The Social Planning Problem with Costly Information Processing: Towards Understanding Production Decisions in Centralized Economies

Naeher, Dominik Article — Published Version The Social Planning Problem with Costly Information Processing: Towards Understanding Production Decisions inCentralized Economies Economica Provided in Cooperation with: John Wiley & Sons Suggested Citation: Naeher, Dominik (2022) : The Social Planning Problem with Costly Information Processing: Towards Understanding Production Decisions inCentralized Economies, Economica, ISSN 1468-0335, Wiley, Hoboken, NJ, Vol. 90, Iss. 357, pp. 285-314, https://doi.org/10.1111/ecca.12442 This Version is available at: https://hdl.handle.net/10419/287875 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Economica (2023) 90, 285–314 doi:10.1111/ecca.12442 The Social Planning Problem with Costly Information Processing: Towards Understanding Production Decisions in Centralized Economies By DOMINIK NAEHER University of Goettingen Final version received 15 July 2022. It has long been argued that market economies feature an inherent advantage over centralized economies because they are subject to fewer information processing needs. However, very little work has investigated the role of these needs in explaining differences in optimal production decisions between centralized and market economies. This paper uses a rational inattention approach to study the decision problem of a social planner who faces uncertainty about households’ preferences and can reduce this uncertainty by allocating scarce resources to processing information. The model shows that costly information processing has important implications for optimal production decisions, including for the trade-off between consumption and leisure, the optimal range of different goods produced, and the role of correlated consumer preferences. Overall, the results suggest that differences in empirical production decisions between socialist and capitalist countries may be driven by the different information needs and associated costs of processing information inherent to each economic system, rather than by differences in ideology or political preferences. INTRODUCTION Since the formulation of the first fundamental theorem of welfare economics by Lerner (1934), Lange (1942) and Arrow (1951),1much of the economic literature has focused on discussing the conditions under which the first direction implicit in this theorem—that markets lead to efficient allocations—holds in practice.2In contrast, much less has been written about the empirical plausibility of the second direction implicit in the first welfare theorem, namely that central planning should, in principle, be able to do just as well as the market. Authors who do discuss possible reasons for why this second direction may not hold in practice typically focus on two arguments: lack of incentives (and the associated lack of innovation) in centralized economies, and the enormous information needs inherent to the social planning problem (Hayek 1945; Stiglitz 1996; Feldman and Serrano 2006).3While some authors discuss these arguments at great length at a qualitative level,4there has been (to the best of my knowledge) no previous attempt to investigate in a structural model the implications of these information needs for optimal production decisions in centralized economies.5 This paper begins to fill this gap by studying the decision problem of a central planner who faces uncertainty about households’ idiosyncratic consumption preferences and can reduce this uncertainty by allocating scarce resources to processing information. Specifically, I assume that each household prefers a particular consumption good out of a continuum of goods, and that the time resources available in the economy can be used for three purposes: to produce consumption goods, to process information about households’ preferences, and for leisure. This feature generates a new link between information-related aspects of the economy (such as the distribution of information and the cost of processing information) and optimal production decisions, with important implications for the trade-off between consumption and leisure, the optimal range of different goods produced, and the role of correlation in consumers’ preferences. ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA 02148, USA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. 286 ECONOMICA [JANUARY In particular, I show that a benevolent social planner who faces opportunity costs associated with reducing uncertainty about households’ preferences will optimally allocate fewer resources to producing consumption goods, and thus generate lower levels of per capita consumption than would be achieved in the first-best case where processing information is free (e.g. the standard social planning problem under full information). The intuition behind this result is straightforward. If it is costly for the planner to choose consumption goods that consumers value highly, because reducing uncertainty about households’ preferences consumes resources, then generating welfare from leisure becomes relatively more attractive compared to generating welfare from producing (and consuming) goods. Moreover, the model implies that optimally, a centrally planned economy will produce a smaller range of different goods than a market economy. This is due to the fact that the values chosen under imperfect information tend to be biased towards the expected value; that is, the average good preferred by households. Finally, the model shows that the level of social welfare that can be achieved in a centralized economy with costly information processing is higher when households’ preferences are correlated. In contrast, in a decentralized market economy, such correlations remain unexploited and do not affect social welfare. Overall, the model predicts that when other factors are equal, economies involving strong elements of planning will feature (1) lower levels of per capita GDP, (2) fewer working hours, and (3) less product variety than economies based on free markets. I argue that these predictions are consistent with anecdotal evidence as well as with empirical evidence obtained from cross-country panel regressions. At the same time, I stress that obtaining direct, causal evidence on the role of the investigated channel remains a challenge. In developing the model, I first define the social planning problem with uncertainty about households’ preferences and costly information processing at a general level, keeping