs are made to maximize profits in each period and over the model horizon. Entry and exit are allowed with regulations to maintain a competitive economy. Therefore, in each period, firms compare prices of inputs and products r i;t ; w h t ; p i;t À Á and determine the optimum level of output that would maximize profits. Implicitly, the level of output depends on the relative prices of inputs and outputs as: The structure of inputs and the type of technology differs for firms operating in different sectorsagriculture, manufacturing and services. Some are capital intensive and others labour intensive, operating on linear, Cobb-Douglas or constant elasticity of substitution technologies. All of them are interested to maximize total profit given the process of capital accumulation K i;t ¼ 1 À δ i;t À Á K i;tÀ1 þ I i;t .

Trade
Economies modelled here are price takers in the global market except that they need to balance their trade over time. Adjustment in the real exchange rates brings such balance in the value of imports Real exchange rate links the ratio of weighted price indices of imports and exports and thus are determined by PE i;t and PM i;t .

Government
Government provides public services like law and order, education and health, social security and pension and protection of environment to households and firms and adds to the public capital by investing in economic infrastructure, health and education. These expenditures enhance productivity and make these economies more competitive in the global market. In a dynamic economy, the public spending 1144 K. Bhattarai should balance with the public revenue as shown in (14).
Government collects revenue through direct taxes on income of households and firms and indirect taxes on their consumption. The optimal level of public expenditure and revenues is set when the benefit from the public spending equals the costs of public funds in the equilibrium (see Mirrlees et al.'s, 2010).

Markets
This dynamic economy is run efficiently by the market-clearing relative price system. There arises a tatonement process in operation to eliminate the excess demand for each commodity in the model. Prices of commodities and services and factors of production continue to adjust until demands are balanced to supplies in each market. The OFDR measures the ratio of capital to output in aggregate. Corresponding measures across sectors are given by optimality conditions guiding the accumulation for these sectors, OFDR i;t ¼ K i;t Y i;t . The real exchange rate links between the domestic and foreign sectors, results of the flow of imports and exports. Equilibrium allocations and arbitrage occur at the core of the economy and are Pareto optimal. In other words, DCGE economy converges towards the competitive equilibrium over time and in each period and are optimal in the sense that all economic agents are doing the best given the amount of assets and time endowments they possess.

VI. Parameters and Results of DCGE Model on Financial Deepening
The DCGE model constructed to assess the prospects of financial development in five economies consisted of 11 sectors of goods and services, capital assets differentiated by sectors and labour differentiated by skills. The microconsistent data sets for these models were taken from the input-output tables published by the OECD in 2006 for Germany, France, the UK, Japan and the USA. These data sets provide information on the actual values for demand and supply balance of firms, revenue and expenditure of the government, saving and investment balance for the private sector and the export-import balance for the economy. For instance, the variation in the capital input tax rates (t k ) by sectors across model economies is presented in Table 3. Other details on data and programme are skipped for space reasons and kept in the Appendix available upon request.
Key parameters of dynamic model such as the elasticity of substitution between consumption and leisure σ h c À Á , inter-temporal subjective discount factor β h À Á , substitution between capital and labour in production (σ y ), elasticity of substitution between domestic goods and imports (σ m ) are based on literature and sensitivity analysis (Robinson, 1991). Acceptable values are assigned for the benchmark rate of growth; benchmark interest and generic rates of depreciation are given in Table 4.

Financial deepening and economic growth
A number of assumptions are made regarding the nature of the steady states among these economies. First, the benchmark rate of return on capital stock is chosen to be the natural rate of interest r ð Þ for each country. Information about the rate of deprecation of capital δ i ð Þ in each sector is obtained from the historical data and tested with sensitivity analyses. The steady state growth rates g i ð Þ are made consistent with the historical growth rates for each sector. The parametric values of r; δ i and g i define the reference path of the economy. Elasticities of substitution in consumption σ c ð Þ and production σ p À Á are based on the literature. In addition to capital input taxes as above, the model contains taxes on consumption, wage income and transfers to households t c ; t w ; R h t È É which are retained for all sectors except for the financial and real estate sectors in the counter factual analyses. Model is applied for policy analysis only after the calibration of the benchmark economies with the micro-consistent data set constructed for the 11 sector general equilibrium model from the inputoutput table obtained from the OECD. Fundamentals to all these rest on the optimizing behaviour of households regarding the division of labour between leisure L h t À Á and work and division of income between consumption C h t À Á and saving S h t À Á . Accumulation capital drives the rate of economic growth.

