Ali Kakhbod, Joseph C. Koo, Demosthenis Teneketzis
We present a decentralized message exchange process (tatonnement process) for determining the level at which a certain public good will be provided to a set of individuals who finance the cost of attaining that level. The message exchange process we propose requires minimal coordination overhead and converges to the optimal solution of the corresponding centralized problem.
Performing a valuation exercise of decentralized companies that explore and exploit natural resources (such as Pemex) interpreted from the perspective of a “Special Purpose Vehicle” (SPV), modeled as structured debt, allowing a deeper analysis when the entity does not own the generating assets of its operating cash flow when capital has a negative book value, the generation of free cash flows is negative and it is subject to tax royalty payments that do not allow for deductibility of debt. Moreover, given its high tax burden and that it is forced to issue debt to finance their capital investments, it is unclear whether it can generate resources to meet its labor and/or financial liabilities, particularly if energy prices would fall. These obligations are modeled as options. In summary, this exercise helps to identify key factors in its operations and finances
Decentralized internal rating based models (self-regulation) which are substituted to public regulation are not able to hold a long-term macroeconomic vision or to take into account interdependencies among private units and markets. Therefore, they seem to be unable to tackle with systemic crises. Moreover, liberal supervision schemes reduce the field of action of monetary authorities and limit the systemic range of their interventions in case of large crisis. Then the absence of macro-regulatory schemes reveals to be one of the causes of the appearance and the persistency of generalized financial crises. A reappraisal of the Minskian financial instability hypothesis and the results of models of conventions, of financing by LBO and of cognitive dissonance points out that the current financial crisis casts doubt on two principles of the way of regulation of modern capitalism: 1) The capacity of market mechanisms for correcting errors of judgment of decentralized actors without structural public interventions; 2) The efficiency of the self-regulation of markets regarding public regulation schemes. These principles turn out to be unable to ensure the continuity in market relations under their present form. So, new research becomes compulsory in order to imagine new macro-prudential mechanisms seeking to strengthen the viability of economic and monetary relations.
The paper presents the main modelling features of the Eurace agent-based macroeconomic simulator. Eurace is a large-scale agent-based model and simulator representing a fully integrated macroeconomy consisting of three economic spheres: the real sphere (consumption goods, investment goods, and labour markets), the financial sphere (credit and financial markets), and the public sector (Government and Central Bank). Following the agent-based approach, Eurace economic agents are characterized by bounded rationality and adaptive behavior as well as pairwise interactions in decentralized markets. The balance-sheet approach and the stock flow consistency checks has been followed as a modeling paradigm, A set of computational results realized by the simulator has been also presented. In particular, results show the real effects on the Eurace economy of the dynamics of monetary aggregates, i.e., endogenous credit money supplied by commercial banks as loans to firms and fiat money created by the central bank by means of quantitative easing. Generally speaking, a quantity easing monetary policy coupled with a loose fiscal policy has been shown to generally provide better macroeconomic performance in terms of real variables, despite higher price and wage inflation rates. Computational results also show the emergence of endogenous business cycles which are mainly due to the interplay between the real economic activity and its financing through the credit market.
We study overborrowing and financial crises in an equilibrium model of business cycles and asset prices with collateral constraints. Private agents in a decentralized competitive equilibrium do not internalize the effects of their individual borrowing plans on the market price of assets at which collateral is valued and on the wage costs relevant for working capital financing. Compared with a constrained social planner who internalizes these effects, they undervalue the benefits of an increase in net worth when the constraint binds and hence they borrow "too much" ex ante. Quantitatively, average debt and leverage ratios are only slightly larger in the competitive equilibrium, but the incidence and magnitude of financial crises is much larger. Excess asset returns, Sharpe ratios and the market price of risk are also much larger. A state-contingent tax on debt of about 1 percent on average supports the planner's allocations as a competitive equilibrium and increases social welfare.
In this paper we study the optimal monetary and fiscal policies of a general equilibrium model of unemployment and money with search frictions both in labor and goods markets\nas in Berentsen, Menzio and Wright (2010). We abstract from revenue-raising motives to focus on the welfare-enhancing properties of optimal policies. We show that some of the\ninefficiencies in the Berentsen, Menzio and Wright (2010) framework can be restored with appropriate fiscal policies. In particular, when lump sum monetary transfers are possible,\na production subsidy financed by money printing can increase output in the decentralized market and a vacancy subsidy financed by a dividend tax even when the Hosios’ rule does\nnot hold.