both the utility and production function as well as the information technology open as much as possible. To solve the model, I focus on a particular type of information technology based on entropy reduction that is used widely in the literature on rational inattention (Sims 2003;Ma ´ ckowiak and Wiederholt 2009;Mat ˘ ejka and McKay 2015).6I then use the constructed model to compare the optimal outcomes obtained under central planning with those achieved in a market economy. For this purpose, I consider two different cases of a market economy. The first case is equivalent to the market economy in a standard general equilibrium model with competitive markets and full information. In particular, this benchmark case assumes that firms know the preferences of each household (or that households can transmit information about their preferences to firms for free). The second case relaxes the assumption of full information in the market by incorporating the possibility that firms face uncertainty about households’ preferences, and transmitting information between households and firms is costly.7The solutions for these two cases show that the main qualitative results of the model do not depend on the (strong) assumption that the presence of markets reduces the cost of transmitting information in the economy to zero. All that is needed for the results to hold is that the social cost associated with transmitting and processing information about households’ preferences is smaller in a decentralized market economy than it is under central planning. Intuitively, this will hold as long as it is generally easier for households to transmit information about their own preferences to firms through markets than it is for a planner to collect information about each household’s preferences, process the information in a central place, and assign production and consumption plans to firms and households in the absence of markets. It should be noted that the model that I build is highly stylized and designed to capture one particular aspect in which centrally planned economies and market economies differ. It is not constructed to provide a complete or realistic picture of either type of economic system. One benefit of this approach is that the mechanisms that I identify are based exclusively Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 287 on differences in the aggregate cost of processing information across different types of economic systems. The predictions of the model thus arise independently of other channels through which market economies and centralized economies have been argued to differ in the real world, including the roles of a shortage of basic goods (Roland 2000), lower total factor productivity (Weil 2013), highly specific relations between firms (Blanchard and Kremer 1997), and soft budget constraints (Kornai 1998). Many recent studies use the rational inattention approach proposed by Sims (2003) to study the role of costly information processing for economic decision making (see the surveys by Handel and Schwartzstein 2018;Ma ´ ckowiak et al. 2018). The two papers most closely related from that literature are perhaps Angeletos and Sastry (2019) and Lipnowski et al. (2020). Both study the role of interventions by a social planner (or principal) in influencing the choices of rationally inattentive agents to increase social welfare. As in most papers in the rational inattention literature, the focus is on the role of costly information processing among market participants (agents) rather than on costly information processing among central authorities (planners). While the latter has been discussed at great length at a qualitative level in the existing literature (see the studies cited in note 4), I present a structural framework that captures, for the first time, the main insights developed in that literature using the rational inattention framework. The remainder of the paper is organized as follows. Section Iexplores differences in empirical production decisions across countries with different types of economic organization. Guided by the empirical findings, Section II constructs a model based on costly information processing that offers a possibility to account for some of the observed differences. Section III concludes by discussing the implications and limitations of the model. I. EMPIRICAL MOTIVATION This section documents three ways in which production decisions tend to differ across countries with different types of economic organization, that is, across economic systems that involve different degrees of government intervention in markets. Specifically, I find that when keeping other factors fixed, economies involving stronger elements of planning tend to feature (1) lower levels of per capita GDP, (2) fewer working hours, and (3) less product variety than economies based on free markets. I consider three approaches used in the literature to classify countries according to their type of economic organization. The first approach is based on the classification in the ‘Freedom in the World’ reports (e.g. Freedom House 2003), which essentially relies on a dashboard approach of various indicators and assessment by country experts. This classification has been used widely in the economic literature (including in the seminal works of Barro 1991; Hall and Jones 1996; Sala-i Martin 1997) and is available for the years 1978 to 2002. In each year, countries are assigned into six categories of economic systems ranging from ‘statist’ to ‘capitalist’.8To reduce noise, I combine each two consecutive categories into one.9The indicator used in the analysis thus captures three types of economic system: statist, mixed and capitalist. An exemplary list of countries and their assigned categories in some years is provided in Table A2 in the Appendix. The second approach uses the Index of Economic Freedom (Heritage Foundation 2019). This index has been available since 1995 and provides a continuous rating of the degree of free markets and absence of government control in each country based on 12 quantitative and qualitative subindicators. I do not make any transformations to the original values of the Index of Economic Freedom, so that the variable used in the analysis ranges from 0 to 100, with higher scores indicating greater economic freedom (see the Appendix for examples). Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 288 ECONOMICA [JANUARY The third approach is based on the view that strong government intervention into markets implies a large share of employees working in the public sector (Fuchs-Sch¨ undeln and Sch¨ undeln 2020). I therefore consider the share of public sector employment in total employment as another proxy of countries’ type of economic organization. The underlying data come from the World Bank’s Worldwide Bureaucracy Indicators database,10 which covers the years since 2000. Since for many countries these data are available only in some years—e.g. sporadically every three or four years—I interpolate missing values using a linear trend (I do not extrapolate values, that is, I do not impute missing values beyond the period for which data are available for each country). The other variables used in the analysis are Log GDP per capita,Hours worked, and Product variety. The construction and data sources of these variables are described in the Appendix, together with basic summary statistics (Table A1). Figure 1provides a first impression of how the three considered outcomes differ across economies classified as capitalist, statist or mixed. As shown in panels A and C, statist economies feature lower levels of GDP per capita and less product variety. At the same time, panel B shows that statist economies feature more working hours than capitalist economies. I obtain the same results for more recent years when using the Index of Economic Freedom as a proxy for countries’ economic organization.11 Of course, these correlations only represent unconditional relations that are likely driven by many other factors. For example, most of the statist economies in my sample represent lowor middle-income countries, which are known to feature more working hours (Bick et al. 2018) and less product variety (Imbs and Wacziarg 2003; Cadot et al. 2011) than advanced economies. As Figure A1 in the Appendix shows, this also applies to my sample. Thus the findings from Figure 1that statism (lack of economic freedom) is positively associated with working hours and negatively associated with product variety could also be driven by differences in other third factors. To test whether or not this is the case, I estimate the empirical role of the type of economic organization in a panel regression framework that controls for time-invariant country characteristics (e.g. geographical features, history, cultural traits) through the inclusion of country fixed effects. Of course, the observational nature of the underlying data limits my ability to identify causal links, but the results will nevertheless be useful in informing the construction of the model in Section II. The regression model that I estimate can be written as (1) Yct =α+βxXct +γc+ψt+εct , where the dependent variable is either Hours worked or Product variety of country cin year t,Xct is one of the three proxies for a country’s type of economic organization (i.e. the classification into statist, capitalist or mixed, the Index of Economic Freedom,orPublic sector employment), γcdenotes country fixed effects, ψtdenotes year fixed effects, and αis a constant. The error term εct is estimated using robust standard errors. The hypotheses that I test are that economies based on strong government intervention in markets feature fewer working hours and less product variety than market-based economies when other factors are equal (corresponding to the predictions of the model discussed below). Therefore I test whether the coefficient βxin equation (1) is negative for both Hours worked and Product variety when Xct is Public sector employment or a dummy for a Statist economy, and βxis positive when Xct is the Index of Economic Freedom. Table 1reports estimates of the regression model specified in equation (1). The dependent variable in columns (1)–(3) is Hours worked, and the dependent variable in columns (4)–(6) is Product variety. The results in column (1) show that once fixed country characteristics Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 289 (a) Per capita income 8.0 8.5 9.0 9.5 Log GDP per capita 1990 1995 2000 2005 Statist Mixed Capitalist (b) Hours worked 32 34 36 38 40 42 44 Hours per week 1990 1995 2000 2005 Statist Mixed Capitalist (c) Product variety 0 500 1,000 1,500 Number of products exported 1990 1995 2000 2005 Statist Mixed Capitalist FIGURE 1. Economic outcomes for different types of economic organization. Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 290 ECONOMICA [JANUARY TABLE 1 RESULTS OF PANEL DATA ANALYSIS Hours worked Product variety (1) (2) (3) (4) (5) (6) Statist −0.920*** −174.297*** (0.001) (0.000) Capitalist −0.332 6.076 (0.263) (0.860) Index of Economic Freedom 0.024*** 5.696*** (0.003) (0.000) Public sector employment −0.034** −11.736*** (0.023) (0.000) Country fixed effects Yes Yes Yes Yes Yes Yes Year fixed effects Yes Yes Yes Yes Yes Yes Observations 821 1489 611 2246 3591 1163 R-squared 0.980 0.966 0.976 0.982 0.988 0.995 Notes: p-values are shown in parentheses. The omitted category for type of economic system (in columns (1) and (4)) is ‘Mixed’. *, **, *** indicate p<0.10, p<0.05, p<0.01, respectively. are accounted for, statist economies feature fewer working hours than mixed economies (the omitted category), reversing the direction of the unconditional relationship shown in panel B of Figure 1.12 At the same time, the results in column (4) confirm that statist countries tend to feature less product variety.13 The same conclusions arise from using the other two ways of classifying economic systems. Specifically, the results in columns (2) and (5) of Table 1show that higher economic freedom is associated with more working hours and greater product variety. The results in columns (3) and (6) show that higher rates of public sector employment are associated with fewer working hours and less product variety, which is in line with the insights obtained from columns (1) and (4). In addition, I obtain largely consistent results when using the logarithms of Hours worked and Product variety as dependent variables, and when estimating weighted regressions that weight for countries’ population size. I stress again that the observational nature of my data limits my ability to interpret the estimates in Table 1as causal. At the same time, there appear to be important differences in empirical production decisions across market-based and more centralized economies that are relatively robust across different measures and specifications. The model constructed in the next section focuses on providing a possible way to rationalize these differences. II. MODEL Setup I study a static economy with Nhouseholds and the same number of firms. There is a continuum of