public provision of private goods as a redistributive device in an optimum income tax model public provision of a private good can alleviate the informational problems that restrict redistribution through the tax transfer system when the identity of high and low skill persons is hidden a pareto improvement may be achieved by publicly providin

effects of indirect taxation in a mixed oligopoly abstract in a mixed oligopoly an increase in tax ad valorem or specific does not change total output but increases the output of the public firm and the tax revenue also privatization may increase both tax revenue and welfare

on the political economy of educational vouchers abstract two significant challenges hamper the analyses of the collective choice of educational vouchers one is the multi dimensional choice set arising from the interdependence of the voucher public education spending and taxation second even absent a voucher preferences over public spending are not single peaked a middling level of public school spending may be less attractive to a household than either high public school spending or private education coupled with low public spending we show that besley and coate s 1997 representative democracy model provides a viable approach to overcome these hurdles we provide a complete characterization of equilibria with an endogenous voucher a voucher is adopted in political equilibrium provided the coefficient of variation of income is sufficiently small we undertake a parallel quantitative analysis and we find that no voucher arises in equilibrium for the u s income distribution which exhibits too much heterogeneity for a tighter income distribution including those in douglas county colorado where a voucher was recently adopted our model predicts a positive voucher public support for a not too large voucher arises because the cross subsidy to public school expenditure from those switching to private schools outweighs the subsidy to those who attend private school in the absence of a voucher

vouchers and voting an initial estimate based on the median voter model one of the major concerns associated with school voucher proposals is that they will only benefit those who leave public schools in favor of private schools the children left behind in public schools are expected to necessarily be made worse off by such policies this paper provides evidence suggesting that children left behind in public schools will also benefit the available evidence suggests that after a voucher plan voter support for public schools will rise enough to increase the human capital of public school students despite the presence of negative peer group effects

time based competition and the privatization of services the authors develop a vertical differentiation model which exhibits congestion externalities and in which firms choose service time this model is formulated for a variety of industry and ownership structures the authors find an ownership effect in which service speeds decrease when public services are privatized this effect however could be offset and even reversed by increased efficiency of private services the authors also investigate how prices and commitment ability with respect to service times will affect the service speeds in equilibrium copyright 1995 by blackwell publishing ltd

an optimal r q policy in a stochastic inventory system with all units quantity discount and limited sharable resource abstract this paper is concerned with the single item inventory system with resource constraint and all units quantity discount under continuous review where demand is stochastic and discrete in most actual inventory systems the resource available for inventory management is limited and the system is able to confront the resource shortage by charging more costs considering the resource constraint as a soft constraint beside a quantity discount opportunity makes the model more practical an optimization problem is formulated for finding an optimal r q policy for the problem in which the per unit resource usage depends on unit purchasing price the properties of the cost function are investigated and then an algorithm based on a one dimensional search procedure is proposed for finding an optimal r q policy which minimizes the expected system costs and converges to a global optimum based on the properties of the partially conditioned cost functions the presented algorithm is modified such that its search path to optimal policy is changed experimental results show that the performance of the modified version of the presented algorithm is much better than the original algorithm in various environments of test problems

competition between private and public schools vouchers and peer group effects a theoretical and computational model with tax financed tuition free public schools and competitive tuition financed private schools is developed students differ by ability and income achievement depends on own ability and on peers abilities equilibrium has a strict hierarchy of school qualities and two dimensional student sorting with stratification by ability and income in private schools high ability low income students receive tuition discounts while low ability high income students pay tuition premia tuition vouchers increase the relative size of the private sector and the extent of student sorting and benefit high ability students relative to low ability students

a single item continuous review inventory problem with space restriction abstract with today s high cost of land acquisition observed in most countries management can no longer afford large storage facilities to house their products for this reason one of the main concerns of inventory managers is to ensure that enough storage space is available upon the delivery of a product they therefore have to determine the least cost ordering policy so as to meet demand and taking into account the space limitation in this paper we investigate the problem of a single item continuous review inventory problem in the presence of a space restriction when the demand is stochastic through a sensitivity analysis study it is found that ordering cost storage capacity average demand per unit time and holding cost rate are the most influential problem parameters on the expected inventory costs it is also shown numerically that a simple solution based on economic order quantity is performing very well with an average and maximum cost penalty of 0 19 and 2 03 respectively