In OLG economies with life-cycle saving and exogenous growth, competitive equilibria in general
fail to achieve optimality because individuals accumulate amounts of physical capital that differ from the one that maximizes welfare along a balanced growth path (the Golden Rule). With human capital, a second potential source of departure from optimality arises, related to education decisions. We propose to recover the Golden Rule of physical and also human capital accumu-
lation. We characterize the optimal policy to decentralize the Golden Rule balanced growth path
when there are no constraints for individuals to finance their education investments, and show that
it involves education taxes. Also, when the government subsidizes the repayment of education
loans, optimal pensions are positive
Traditionally, the geography of money has been a topic of only marginal or peripheral interest to economists. To be sure, economists have long studied banking, the operation of national financial and monetary systems, international capital movements and the like; but in typical economics fashion, the spatial frames and contexts within which banking, financial systems and capital markets operate have not of themselves been of interest and have typically been considered as exogenous and pre-given. Even geographers tended largely to ignore the spatialities of finance. Admittedly, in the 1970s and 1980s, there were some studies of regional banking structures, urban mortgage markets, regional credit availability and regional interest rate differentials; but the studies that appeared hardly added up to a substantial or coherent body of theoretical or empirical research. During the 1990s, however, the relationship between money and space began to attract increasing attention, with a succession of books and papers by economists and geographers (for example, Cohen, 1998; Corbridge et al., 1994; Dow, 1990; Eichengreen and Flandreau, 1996; Laulajainen, 1998; Leyshon and Thrift, 1997; Martin, 1999; O'Brien, 1990, 1992; Porteous, 1995). Ironically, this flurry of publication occurred at the very time that developments in the world of finance were leading some of the new commentators to argue that if geography had once been of relevance for understanding money, it was rapidly becoming irrelevant. O'Brien (1990, 1992) in particular claimed that various processes, especially technological advances in information and communication technologies (ICT), the wave of financial deregulation that had begun in the 1980s in the USA and UK and a new trend of financial innovation, were together facilitating—indeed promoting—accelerating financial integration at a global scale, rendering geography and location of rapidly declining significance for financial firms, financial flows and access to financial products and services. The globalization of money, it was contended, was annihilating space. Not only was financial globalization undermining national economic sovereignty (Cohen, 1998), by going global banks were free to locate wherever they chose, and money having become electronic, and hence hyper-fungible and hyper-mobile, could now move anywhere almost instantaneously. In this brave new world of global finance, money had escaped space. Geographers on the whole have been much more cautious in pronouncing what O'Brien called the ‘end of geography’ with respect to finance. While they acknowledge that distance may have become irrelevant in financial transactions and operations, they have argued that location and place remain of crucial importance (see Leyshon, 1995, 1997, 1998; Martin, 1994, 1999). The spatial concentration of banks, investment houses and other financial institutions in the major national (and global) financial centres has not dramatically lessened: indeed in many respects it has increased, as has the financial specializations of those centres and the competition between them. The outsourcing and offshoring of certain financial functions and services (such as call centres), themselves developments facilitated by ICT and related ‘globalization’ processes, have been highly geographical in their locational dynamics and impacts. Global and national financial centres may be linked together in worldwide networks of financial flows and transactions that ignore national borders, but in so doing they also function as the portals through which monetary fluctuations, perturbations and shocks originating elsewhere are transmitted down through their domestic financial systems and economies, with highly geographically differentiated effects on the economies of different regions and cities (Tobin, 1984). In the other direction, local and regional economic imbalances within nations can trigger off inflationary pressures and house price bubbles that then not only disturb national domestic monetary conditions and management, but through the global interconnections that link financial institutions in world markets can even trigger off global monetary instabilities. And while the banking and financial systems of individual countries have become increasingly and inextricably interconnected, most retain a local or regional dimension in their organization and operation. How these local circuits of money relate to and are entwined with global circuits has major implications for the propagation and impact of financial shocks and perturbations. In short, contrary to what some argued, money remains highly geographical, even in today's globalized world. This special issue of the Cambridge Journal of Regions, Economy and Society brings together a number of papers on this issue, ranging from the geographical organization of financial centres in pre-industrial Europe to the geographical dimensions of today's global ‘credit crunch’. The four papers in this issue that deal with the geographies of finance each