consumption goods, c∈R, of which each household iprefers a specific good, denoted c∗ i. Each household can consume only a single good. Households’ preferences are represented by a utility function of the form (2) u(ci,c∗ i,q(ci),Ri)=m(ci,c∗ i)v(q(ci)) +w(Ri) Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 291 defined over a similarity measure m(ci,c∗ i)between the consumed good ciand the preferred good c∗ i, the quantity q(ci)of the consumed good, and leisure Ri. The similarity measure m(·,·)is defined as the inverse of a distance metric d:R×R→[0,∞), where d(ci,c∗ i)=0 if and only if ci=c∗ i, and d(ci,c∗ i)>0 whenever ci= c∗ i. For ease of exposition, let dbe such that m(ci,c∗ i)∈(1,A), where A>1 is a constant (a distance metric with this feature will be specified below). The functions v(·)and w(·)are non-negative, twice continuously differentiable, and strictly increasing and concave. In other words, the household derives higher utility from consuming goods that are more similar to the preferred good, from consuming more of any good (including when ci=c∗ i), and from taking more leisure. Each household is endowed with Tunits of time, which can be used as either leisure or labour. Each firm juses labour to produce a single consumption good according to an increasing and concave production function f(zj), where zjis the firm’s labour input (in units of time), and f(zj)is the produced quantity. Social planning problem with uncertainty about household preferences I start by studying the decision problem of a benevolent central planner who seeks to maximize the total sum of households’ utility but does not readily observe the parameters c∗ ithat define each household’s consumption preferences.14 For now, I focus on the case where households’ preferences are fully idiosyncratic; that is, c∗ iis drawn independently from some distribution for each i(this assumption will be relaxed in the final subsection of this section). To reduce uncertainty about c∗ i, the planner can assign part of the labour force to collecting and processing information. These activities are costly in that they consume time and represent a second form of labour to households, in addition to the labour spent on producing consumption goods at a firm. Processing information is modelled as generating a signal sion the value of c∗ i. Signals will be specified below. For now, simply consider signals to be a function sof households’ true values of c∗ iand the time allocated to processing information about c∗ i.Letκi denote the amount of time that household ispends on collecting and processing information (about the preferences of any households), and let κ(c∗ i)denote the total amount of time across households that is allocated to processing information about a particular c∗ i.15 The precision of sidepends positively on κ(c∗ i). Moreover, the planner faces the following time budget constraint for each household: (3) κi+li+Ri≤T, where κiis the amount of time that household ispends on processing information, liis the amount of time spent on producing consumption goods, and Riis leisure. The timing of the model is such that the central planner first chooses the allocation of time subject to constraint (3), then receives the signals, and finally decides which good each firm produces and each household consumes, based on the received signals. Formally, the social planner’s problem with uncertainty about households’ preferences and costly information processing can be written in the form (4) max κi,li,Ri≥0 N  i=1 Eu(ci,c∗ i,q(ci),Ri) subject to ci=arg min c∈REd(c,c∗ i)|si,(5) Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 292 ECONOMICA [JANUARY si=s(c∗ i,κ(c∗ i)),(6) q(ci)≤ N  j=1 fci(zj),(7) N  j=1 zj≤ N  i=1 li,(8) N  i=1 κ(c∗ i)≤ N  i=1 κi,(9) κi+li+Ri≤T.(10) Condition (5) states that the social planner chooses for each household the consumption good that is closest in expectation to the household’s optimal good given the signal obtained according to condition (6).16 Condition (7) describes the available production technology for consumption goods (recall that q(ci)is the quantity of the good ciproduced for household i, and zjis the labour input of firm j, where fciindicates that a firm produces good ci). Conditions (8) and (9) are the time constraints for production and information processing, and condition (10) is the time budget constraint facing each household. The objective of the central planner is to maximize households’ expected welfare defined by expressions (4) and (2). Overall, the central planner allocates the available time resources NT so as to maximize households’ welfare subject to the given constraints. Notice that the central planner faces the following two trade-offs. First, the planner can tailor the economy’s production and consumption schedules to better match individual households’ preferences by allocating more resources to processing information. This comes at the cost of fewer resources being available for the production of the chosen goods (resulting in lower quantities q(ci)) and for leisure. In some sense, this might be seen as a trade-off between the quantity and ‘quality’ of consumption (where quality refers to the match with household-specific consumption preferences, not to any intrinsic attributes that reflect higher quality of the produced goods in general). Second, the economy can enjoy more leisure at the cost of lower utility from consumption (in terms of both quantity and how well the produced goods match households’ preferences).17 Importantly, these two trade-offs will in general be linked to each other. For example, if the central planner chooses to learn relatively much about which good each household values most, then it also becomes attractive to allocate relatively more resources to the production of consumption goods than to leisure. Of course, the optimal allocation of resources will depend on the specifications of the utility function and production function as well as the available information technology (i.e. the structure and cost of signals). To shed some light on the role of each of these components, the next four subsections will proceed in the following way. First, I will characterize the mechanisms underlying the solution of the model for a particular information technology based on entropy reduction (from the rational inattention literature), keeping the utility and production function as general as possible. Next, I will compare the resulting allocation to the allocation achieved in a decentralized market economy. Then I will consider a specific