support for public provision of a private good with top up and opt out a controlled laboratory experiment abstract this paper presents the results of a revealed choice experiment testing the theoretical predictions of political economy models regarding public support for a publicly provided private good financed with proportional income taxes when individuals can purchase the good privately and either continue to consume public provision top up or forego public provision opt out but in each case continue to pay income taxes our laboratory results confirm behavior is consistent with the predicted majority preferred tax rate under mixed financing with top up but we identify preferences for significantly higher rates of public provision than predicted under mixed financing with opt out using non parametric regression analysis we explore the relationship between individuals top up and opt out decisions and both their income levels and the implemented tax rates

the mix of social and private provision of goods and services abstract in some countries health insurance and education are supplied as social goods at near zero prices and funded from taxation this paper analyses a case where such provision competes with income supplements as a redistribution device and where taxes or transferable vouchers influence the take up of the social goods compared to alternatives available in a private market each consumer chooses to consume either social or private goods and a vertical market segmentation results from differential demand for quality

topping up or opting out the optimal design of public provision schemes there is extensive public provision of private goods in all developed countries the public provision scheme is often designed so that individuals can opt out but not top up supplement the publicly provided quantity quality using an optimal income ta

are the rich too rich to be expropriated economic power and the feasibility of constitutional limits to redistribution why is it that in democracies the poor do not expropriate the rich even though they outnumber them in this paper we analyze the commonly held belief that the rich escape expropriation because they are economically powerful we demonstrate that the economically powerful i e the above average income earners are indeed in a position to bribe the small segment of the voters with incomes between the median and the mean to resist the temptation of supporting confiscatory taxation this is true even if compensation payments in cash are politically unfeasible and therefore need to be made in terms of an evenly distributed private good and it may even be true if only pure public goods are available to swing the middle class

a critique of the proposed tests of the natural rate rational expectations hypothesis the failure of the governments of most industrialised countries over the past decade to deal successfully with the problems of rising unemployment and high and often accelerating rates of price and wage inflation has revived interest in the pre keynesian macro economics of sound money and has renewed the controversy of the 1930 s over what governments should or should not do if they wish to achieve full employment with reasonable price stability in democratic societies keynesian economics has come under attack both by those who primarily oppose interventionist policies on ideological grounds and by the so called new classical macro economists who adhere to the natural rate of unemployment hypothesis nrh advanced by phelps i967 and friedman i 968 and the hypothesis of rational formation of expectations reh originally due to muth i96i armed with this joint natural rate rational expectations nr re hypothesis the new classical macro economists have argued that in a world capable of being represented by linear or log linear relations with a unique and stable equilibrium and no information discrepancy between the government and the private sector no government macro economic policy whether monetary or fiscal no matter how ingeniously formulated and how effectively implemented can have any systematic or lasting impact upon the level of output or the rate of unemployment 2 clearly this fundamental policy

selective versus universal vouchers modelling median voter preferences in education under the majority voting rule a system of universally available vouchers uv is politically less feasible than a system of selective vouchers sv confined to families with incomes equal to or less than median voter income after the introduction of uv public expenditure on education will have to be shared with previous private school users per capita expenditure will then drop and or tax will increase since these events will injure the median voter he will reject uv he will be indifferent between the status quo and sv indifference will turn into enthusiasm however if as can be expected the new regime sv brings effective new competition this abstract was borrowed from another version of this item

the scourge of the monetarists kaldor on monetarism and on money nicolas kaldor s critique of monetarism from 1970 until his death is described and evaluated it is argued that subsequent events as well as empirical research vindicate his critique kaldor s critique of john maynard keynes s monetary theory is then seen as an extension of his radcliffe views and this paper argues that in this matter kaldor was mechanical in his modeling of the endogeneity of money supply it is suggested that the work of hyman minsky and victoria chick has the potential for further work in this area copyright 1989 by oxford university press

monetary anticipations and the demand for money abstract it has recently been argued that unanticipated changes in nominal money supply should affect the real demand for money and empirical results which appear to support this hypothesis have been presented unfortunately the econometric techniques used by previous authors yield severel biased estimates in this paper we show how valid estimates may easily be obtained and present some illustrative empirical results using u s data these results do not support the conclusions of earlier authors we find that unanticipated money supply changes do indeed affect the real demand for money but with the wrong sign and that anticipated changes also have a significant effect a consistency test for all models employing the anticipated unanticipated distinction is also proposed