offer a different perspective on the spatiality of financial markets and financial transactions. By taking an historical perspective and by using mid-18th century data that precede the Industrial Revolution, Flandreau et al. (2009) explore the spatial linkages of financial transactions across Europe, circa 1750. The central unit of observation is the city, so that in effect the paper is really about the monetary geography of European cities and in particular about the extent to which ‘local’ or own-city currencies circulated ‘abroad’, that is in other cities. The mapping of the monetary geography of Europe in the paper of Flandreau et al. is inspired by three interdisciplinary approaches. The first concerns the role of states. History shows that before the ascent of the nation state, there was an intricate and almost seamless web of financial relations across Europe. With the rise of nation states, however, the monetary and financial space of Europe was progressively nationalized and compartmentalized into sovereign territories. The second approach upon which the paper builds is economic geography. The description of intra-city linkages across the Europe of the mid-18th century clearly point to the relevance of agglomeration forces. The main financial centre at that time was the city of Amsterdam, though other financial hubs or agglomerations in the European network of currency transactions are also clearly discernible. In southern Europe, the city of Genoa was for instance very important and likewise the city of Hamburg in Northern Europe. But in the hierarchy of financial centres, Amsterdam dominated, with London and Paris also being very important. A third and final approach that can be used to understand the network of financial connection across European cities is (of course) economic history. Here, the authors argue that their main result can be interpreted through the lens of modern or new institutional economic history. Whatever the analytical approach used, however, the main finding of the paper is that in pre-modern Europe, that is prior to the Industrial Revolution, there was already a dense and quite distinct spatial urban network of financial connections in Europe. Local currencies or bills of exchange circulated widely outside their own locality or city. At the same time, not all cities or bills of exchange were equally widespread: the monetary geography of Europe in those days was one in which a few cities dominated, much like in the modern monetary geography of Europe. In the literature on ‘money and space’, the geographical role or relevance of financial intermediation and banks in particular is emphasized. The claim by O'Brien (1992) that geography has become irrelevant in the modern financial system applies most to public capital markets. When it comes to the supply of and demand for bank loans, however, even casual observation suggests that proximity still matters. At the same time, in many countries the banking sector has seen structural change at an unprecedented scale in the last few decades. Banking has gone ‘global’ and this has been accompanied by a very substantial (spatial) concentration of banking. This leads to important questions about the interrelationship between global banking and local credit markets. This interrelationship is at the heart of the paper by Alessandrini et al. (2009). Using O'Brien (1992) as a point of departure, Alessandrini et al. seek to establish if and how distance still matters in the case of the Italian credit and banking market. Distance is a multi-faceted concept and the authors come up with two ways to define distance, namely ‘operational’ and ‘functional distance’, that are subsequently used in their empirical analysis. Three findings stand out. First, geography (still) matters when it comes to the Italian credit market and the way in which firms and banks interact (locally). Second, the impact of distance on the interrelationship between global banking and local credit markets is not unambiguous. This then leads to the third finding or probably more accurately an agenda for future research: geography matters when it comes to local banking structures, and banks’ own territorial strategies, as well as the relevance of the banks’ headquarters for regional development. In these first two papers on the geographies of finance, financial centres play a key role. In his paper, Wójcik (2009) takes the location of financial centres as given and tries to find out whether (non-financial) firms that are located in financial centres are more likely to go public than similar firms that are located in the financial periphery. Going public means taking the firm to the stock market via a so-called ‘initial public offering’ (IPO). Using firm-specific data for 32 countries, Wójcik shows that there is indeed a strong positive correlation between the location of firms and their IPO activity. Firms that are located in financial centres are more likely to go public. Given the high degree of (international) capital mobility and the current technological possibilities for both investors and firms to inform themselves about each other and the functioning of the stock market, one may wonder why in this case geography still matters. The author points out, for instance, that closeness to financial intermediaries may make it easier for firms to go public and also that the specialized labour that is needed for an IPO process is more readily available in financial centres. In this way, it appears that the geography of financial centres influences the capitalization process (via IPOs) of businesses. All three papers introduced so far suggest, somewhat contrary to what O'Brien (1992) claimed, that even with unhampered capital mobility geography is still relevant for many financial transactions. Even with capital free to move