utility and production function that provides an intuitive numerical illustration of the model’s solution. Finally, I will sketch a simplified decision problem that abstracts from Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 299 FIGURE 2. Optimal allocation and expected utility for different values of μ. Notes: Simulations are performed by varying μfor fixed parameter values T=3andσ2 c=1. This equation implicitly pins down the optimal value of lifor given parameter values T,μ and σ2 c. For example, solving equation (20) for fixed parameter values (T,σ2 c)=(3,1)and different values of μleads to the solution shown in Figure 2. When μis very small—i.e. reducing uncertainty about households’ preferences is very costly—it is optimal not to allocate any resources to processing information. In this case, the term in square brackets multiplying ln(1+q(ci)) in equation (17) equals 1 (recall the proof of Proposition 1), so that half of the available resources Tare allocated to production and to leisure, respectively (see the left-hand edge of Figure 2). As μincreases, more resources are allocated to the production of consumption goods so that liincreases (in line with Proposition 1). For example, when μ=μ SP =1, the optimal allocation of resources chosen by the central planner with costly information processing is given by (κi,li,Ri) SP ≈(0.60,1.77,0.63), and the expected welfare of each household is U SP i≈2.2. Let us first compare (κi,li,Ri) SP to the allocation chosen by a central planner who does not face uncertainty about households’ preferences, that is, the standard social planning problem under full information. In this case, the planner is able to set ci=c∗ ifor each household without devoting any resources to processing information. For the numerical example with (T,σ2 c)=(3,1)considered above, the optimal allocation is then given by (κi,li,Ri)SP =0,21 3,2 3, and the associated welfare of each household achieved by the social planner with full information is USP i≈2.9. Of course, it holds that USP i>U SP i.In addition, notice that in line with Proposition 1, the social planner sets a higher value of li under full information than in the case of costly information processing. Next, let us compare (κi,li,Ri) SP to the allocation achieved in a market economy where consumption and production decisions are made locally by those actors who are in the immediate possession of the relevant information (e.g. consumption decisions are made by households according to knowledge about their own preferences), so that there is no need to collect and process information centrally. For the benchmark case of a market economy with full information, the optimal allocation of resources is equivalent to the first-best allocation Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 300 ECONOMICA [JANUARY chosen by the social planner with full information (the first welfare theorem applies). In the example considered above, the market allocation under full information is thus given by (κi,li,Ri)M=0,21 3,2 3, with welfare UM i≈2.9. Now consider the case of a market economy where the assumption of full information is relaxed and μ M>μ  SP =1. For example, if (T,σ2 c)=(3,1)as above, and μ M=2 (i.e. achieving the same reduction in entropy requires twice as many resources under central planning than with decentralized markets), then the optimal allocation for the market economy is given by (κi,li,Ri) M≈(0.48,1.94, 0.58). The associated welfare of each household is U M i≈2.5. In line with the results in Proposition 3, it holds that l SP i<l M i<lM i=lSP i (and thus also q SP (ci)<q M(ci)<qM(ci)=qSP (ci)), and USP i=UM i>U M i>U SP i. Correlated preferences Until now, the analysis has assumed that households’ consumption preferences are fully idiosyncratic, that is, the fundamentals c∗ iare independently distributed across households. To provide some additional insights, I now allow for the possibility that the variables c∗ iare correlated across households.29 In the model, the total uncertainty that the central planner faces about households’ preferences can be quantified by the joint entropy of the set of random variables {c∗ 1,... ,c∗ N}. If c∗ 1,... ,c∗ Nare independent, then the joint entropy is given by the sum of individual entropies (21) H(c∗ 1,... ,c∗ N)= N  i=1 H(c∗ i). If c∗ 1,... ,c∗ Nare correlated, then the joint entropy is given by H(c∗ 1,... ,c∗ N)= N  i=1 H(c∗ i|c∗ i−1,... ,c∗ 1), which is strictly smaller than the joint entropy of independent variables specified in equation (21) (see Cover and Thomas 1991, p. 40). Given the information technology specified in equation (13), it thus holds that correlated preferences are associated with lower information processing costs to achieve the same reduction in uncertainty. The level of social welfare that can be attained in a centrally planned economy is thus higher if consumption preferences are correlated across households (and planners are aware of it). In contrast, in a decentralized market economy, correlated preferences do not necessarily increase the level of social welfare. If the good that a firm produces for a particular household depends only on the signal received from that household (i.e. there are no spillovers in uncertainty reduction), then the correlation in households’ preferences remains unexploited, and the level of social welfare is the same as with fully idiosyncratic preferences. Proposition 4. The level of social welfare that can be achieved in a centralized economy with costly information processing is higher when households’ preferences are correlated, that is, when c∗ 1,... ,c∗ Nare not independent. In a market economy with costly information processing and no spillovers in uncertainty reduction, social welfare is the same irrespective of whether preferences are correlated or not. Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 301 Proof. The result for centralized economies follows from the fact that with correlated preferences, the social planner can achieve any allocation that can be achieved with independent preferences, but with fewer resources κ. The statement about market economies holds because, in the absence of spillovers, every household and firm behaves identically to the situation when preferences are not correlated.  