Optimal and actual financial deepening
The general equilibrium theory provides a very clear framework for analysis of results obtained by solving equations with more than 14 thousands variables simultaneously for each of the five model economies: France, Germany, UK, Japan and the USA with a lifetime horizon of 86 years between 2006 and 1992. The OFDRs in the steady state are based on DCGE , the ratios of AFDRs are ratios of stock of assets from the OECD balance sheet to the  Table 5. 4 The overall optimal real FDRs from the general equilibrium models are consistent across countries; these are found to be around 3.16 in France, 3.31 in Germany, 3.24 in the UK, 1.51 in Japan and 3.19 in the USA. These are sensible results and consistent to the converging patterns of economic growth across these countries. The actual ratios of FD reported in the OECD nonconsolidated balance sheets exceeded by factors 10.98, 8.02, 19.1, 17.48 and 15.53 than the optimal ratios 3.5, 2.4, 5.1, 11.6 and 4.8 computed from the solutions of the DCGE models of France, Germany, the UK, Japan and the USA, respectively, as shown in Table 5. These are easier to compare and appraise in Fig. 8.
The discrepancy between the real and the nominal magnitudes of FD gives credibility to over-financing hypothesis that UK economy is more vulnerable to financial crises as it has more assets originating from the financial derivatives and is more subject to the problems caused by asymmetric information. Japan is in a similar situation. Sectoral impacts of financial sector reforms are different for each of the three countries. Despite this, economic growth rates in these models are driven by fundamentals of the financial markets based on the net present value calculations and portfolio selections satisfying the arbitrage across markets, risk-return analysis to minimize risks and maximize returns in anticipation of insurances to cover unforeseen contingencies. Supply of funds arises from inter-temporal utility maximizing consumers, and demand of funds for investment originates from profit maximizing producers. Subjective discount factors of consumers and depreciation rates of capital of firms are balanced by the real interest rates so that funds are allocated according to the marginal utilities of households or productivities across various sectors leaving regulatory roles to the

Policy implications
On-going financial sector reforms, including the mortgage to income ratios announced recently for the housing markets at 4.5 or tax-free ISA in the UK, can be expected to make these economies more efficient so that the costs of funds decline in the counter factual experiments, where the taxes on the financial sectors are set to minimize distortions relative to the benchmark. Such measures will then result in the higher rate of growth of output, employment and capital stock in almost all sectors even with lower capital output ratios. By designing measures to counter inefficiencies due to the asymmetric information problem, the financial liberalization pays for itself, welfare of consumers improves with reforms rather than without it. The proper reform of financial markets not only improves the efficiency of financial intermediation but also brings speedier rate of economic growth by linking the lending and borrowing rates to the fundamentals of demand and supply of funds, by removing controls on credits, by creating right structure of incentives for investors and depositors, by freeing up the foreign exchange market from arbitrary decisions and by making it subject to fundamentals of domestic and foreign asset markets. These mechanisms remove repressionary regimes with noninflationary public finance for smooth process of capital accumulation, increased liquidity, technical advancement and economic growth, elimination of parallel markets and reducing the proportion of toxic nonperforming assets. Liberalization and reform mechanisms thus are instrumental in reversing repressionary and distortionary financial regimes towards more classical free enterprise economy that would promote accumulation and growth in these model economies.
Monetary policy was not effective in containing the current crisis because of excess FDR due to the excess leveraging and collateral debt obligations in the financial markets made possible by financial liberalization and deregulations that led to proliferation of toxic assets in these economies. Further analyses of these are found in seminal and most recent papers such as Fama (2014), Shiller (2014), Hansen (2012), Taylor (2010), Brunnermeier and Sannikov (2014) and Nordhaus (1994). Bhattarai (2014) proves the neutrality of money both cash in advance and money in utility models. This provides validity to the analysis of the real financial sector as presented in this article.
Competitive financial markets are perfect in allocating assets only when all agents have complete information and are efficient in processing such information. Financial markets are full of asymmetric information, activities of one set of players depend on actions taken by another set of players and the amount of information they impact on the likely choices of others. This requires state contingent

VII. Conclusion
The core of Shapley-Shubik games and general equilibrium models with a Venn diagram is applied for a theory on the role of real finance in economic growth among advanced economies. Then the DCGE models for Germany, France, the UK, Japan and the USA are constructed to assess the validity of the over-financing hypothesis that has reappeared after the financial crisis of 2008. AFDRs observed in the nonconsolidated balance sheet of the OECD exceeded by factors of 3.5, 2.4, 5.1, 11.6 and 4.8 than the OFDRs implied by DCGE models, respectively, in these countries because of excessive leveraging and bubbles up to 19 times of the GDP which were responsible for this great recession. Containing such massive fluctuations for macroeconomic stability and growth in these economies are not possible in conventional fiscal and monetary policy models and require a DCGE analysis like this along with adoption of separating equilibrium strategy in line of Miller-Stiglitz-Roth mechanisms to avoid the problem of asymmetric information in the process of financial intermediation so that the gaps between actual and optimal ratios of FD remain as small as possible.
The DCGE model results used in measuring the gap between the AFDR and OFDR is a unique contribution for this article to the literature on FD and economic growth. It takes account of wide-ranging interactions among a large number of consumers and producers and mimics the real-world situations in model economies. 5