within or between countries, the bulk of financial transactions is or remains spatially bounded and has a distinct geographical footprint. From an international macro-economic perspective, the idea that free international capital mobility does not seem to go along with a de-nationalization of capital flows is known as the Feldstein–Horioka paradox. More specifically, the paradox here is that with free capital mobility, one would expect that national savings and national investment are no longer positively correlated. Without capital mobility, national investment is inevitably constrained by the amount of national savings. But with capital mobility, this is in principle no longer the case. However, following the seminal study by Feldstein and Horioka (1980), scores of researchers have found that for almost every country national savings and national investment are still strongly correlated. The paper by Kool and Keijzer (2009) throws new light on this issue. Using new (panel) estimations and estimation techniques for a sample of 23 countries for the period 1973–2003, they find that the Feldstein-Horioka (FH) coefficient that measures the relationship between savings and investment has in fact dropped significantly in recent years. Indeed, around the year 2000, the coefficient is no longer significantly different from zero. This suggests that economic and financial integration has increased markedly in recent years. As to the reasons behind the de-coupling between national savings and investment, the authors single out increased trade openness and especially a fall in the so-called ‘home equity bias’. The latter refers to the stylized fact that investors typically have a tendency to underinvest in foreign equity. According to Kool and Keijzer, with this bias getting weaker, the correlation between national savings and investment also has weakened. Since it is only fairly recently that the FH coefficient has fallen so strongly, it remains to be seen if this is merely a temporary phenomenon or if national savings and investment have really started to move independently of one another. The current financial crisis is a first real test in this respect. The spatial dimensions of finance have been highlighted by the current financial crisis—where a shock ostensibly emanating from the US housing market was rapidly transmitted into a global recession. In their paper, O'Brien and Keith (2009) argue that the crisis has been facilitated by the ‘end of geography’ with ICT and lightly regulated finance enabling ultra-rapid and highly complex flows of financial capital across borders. However, when reviewing the future of finance, O'Brien and Keith suggest that it is likely that the drive towards the ‘end of geography’ will be slowed by the crisis; as the level of financial regulation is likely to increase, developments in ICT may help improve the management of information, a feature that has been manifestly lacking in modern global financial markets. In any case, as discussed above, the ‘end of geography’ thesis should not be exaggerated: deregulation and ICT may promote and facilitate the movement of money and capital across space, but they do not necessarily result in a ‘geography-free’ world of finance. Furthermore, it can be argued that globalized financial markets have intensified geography by sustaining and, in some cases, intensifying spatial differences in economic prosperity and social welfare. Global capital markets have enabled countries such as the USA and the UK to run persistent balance of payments deficits by facilitating circulation of finance from those countries that have maintained persistent balance of payments surpluses. And within both the USA and UK, the recession that the credit crunch sparked off has been anything but spatially even in its impacts. The notion of the ‘end of geography’ is subject to a powerful critique by Dymski. Tellingly, Dymski (2009) argues that O'Brien's argument is a repackaging of efficient markets theory—a theory that has been left in tatters by the recent behaviour of financial markets. Dymski constructs an alternative counter-narrative where government policy is fundamental to the construction of financial markets—not only through the regulatory framework but also through the macroeconomic and industrial policies which shape the opportunities for doing business and generating profits. Furthermore, Dymski argues that global finance has not led to the emergence of a ‘global banking customer’ but has instead created a spectrum of different financial customers, which has contributed to the global divisions in income and wealth. Customers from poorer parts of the spectrum are charged higher interest rates, are more likely to suffer from foreclosure and are the first to be deprived of liquidity when crisis strikes. But, of course, the most impoverished, such as many of those in Africa, are completely disconnected from the financial system. According to French et al. (2009), the credit crunch is a ‘very geographical crisis’. They argue that the crisis has arisen from an active use of space at a range of scales and along networks of varying length which connect individuals and institutions to the financial system. Thus, the crisis has been characterized by different geographies of financial flows, wealth effects and impacts. It should also be emphasized that the financial crises has led to an economic crisis—and the geographies of the two crises are likely to be different and will be determined by the mechanisms through which the former is transmitted to the latter—as the decline in world income and trade and the inability of producers and consumers to borrow to invest and consume will have different spatial impacts and amplitudes. The paper by Bieri (2009) also counters the O'Brien position, on the grounds that the globalization