Obviously, with full information it does not matter whether preferences are correlated or not. When information processing is costly, however, Proposition 4implies that consumers in centralized economies will tend to fare better if their preferences happen to be correlated. Intuitively, this is due to the fact that with correlated preferences, learning which good a particular household prefers also helps the planner to learn about the preferred goods of other households.30 In contrast, in market economies, this feature will remain unexploited given that the good produced for a particular household depends only on the signal sent by that household, which remains unaffected by the presence of correlation in preferences.31 Proposition 4also implies that if it were possible to influence consumers in a way that induces correlation in their preferences, then this might indeed be optimal for a benevolent social planner to do (including some scope for potential costs associated with such efforts). While I do not have data to test this prediction empirically, it appears to correspond well to anecdotal evidence of efforts of leaders in socialist countries to promote certain ideals that make consumer behaviour more uniform.32 While in practice such efforts are often justified with a particular ideology (e.g. communism), the model suggests that these efforts may also be the result of mere economic considerations. Specifically, the fundamental structure of economic organization in socialist countries with strong elements of central planning may give rise to an intrinsic motive for aligning consumer preferences in order to reduce the costs for planners associated with choosing goods that consumers value highly.33 III. CONCLUSION To the best of my knowledge, this paper presents the first attempt to investigate in a structural model the implications of the long-held view that the presence of markets provides a natural advantage over economic systems based on central planning, as the latter involve large information processing needs that do not (or only to a lesser extent) exist in market economies. The stylized model that I construct shows that the optimal outcome in a centralized economy with uncertainty about households’ preferences and costly information processing is characterized by a lower share of resources allocated to production, lower levels of per capita consumption and welfare, and a smaller range of different goods produced than the optimal outcome achieved in a market economy. Importantly, these results do not depend on the assumption that information processing is free in market economies, or that costly information is not a binding constraint also in market economies. All that is needed is that centralized learning is costlier than localized learning—that is, that collecting information about households’ idiosyncratic preferences, processing the information in a central place, and assigning production and consumption plans to firms and households requires more resources than allowing households and firms to coordinate their consumption and productions plans directly with each other through interaction on markets. The intuition behind the results is based on two effects in the model. First, centralized economies require more resources to attain production and consumption schedules that are associated with the same expected level of utility as the outcomes attained in market economies (income effect). Second, given that it is more costly for centralized economies to produce goods that consumers value highly, optimally, centralized economies generate Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 302 ECONOMICA [JANUARY relatively more utility from leisure than from producing goods compared to market economies (substitution effect). The magnitudes of these effects are less severe when central authorities require fewer resources to reduce uncertainty about households’ preferences, for example, when better information infrastructure is available or when households’ preferences are correlated with each other. Overall, the model captures the idea that the existence of markets in an economy helps to align decision-making with the way information is distributed. For example, consumption decisions are made by those actors (households) who are also in the immediate possession of information about consumption preferences. In contrast, in centralized economies, parts of these decisions are made by central authorities who are not in immediate possession of the needed information and thus have to engage in costly activities to collect, process and transmit information.34 Hence the model reflects the view that in addition to their role in balancing demand and supply through prices, markets also constitute an effective mechanism to minimize the social cost of information processing in an economy. This feature also implies that the existence of markets is part of a society’s information infrastructure, and societies that start to replace elements of central planning with market interaction should be expected to achieve reductions in the economic burden associated with information processing, leading to higher levels of overall efficiency and welfare. At the same time, it is important to stress that this paper focuses on a highly stylized, static model of a centrally planned economy with a single factor of production, which abstracts from many features of centralized economies that are likely crucial in shaping real-world production decisions. Future work concerned with developing a more general theory of optimal production decisions in centralized economies with costly information processing might consider embedding some of the ideas presented here into a dynamic framework, and incorporating additional aspects (such as sticky production decisions or soft budget constraints) that have been argued to play important roles in centralized economies. APPENDIX Data Data on average annual hours worked by persons engaged are taken from the Penn World Table (version 9.1; see Feenstra et al. 2015). These data are available for about 65 countries in every year since 1990 (for earlier years, significantly fewer countries are covered). Measuring product variety, that is, the range of different goods produced in an economy, is less straightforward as there are no indicators readily available that are comparable across countries. I therefore follow the approach taken by Funke and Ruhwedel (2001) and use the number of different products exported (counted at the 6-digit HS level) as reported in trade data to proxy domestic product variety. This approach has the limitation that some intermediate goods produced at home may not be traded internationally and are thus not captured in trade data. On the other hand, using trade data has the benefits that the classification of