public goods and the value of product quality regulations the case of food safety abstract the paper examines preferences for product quality regulations our premise is that preferences for product quality regulations derive from preferences for both private and public goods the model is used to explain public attitudes toward a referendum measure to eliminate pesticide residues on food results from a survey of consumers are consistent with the conceptual model and show that preferences for public goods influence support for the product quality regulation the results help explain why consumption behavior is a poor predictor of political behavior and have implications for methods that use voting and market behavior to value public goods

the religious factor in private education abstract we quantify the religious factor in private education in the united states by calibrating a political economy model of school choice among public private nonsectarian and subsidized religious schools in which parents differ in their incomes and in their preferences for religious education this shows strong latent demand for religious schooling conditioned on current subsidized tuition levels in parochial schools which is suppressed by the need to pay twice for private education applying the results of the calibration to gauge the effect of means tested school voucher programs on religious and nonsectarian private enrollment we find that when the amount of the voucher is relatively small including subsidized religious schools in such programs is necessary for providing low income families effective access to private education

why are tax expenditures for giving embodied in fiscal constitutions abstract this paper asks whether there can be unanimous support for a fiscal constitution that includes tax expenditures for giving among the set of available fiscal instruments we describe a political economy model of fiscal choices where individuals with different incomes vote over levels of proportional income taxation and over tax incentives for giving and investigate how the availability of such incentives affects political equilibrium outcomes and welfare for different income groups we find that unanimous support for the use of tax incentives for giving can indeed arise and is more likely the more unequal is the distribution of income

the search for a stable money demand function a survey of the post 1973 literature phe question of whether the demnand function for money is stable is one of the most important recurring issues in the theory and application of macroeconomic policy what is being sought in a stable demand function is a set of necessary conditions for money to exert a predictable influence on the economy so that the central bank s control of the money supply can be a useful instrument of economic policy as such the notion of a stable demand function appears to involve three key elements e g john t boorman 1976 p 317 and david e w laidler 1977 pp 34 35 first the demand for money relation should be highly predictable in a statistical sense as measured by the usual goodness of fit statistics precision of estimated coefficients and presumably its ability to forecast accurately out of sample second a stable demand function for money has relatively few arguments a relationship that requires knowledge about a large number of variables in order to pin it down is in effect not predictable finally the variables that appear as arguments in the function should represent significant links to spending and economic activity in the real sector in sum a stable demand function for money means that the quantity of money is predictably related to a small set of key variables linking money to the real sector of the economy the importance of a stable money demand function for monetary policy can be illustrated with the standard is lm model in which interest rates and gnp are the endogenous variables in this context a stable demand function for money translates into a well defined relationship between money interest rates and gnp

educational vouchers and cream skimming educational vouchers and cream skimming epple and romano 1998 show equilibrium provision of education by public and private schools has the latter skim off the wealthiest and most able students and universal vouchers lead to further cream skimming here we study voucher design that injects private school competition and increases technical efficiencies without cream skimming conditioning vouchers on student ability without restriction on participating schools policies fails to affect significantly cream skimming however by adding restrictions like tuition constraints such vouchers can reap the benefits of school competition without increased stratification this is accomplished while allowing voluntary participation in the voucher system and without tax increases

erratum to competition incentives and public school productivity abstract when competition increases it is often presumed that public schools will be forced to become more efficient this paper challenges that presumption showing that in well defined circumstances rent seeking public schools find it optimal to reduce productivity when a voucher is introduced this occurs for incentive reasons alone more generally the productivity effects of vouchers are shown to be non uniform varying systematically according to household and community characteristics and the form the voucher takes when the voucher is targeted perverse productivity outcomes do not arise the analysis has relevance to the policy issue of voucher design

a keynesian macro econometric model of the uk 1955 1984 summary in this paper we present a small keynesian macro economic model in which wage price determination is linked to the working of goods and money markets by explicitly treating the keynes effect we derive a general expression for the employment money supply elasticity and draw the is lm loci in the employment interest rate space our empirical specification allows for short run disequilibrium dynamic adjustments around the static long run relations predicted by the theoretical model by careful use of our specification search strategy we obtain a statistically sound econometric model which exhibits sensible long run properties a remarkable finding implied by our estimates is that equilibrium unemployment is negatively affected by both money supply and incomes policy