of financial markets has led to a change in geography rather than its demise. Bieri contrasts the ‘old’ geography characterized by competing nation states with the ‘new’ geography comprising globally dispersed creditors and debtors with both strong local and global connections and drivers. Furthermore, such bi-polar processes will continue in the future and global financial markets will become more ‘curved and spiky, not flat’. This will create challenges for regulation and global financial architecture: the Bretton Woods system, which was established after the Second World War in the era of dominant nation states, largely remains in place today. Thus, there is a need to re-evaluate the global financial architecture and balance the need for decentralized local regulation and centralized interventions and coordination. The issue of the relationship between financial liberalization and poverty is analysed by Arestis and Caner (2009). The conventional focus is on the link between financial liberalization and growth and how the latter may influence poverty through ‘trickle down’ effects. Arestis and Caner analyse three further channels: the crises channel, the access to credit and financial services channel and the income share of labour channel. They show that although the relationships between financial liberalization and poverty are complex, the former often causes increases in the latter. Although the paper of O'Brien and Keith provides an updated view of the ‘end of geography’ thesis, the majority of papers in this issue suggest that to characterize the contemporary global financial landscape in such terms is to capture at best only certain facets of today's monetary reality. There is in fact considerable evidence that ‘money and space’ are still closely intertwined—geography has evolved and changed but its ‘end’ is not in sight. This will become even more apparent as the fallout and complex repercussions of the current financial crisis continue to feed through to the real economy throughout the globe: including house repossessions across numerous cities in the USA and UK; major plant closures, job losses and unemployment in many local communities; future major cutbacks in public sector spending programmes, to help reduce the government debt incurred by bailing out failed banks and mortgage lenders; the collapse of the Iceland economy and the need for IMF support; and the contraction of world trade, which is affecting the German and Japanese economies to such an extent that these two countries are forecast to have much deeper recessions than those countries from where the crisis emanated in the first place (IMF, 2009, 10). What recent events demonstrate so clearly is that finance may have gone global but its complex circuits are profoundly spatial in their operation and impact.
Martin Altemeyer-Bartscher, Dirk Rübbelke, Eytan Sheshinski
International environmental protection like the combat of global warming exhibits properties of public goods. In the international arena, no coercive authority exists that can enforce measures to overcome free‐rider incentives. Therefore decentralized negotiations between individual regions serve as an approach to pursue efficient international environmental protection. We propose a scheme which is based on the ideas of Coasean negotiations and Pigouvian taxes. The negotiating entities offer side‐payments to counterparts in order to influence their taxation of polluting consumption. Side‐payments, in turn, are self‐financed by means of externality‐correcting taxes. As we show, a Pareto‐efficient outcome can be attained.
Aleksander Berentsen, Mariana Rojas Breu, Shouyong Shi
Many countries simultaneously suffer from high inflation, low growth and poorly developed financial sectors. In this paper, we integrate a microfounded model of money and finance into a model of endogenous growth to examine the effects of inflation on welfare, growth and the size of the financial sector. A novel feature is that the innovation sector is decentralized. Financial intermediaries arise endogenously to provide liquidity to this sector. Consistent with the data but in contrast to previous work, reducing inflation generates large growth gains. These large gains cannot be easily reproduced by imposing a cash-in-advance constraint in the innovation sector.
The paper considers a three-tier credit chain consisting of a bank, a logistic company and a retailer. The paper analysis the following outsourcing structure implemented by top-tier bank: inhouse consignment, under which the bank signs independent contracts with the logistic company and the retailer. Under the assumption that the logistic company cannot change its decision, the paper investigates the equilibrium behavior of the decentralized credit chain with non-cooperation newsvendor under demand uncertainty. The model includes the case of a risk-neutral bank offering loan to a noncooperative risk-neutral retailer who mortgages its inventory to the bank to finance more inventory. The retailer faces a random demand in a single sales season as in the classical newsvendor problem. By game theorem, the paper gives the optimal loan to value to retailers with different initial capital.
A family of core extensions for cooperative TU-games is introduced. These solution concepts are non-empty when applied to non-balanced games yet coincide with the core whenever the core is non-empty. The extensions suggest how an exogenous regulator can sustain a stable and efficient outcome, financing a subsidy via individual taxes. Economic and geometric properties of the solution concepts are studied. When taxes are proportional, the proportional prenucleolus is proposed as a single-valued selection device. An application of these concepts to the decentralization of a public goods economy is discussed.