goods is largely consistent across countries, and that data are available for a wide range of countries and years. Funke and Ruhwedel (2001) also argue that most of the important goods in a country are probably either exported or imported (see also Feenstra and Kee 2004). The indicator that I use in the analysis is the number of different products (at the 6-digit HS level) exported in a given year from the World Integrated Trade Solution database, which is based mainly on data from the United Nations’ Commodity Trade Statistics. Data on this indicator are available for at least 150 countries each year since 1988. Data on GDP per capita come from the World Bank and are measured in constant international 2011 dollars (PPP adjusted). These data are available from 1990 onwards. Table A1 reports summary statistics together with the number of countries and years covered in the analysis. Table A2 provide an exemplary list of countries and the data used to classify different types of economic organization (the list includes all countries in the sample with a population of more than Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 303 TABLE A1 SUMMARY STATISTICS Variable Mean Min Max S.D. Countries Years Economic system Statist 0.271 0 1 190 1978–2002 Mixed 0.311 0 1 190 1978–2002 Capitalist 0.418 0 1 190 1978–2002 Index of Economic Freedom 59.813 15.600 90.500 10.764 180 1995–2017 Public sector employment 15.926 1.862 47.261 10.347 126 2000–2017 Log GDP per capita 8.991 5.889 11.813 1.222 191 1990–2017 Hours worked 36.770 26.036 51.476 5.22 70 1990–2017 Product variety 1002.895 1 4883 1308 187 1988–2017 TABLE A2 DATA ON TYPE OF ECONOMIC ORGANIZATION Economic system (Freedom House) Index of Economic Freedom Public sector employment Country 1990 2002 2010 2010 Afghanistan Statist Statist 9.7 Algeria Statist Statist 56.9 Angola Statist Statist 48.4 9.2 Argentina Mixed Capitalist 51.2 16.2 Australia Capitalist Capitalist 82.6 Bangladesh Mixed Mixed 51.1 4.5 Belgium Capitalist Capitalist 70.1 33.7 Bolivia Mixed Capitalist 49.4 10.0 Brazil Mixed Mixed 55.6 12.0 Burkina Faso Statist Statist 59.4 2.2 Cambodia Statist Statist 56.6 5.6 Cameroon Capitalist Capitalist 52.3 6.2 Canada Capitalist Capitalist 80.4 21.0 Chad Capitalist Capitalist 47.5 4.3 Chile Capitalist Capitalist 77.2 7.3 China Statist Statist 51.0 15.7 Colombia Mixed Mixed 65.5 4.3 Congo, Dem. Rep. Mixed Statist 41.4 5.6 Cˆ ote d’Ivoire Capitalist Capitalist 54.1 Czech Republic Statist Capitalist 69.8 18.4 Ecuador Mixed Mixed 49.3 9.4 Egypt Statist Statist 59.0 24.2 Ethiopia Statist Statist 51.2 4.4 France Capitalist Capitalist 64.2 31.8 Germany Capitalist Capitalist 71.1 Ghana Mixed Mixed 60.2 5.8 Greece Capitalist Capitalist 62.7 22.0 Guatemala Mixed Mixed 61.0 5.5 Guinea Capitalist Capitalist 51.8 3.2 Hungary Statist Capitalist 66.1 22.9 Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 304 ECONOMICA [JANUARY TABLE A2 (CONTINUED) Economic system (Freedom House) Index of Economic Freedom Public sector employment Country 1990 2002 2010 2010 India Mixed Mixed 53.8 Indonesia Mixed Mixed 55.5 Iran Mixed Mixed 43.4 Iraq Statist Statist Italy Mixed Mixed 62.7 18.6 Japan Capitalist Capitalist 72.9 Kazakhstan Statist Mixed 61.0 37.2 Kenya Capitalist Capitalist 57.5 Korea, Rep. Mixed Mixed 69.9 Madagascar Statist Statist 63.2 2.6 Malawi Capitalist Capitalist 54.1 5.2 Malaysia Capitalist Capitalist 64.8 Mali Statist Statist 55.6 Mexico Mixed Mixed 68.3 11.3 Morocco Mixed Mixed 59.2 Mozambique Statist Statist 56.0 3.3 Myanmar Statist Statist 36.7 Nepal Capitalist Capitalist 52.7 Netherlands Capitalist Capitalist 75.0 Niger Capitalist Capitalist 52.9 4.0 Nigeria Mixed Capitalist 56.8 Pakistan Mixed Mixed 55.2 7.7 Peru Mixed Mixed 67.6 8.2 Philippines Mixed Mixed 56.3 8.2 Poland Statist Capitalist 63.2 19.1 Portugal Capitalist Capitalist 64.4 21.5 Romania Statist Mixed 64.2 12.8 Russia Statist Mixed 50.3 Rwanda Statist Statist 59.1 Saudi Arabia Mixed Mixed 64.1 Senegal Capitalist Capitalist 54.6 South Africa Mixed Mixed 62.8 15.7 Spain Capitalist Capitalist 69.6 22.7 Sri Lanka Mixed Capitalist 54.6 16.1 Sudan Capitalist Mixed Tanzania Statist Statist 58.3 2.7 Thailand Capitalist Mixed 64.1 9.1 Tunisia Capitalist Capitalist 58.9 21.1 Turkey Mixed Mixed 63.8 12.9 Uganda Mixed Mixed 62.2 3.0 UK Capitalist Capitalist 76.5 30.1 Ukraine Statist Capitalist 46.4 38.8 USA Capitalist Capitalist 78.0 Uzbekistan Statist Statist 47.5 Venezuela Mixed Mixed 37.1 Vietnam Statist Statist 49.8 9.6 Yemen Mixed Mixed 54.4 Zambia Statist Statist 58.0 6.2 Zimbabwe Mixed Mixed 21.4 6.1 Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 305 (a) Hours worked 25 30 35 40 45 Hours per week 6 8 10 12 Log GDP per capita (b) Product variety 0 2,000 4,000 6,000 Number of products exported 6 8 10 12 Log GDP per capita FIGURE A1. Hours worked and product variety for different income levels, 2010. Notes: Product variety is measured as the number of different products exported in a given year, counted at the 6digit HS level. GDP per capita is measured in constant international 2011 dollars, purchasing power parity adjusted. All values are for the year 2010. 10 million people). Figure A1 shows the correlations between per capita GDP and working hours as well as product variety. Proof of Proposition 1 With the assumptions made in Section II, including the properties of v(·)and w(·)specified in the first subsection, the decision problem of the rationally inattentive central planner can be written as (A1) max κi,li,Ri≥0 N  i=1 EA−(ci−c∗ i)2v(q(ci)) +w(Ri) subject to ci=argmin c∈RE(ci−c∗ i)2|si, with c∗ i∼N(0,σ2 c),(A2) Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 306 ECONOMICA [JANUARY si=c∗ i+εi, with εi∼N(0, σ2 ε),(A3) μκ(c∗ i)=H(c∗ i)−E[H(c∗ i|si)],(A4) q(ci)≤ N  j=1 fci(zj),(A5) N  j=1 zj≤ N  i=1 li,(A6) N  i=1 κ(c∗ i)≤ N  i=1 κi,(A7) κi+li+Ri≤T.