education choice endogenous growth and income distribution this paper studies an economy where parents can choose between public or private schools and can vote on taxes used to fund public schools the model is calibrated to us data and studied using simulations a bimodal income distribution emerges where public education students converge to a low income equilibrium while private education students experience endogenous growth with higher incomes however public education students experience long run growth through a spillover from private education students possible problems with the existence of a private alternative to pubic education such as the emergence of a education based class structure are identified

ends against the middle determining public service provision when there are private alternatives abstract public provision of a service coexists with private market provision the quality of public provision is determined by majority vote preferences are not single peaked owing to the presence of private alternatives we identify two cases in one majority voting equilibrium always exists and the median income voter is pivotal in the other a necessary condition for equilibrium indentifies the pivotal voter who must have income below the median when equilibrium exists a coalition of middle income households who consume the public alternative will be opposed by a coalition of rich and poor households with the rich choosing private consumption

on limiting the market for status signals abstract in many societies individuals care how others view them in their efforts to impress such individuals engage in consumption signals which attempt to establish higher levels of status than their true status type this leads to an inefficiency which offers the possibility of a pareto improving tax policy the impacts of tax policy and benefits on the signalling equilibrium are considered

school vouchers a survey of the economics literature we review the theoretical computational and empirical research on school vouchers with a focus on the latter our assessment is that the evidence to date is not sufficient to warrant recommending that vouchers be adopted on a widespread basis however multiple positive findings support continued exploration specifically the empirical research on small scale programs does not suggest that awarding students a voucher is a systematically reliable way to improve educational outcomes and some detrimental effects have been found nevertheless in some settings or for some subgroups or outcomes vouchers can have a substantial positive effect on those who use them studies of large scale voucher programs find student sorting as a result of their implementation although of varying magnitude evidence on both small scale and large scale programs suggests that competition induced by vouchers leads public schools to improve moreover research is making progress on understanding how vouchers may be designed to limit adverse effects from sorting while preserving positive effects related to competition finally our sense is that work originating in a single case e g a given country or in a single research approach e g experimental designs will not provide a full understanding of voucher effects fairly wide ranging empirical and theoretical work will be necessary to make progress

shared resource capacity expansion decisions for multiple products with quantity discounts abstract when multiple products compete for the same storage space their optimal individual lot sizes may need to be reduced to accommodate the storage needs of other products this challenge is exacerbated with the presence of quantity discounts which tend to entice larger lot sizes under such circumstances firms may wish to consider storage capacity expansion as an option to take full advantage of quantity discounts this paper aims to simultaneously determine the optimal storage capacity level along with individual lot sizes for multiple products being offered quantity discounts either all units discounts incremental discounts or a mixture of both by utilizing lagrangian techniques along with a piecewise linear approximation for capacity cost our algorithms can generate precise solutions regardless of the functional form of capacity cost i e concave or convex the algorithms can incorporate simultaneous lot sizing decisions for thousands of products in a reasonable solution time we utilize numerical examples and sensitivity analysis to understand the key factors that influence the capacity expansion decision and the performance of the algorithms the primary characteristic that influences the capacity expansion decision is the size of the quantity discount offered but variability in demand and capacity per unit influence the expansion decision as well furthermore we discover that all units quantity discounts are more likely to lead to capacity expansion compared to incremental quantity discounts our analysis illuminates the potential for significant savings available to companies willing to explore the option of increasing storage capacity to take advantage of quantity discount offerings for their purchased products

income inequality and the quality of public services a developing country perspective abstract that the high growth rising inequality phenomenon experienced by many high growth developing economies in the recent past may lead to discontent among the poor is a well discussed issue this paper argues that as the demand for higher quality of the public services rises with income the rising income in the hands of the rich leads to their dissatisfaction about the quality of available public services thereby inducing them to form their own club for self provision of the public services whenever possible this however results in a lowering of quality available to the others outside the club despite less crowding at the publicly provided facilities

capitalization and the voucher an analysis of precinct returns from california s proposition 174 abstract in 1993 californians voted on a school voucher initiative we hypothesize that homeowners in good school districts understood the voucher to be a threat to their property values and thus voted against it precinct returns from los angeles county confirm this hypothesis we also examine an alternative hypothesis explaining the relationship between school quality and precinct returns according to the alternative voters perceived the initiative to be a referendum on public school quality to distinguish between the two hypotheses we compare the voting patterns of homeowners and renters the comparison does not favor one hypothesis over the other