This paper studies the effects of stock market valuation on research investment, the rate of innovation, and welfare. In the presence of financing constraints for R&D investment, episodes of high market valuation can ease these constraints and raise the economy-wide investment in R&D and the rate of innovation. If the decentralized equilibrium rate of innovation is inefficiently low, then such episodes may lead to an increase in aggregate welfare even if the higher valuation is not entirely justified by fundamentals. We present a Schumpeterian-style growth model with a costly financial intermediation process to characterize the relationship between market value, entry of new firms, and the aggregate rate of innovation. We use the model to measure the welfare consequences of a stock market run-up that may only partly be justified by fundamentals. In particular, we apply the model to the US economy in the 1990s and calibrate the impact of the NASDAQ boom on the rate of innovation, growth and welfare. The welfare effect depends on the underlying change in fundamentals. We find that with an acceleration in US trend productivity growth from a pre-1995 rate of 1.4% to a rate of 2.0% per annum, the NASDAQ boom will have resulted in a net welfare gain of 0.55%. If the new growth rate is as high as 3%, the net gain was 1.35% of the present discounted value of consumption.
This paper develops a two‐period overlapping generations model with heterogeneous agents aiming at analysing how decentralization in the provision of public education affects growth and personal inequality via human capital investment. Education is financed by a tax levied by either national or local authorities. The tax rate is chosen according to a median voter mechanism. During their working period of life, individuals look after their offspring by providing them with a high level of school education stemming from taxation. In addition parent's contributions to the social security system provide them with retirement income. Heterogeneity accounts for the differences in the optimal taxation mechanism, linking the income distribution to the tax rate, and hence to human capital accumulation, growth and income inequality. In this way we relate differences among agents to the tax rate. We show that decentralization induces growth rate disparities among local communities but it can be ruled out by a proper fiscal substitution between social security and locally provided education. Unlike in the literature, this type of fiscal design allows local economies to grow faster and more equally than the national design.
Capital adequacy regulations specify a minimum capital-to-assets ratio for banks in the economy. The effects of these regulations on the level of economic activity have not been thoroughly studied by the banking regulation literature. Specifically, the fact that as proposed by the Basle Accords, a constant ratio tying bank lending to bank equity may reinforce macroeconomic fluctuations has been looked at by only a few existing theoretical papers. This paper proposes a stochastic dynamic general equilibrium model to study the interactions between the banking sector and the aggregate level of economic activity. Banks behavior is fully micro-founded. Banks financing decisions (equity versus deposits) are constrained not only by the regulation but also by a financial imperfection arising from the fact that during bad times banks find it difficult to recapitalize by raising equity. Thus, higher borrower bankruptcy rates during recessions imply that banks have to cut new loans until the ratio is restored to the required level. Since production firms can only imperfectly substitute bank lending with other forms of financing, a negative macroeconomic shock affects production and investment both directly and indirectly through the bank loan supply. This banking regulation and the financial imperfection imply two different constraints to the banks problem that bind only occasionally in the stochastic steady state. This prevents the use of standard linearization techniques to solve the model numerically. Alternatively, using some discretization of the state space methods such as Value Function Iteration is difficult because the model cannot be written in terms of a central planner problem. Following Fackler (2003) I solve the decentralized general equilibrium problem by using a very general Function Approximation technique that nests the Parameterized Expectation Approach as a particular case. The method allows to approximate numerically either the policy functions or the expectation functions. It is also flexible as regards the choice of approximating functions, including Chebyshev polynomials and piecewise polynomial splines. The technique relies on the Collocation Method to solve for the polynomial coefficients in combination with either generic root-finding algorithms or a fixed-point iteration scheme. Numerical results suggest that banks try to anticipate aggregate shocks by accumulating a buffer of capital over the regulatory minimum. Nevertheless, a series of bad shocks may be strong enough to eventually undermine these "reserves" and to make banks cut back on lending. This suggests the existence of a financial accelerator, since the supply of loans shrinks together with the demand during recessions. This mechanism has interesting policy implications and provides grounds for a procyclical value of the required capital-to-assets ratio