(A8) Notice that households are identical except for their value of c∗ i, and firms are identical except for the type of good that they produce. Given that each household can consume only a single good, each firm can produce only a single good, and both the production function and the information technology are the same for all goods, the optimal production and consumption plans must be symmetric across firms and households. Specifically, it will hold that each firm uses the same amount of labour input z, each household consumes the same quantity qof some consumption good (where the good itself and the derived utility may differ across households, whereas the ex ante expected utility is the same for all households), and the central planner allocates an equal amount of time κto reducing uncertainty about the consumption preferences of each household. To see that this must hold, suppose that there was a solution that featured two households with different values of q(ci)and Ri. Since the central planner does not observe the values of c∗ i(which are drawn independently from the same distribution for all households) when deciding on the allocation of time for each household, all households are ex ante (i.e. before processing any information) identical to the planner. Given that the utility function is strictly concave and the choice set is convex for each household, there exists a unique solution for each household. Therefore any allocation that features two households with different values of q(ci)and Ricould be improved by adjusting these values for at least one household. Since firms and goods are homogeneous in the sense that the production function f(zj)is the same for every good, it does not matter by which particular firm a given good is produced, nor for which firm a household works (recall that each firm can produce only a single good, and there are as many firms as households). In the same way, it does not matter which households are assigned to process information about any particular household’s consumption preferences. Without loss of generality, let firm j=1 produce the good consumed by household i=1 with the labour supplied by household 1, let firm 2 produce the good consumed by household 2 with the labour supplied by household 2, and so on, such that it holds that zj=lifor j=i,andq(ci)=f(li). Analogously, since the total amount of time allocated to processing information about c∗ iis the same as the time spent by household ion processing information (about the preferences of any household), let κireplace κ(c∗ i)in the notation used so far.35 Also notice that equation (A2) implies that the value of cichosen by the central planner equals E[c∗ i|si]. Using this property, the term E[(ci−c∗ i)2] from the planner’s objective (A1) can be written as (A9) E[(E[c∗ i|si]−c∗ i)2]=σ2 c|s, where σ2 c|sis the conditional variance of c∗ igiven si. In addition, using the definition of entropy, equation (A4) can be written as (A10) μκi=1 2lnσ2 c σ2 c|s. Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 2023] THE SOCIAL PLANNING PROBLEM 307 Combining equations (A9)and(A10) shows that (A11) E[(ci−c∗ i)2]=σ2 ce−2μκi. Notice that this implies that the term E[A−(ci−c∗ i)2] in the planner’s objective (A1) takes values that range between A−σ2 c(when κi=0) and A(when κi→∞). The parameter value restriction σ2 c∈(0,A)specified in the third subsection of Section II thus guarantees that the term multiplying v(q(ci)) in the planner’s objective is positive (and bounded from above by A). The planner’s problem given by expressions (A1)–(A8) can thus be written as max κi,li,Ri≥0 N  i=1 (A−σ2 ce−2μκi)v(f(li)) +w(Ri)(A12) subject to κi+li+Ri≤T.(A13) Using λto denote the Lagrange multiplier of the time budget constraint (A13), the first-order conditions of this problem are given by {κi}:2σ2 cμe−2μκiv(f(li)) =λ, {li}:(A−σ2 ce−2μκi)vl(f(li)) =λ, {Ri}:wR(Ri)=λ, where vl(f(li)) is the partial derivative of v(f(li)) with respect to li,andwR(Ri)is the partial derivative of w(Ri)with respect to Ri. Combining these three equations and simplifying the results leads to the two optimality conditions κi=1 2μln2σ2 cμv(f(li)) wR(Ri),(A14) wR(Ri)=2Aμv(f(li)) vl(f(li)) 2μv(f(li)) +vl(f(li)).(A15) The result in Proposition 1can now be derived as follows. First, rewrite equations (A14)and(A15) in the form f(l,R,μ) =2σ2 cμe−2μ(T−l−R)v(f(li)) −wR(Ri)=0,(A16) g(l,R,μ) =1 wR(Ri)−1 Av l(f(li)) −1 2Aμv(f(li)) =0,(A17) where land Rare variables, μis a parameter that may change, and A,Tand σ2 care constants. Equations (A16)and(A17) define implicitly land Ras functions of μ. A solution (l∗(μ),R∗(μ)) to the system of two equations (A16)and(A17) must fulfil f(l∗(μ),R∗(μ),μ) =0andg(l∗(μ),R∗(μ),μ) =0 for all μ. Without specifying explicitly v(·)and f(·), it is not possible to solve for l∗(μ) and R∗(μ). Nevertheless, for small changes in μ, the associated changes in l∗(μ) and R∗(μ) can be calculated using the implicit function theorem. Taking the differential on each side of the equations f(l,R,μ) =0 and g(l,R,μ) =0gives fl·dl+fR·dR+fμ·dμ=0, gl·dl+gR·dR+gμ·dμ=0. Solving for dl/dμleads to the expression (A18) dl dμ=−gR·fμ+fR·gμ fl·gR−fR·gl. Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science 308 ECONOMICA [JANUARY The partial derivatives in equation (A18) can be calculated as fl=2μσ2 ce−2μ(T−l−R)(2μv+vl), fR=4μ2σ2 cve−2μ(T−l−R)−wRR, fμ=2σ2 cve−2μ(T−l−R)[1 −2μ(T−l−R)], gl=vll Av2 l+vl 2Aμv2, gR=−wRR w2 R , gμ=1 2Avμ2. Note that, for ease of exposition, the arguments of the partial derivatives have been omitted. Strict concavity of the functions v(·)and w(·)implies that fl,fR,gR,gμ>0. In addition, the parameter restrictions introduced in the third subsection of Section II ensure that fμ,gl<0. Using these results in equation (A18) shows that dl/dμ>0. Thus the equilibrium value of ldepends positively on the parameter μ. This completes the proof of Proposition 1. Proof of Proposition 2 As shown in the previous subsection, the rationally inattentive social planner selects for each household i the consumption good ci=E[c∗ i|si], where the signal is given by si=c∗ i+εiwith c∗ i∼N(0, σ2 c)and εi∼N(0,σ2 ε). In the Gaussian case considered here, the conditional expectation of c∗ igiven siequals (A19) E[c∗ i|si]=E[c∗ i]+cov(si,c∗ i) var(si)(si−E[si]). It can be verified easily that cov(si,c∗ i)=σ2 cand var(si)=σ2 c+σ2 ε. Plugging these expressions into equation (A19)gives (A20) E[c∗ i|si]=σ2 c σ2 c+σ2 ε (c∗ i+εi). For normally distributed random variables, it also holds that the conditional variance takes the form (A21) σ2 c|s=var(c∗ i)−cov(c∗ i,si)cov(si,c∗ i) var(si). Using the results from above to simplify equation (A21)leadsto (A22) σ2 c|s=σ2 c1−σ2 c σ2 c+σ2 ε. From equation (A10), we also know that (A23) σ2 c|s=σ2 ce−2μκi. Finally, using equations (A22)and(A23) to simplify the expression in equation (A20) shows that the good consumed by household iis given by ci=(1−e−2μκi)(c∗ i+εi), where (1−e−2μκi)∈[0,1]. Hence the value of ciis dampened in magnitude relative to c∗ i(as well as noisy), and the range of goods ciconsumed across all households is smaller than the range of the preferred goods c∗ i. This completes the proof of Proposition 2. Economica ©2022 The Author. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science