educational vouchers welfare effects and voting abstract we analyze the welfare effects and voting equilibrium with vouchers for private schools the analysis shows that vouchers like tuition tax credits martinello and west 1988 frey 1983 may improve the welfare of all families including families whose children remain in public schools if public educational quality is unchanged and publicly financed educational expenditures decrease thus voters will approve a voucher only if it reduces taxes when public educational quality is endogenous a voucher may increase quality by reducing the tax cost of quality and will be approved if it reduce taxes at the pre voucher public education level and increases quality

computing the optimal replenishment policy for inventory systems with random discount opportunities the paper considers the optimal control of a single item continuous review inventory system with random demand and discount opportunities items can always be purchased with the regular order setup and variable costs however when a discount opportunity occurs they can also be purchased with a different setup cost and a lower variable cost demands for individual items and discount opportunities occur according to independent poisson processes the paper proposes an algorithm to compute the parameters of the optimal replenishment policy which has been shown to be an r r d d policy wherer

equilibrium and inefficiency in a community model with peer group effects public service output depends on input expenditures on own personal characteristics and on the characteristics of the other residents in the community the peer group effect in a community model with public expenditures set by voting with migration between communities and with land price differentials capitalization it is shown that communities may become heterogeneous in composition and second best inefficient this equilibrium occurs when the peer group effect is neither too strong nor too weak the inefficiency arises because an externality is created by migration the land price differential does not play the part of the price of the better peer group but of a transfer payment

monetary anticipation and the demand for money in the u k testing rationality in the shock absorber hypothesis the carr darby shock absorber hypothesis that unanticipated changes in the money supply cause changes in real balances but anticipated changes have a unit impact on the price level and therefore leave real balances unchanged is tested using two step and joint estimation techniques for the u k two step methods appear to support the shock absorber hypothesis but the superior joint estimation technique decisively rejects the hypothesis particularly the implicit rational expectations cross equation restrictions

an inventory model for perishable items in a lot size system with quantity discounts abstract an inventory model continuous in units and time is developed for perishable items in a lot size system that allows partially backlogging and partially lost saling the model is developed under deterministic demand instantaneous delivery with quantity discounts and any well behaved probability distribution for the time to perishables an example is given to illustrate the model

the economic consequences of mrs thatcher it is difficult to overstate the gravity of what is now happening to the uk economy output is falling at an almost unprecedented rate unemployment is at a level not experienced since the inter war depression and is increasing rapidly towards 3 million or 13 of the labour force which would mean rates of 20 and above in large areas of the country nor is there any foreseeable prospect of unemployment coming down instead all the signs point to a continuing increase the social and political consequences of this are hard to predict it is not just a matter of the financial hardship suffered by those thrown out of work through no fault of their own or of the strain imposed on the social welfare system but also of the mental stress disillusionment and sense of helpless ness which come with not being able to find a job this could well have a damaging long term effect on attitudes towards work particularly among young people con demned to go straight from school on to the unemployment register and is almost bound to lead to widespread social unrest the cause of sudden high unemployment in the uk is a backlash against the consensus post war economic policies of demand management to maintain near full employment of support for industry and of income maintenance and the provision of social welfare services policies which were developed in response to the experience of the inter war years the thatcher government has embraced a doctrinaire monetarist strategy for controlling inflation and a laissez faire approach to the supply side of the economy without attempting to moderate the social impact of these policies the extreme nature of this experiment has attracted a good deal of interest from abroad especially from the usa where the new president has expressed admiration for mrs thatcher and what her government is doing if only because there is a danger of governments in other countries following the uk s lead particularly as both the imf and oecd have recently urged the pursuit of similar policies it is important to draw international attention to what is now happening in the uk 18 months after mrs thatcher took office in sharp contrast to the rhetoric of government promises the economy has been set on a path of rapidly falling output and escalating unemployment leading towards the destruction of large parts of manufacturing industry at the same time the social advances which have been made during the post war period are beginning to be reversed as the government desperately seeks ways of offsetting the increase in its budget deficit caused by deepening recession

the buffer stock notion in monetary economics not the least of harry johnson s contributions to our discipline was the series of survey papers on aspects of monetary economics which he produced during the 1960s and 1970s it is no exaggeration to say that the way in which the literature of monetary economics is organised even now follows the design which johnson first set out in 1962 and built on in the subsequent decade nevertheless the subject has not stood still and one of the more interesting developments of the last few years has involved the introduction of what has been variously described as the disequilibrium money shock absorber or as i prefer to call it the buffer stock notion into monetary economics the development of this line of thought was getting underway before johnson s untimely death much of it indeed under the auspices of his own research programme at the lse even so this did not happen early enough for the relevant ideas to be put in their proper niche by johnson himself and the buffer stock approach is not yet fully integrated into the mainstream of monetary economics much of the relevant literature is empirical concerned with modelling particular aspects of the monetary experience of specific countries hence it is fragmented and contains few papers which attempt a coherent account of just what this approach is and how it relates to other approaches to monetary analysis although essays by goodhart 1982a b and judd and scadding 1982a contain important discussions of these issues to the best of my knowledge only jonson 1976a and knoester 1979a have devoted entire papers solely to this central issue and these are relatively inaccessible and hence less widely known than they deserve to be in this lecture i shall attempt to give a brief account of the essentials of this approach first i shall sketch out its microeconomic background and then i shall discuss those theoretical characteristics which differentiate it from the neo austrian analysis of robert e lucas jr and his associates as well as from the conventional keynesian approach to monetary economics last but by no means least i shall argue that it yields simple theoretical insights into important empirical and policy related issues which are troublesome for alternative approaches in short my tribute to harry

family policy with non cooperative families we consider a non cooperative model of a family s time allocation between market work and providing a home produced family public good such as child care or care for the elderly the model predicts underprovision of the public good because of crowding out this does not necessarily warrant public provision in contrast to other approaches in family economics we find that attempts to redistribute between spouses may alter the final distribution within the marriage and that such a policy may be pareto improving we also find that some degree of progressivity of the income tax can be welfare improving

the melting pot and school choice abstract in the united states parents are offered free public education in the mainstream culture but must pay the full cost of tuition if they educate their children privately this creates strong economic incentives for remaining within the public system which promotes the assimilation of minorities a pareto improvement can be achieved by subsidizing private education in exchange for modifying its social content so as to reduce polarization popular opposition to voucher programs that facilitate school choice without regulating cultural content may partly reflect voters concerns that such programs threaten to erode the common ground created by public education

alternative systems of health care provision health care timothy besley and miguel gouveia rising costs of health care provision throughout the world have provoked a vigorous debate about the design of health care systems and the role of government in health care in countries that have predominantly social provision there are moves towards privatization and direct attempts to regulate prices in more market oriented systems rising costs seem to be resulting in a greater uninsured problem and this too is provoking demands for reform technological change will put constant pressure on health care systems public or private generating serious social choice problems however this must be seen in the context of a traditional commitment to providing a non trivial minimum level of health care to all members of society we discuss evidence from oecd countries from 1960 to 1990 then focus on more recent changes the central problems of market oriented systems are cost containment and the fact that significant segments of the population go uninsured most oecd countries have therefore chosen solutions where health care financing is socialized we present a simple political economy model to describe the conditions under which this will arise the hallmark of public provision is relative uniformity in access to health care this creates incentives for individuals to opt out of the public sector and to supplement what they can obtain through the state in addition it creates a significant commitment to redistribution through public provision rising costs strain the generosity of taxpayers we expect to see increased reliance on market solutions and moves towards multi tiered health systems that differentiate health care provisions made for different groups in society

incorporating quantity discounts and their inventory impacts into the centralized purchasing decision multi site organizations must balance conflicting forces to determine the appropriate degree of purchasing centralization for their respective supplies the ability to garner quantity discounts represents one of the primary reasons that organizations centralize procurement this paper provides methodologies to calculate optimal order quantities and compute total purchasing and inventory costs when products have quantity discount pricing procedures for both all units and incremental quantity discount schedules are provided for four different strategic purchasing configurations scenarios complete decentralization centralized pricing with decentralized purchasing centralized purchasing with local distribution and centralized purchasing and warehousing for ordering decisions under local distribution procedures to determine optimal order quantities and costs are presented in a precise form that could be easily implemented into spreadsheets by practicing managers for the more complicated multi echelon scenarios we introduce a single cycle policy with a tailored aggregation refinement step that performs very well under experimentation when compared to a conservative bound