Theoretical background on aggregation: Micro-macro debates M.A. Keyzer

1. Lecture 2 Theoretical background on aggregation:Micro-macro debatesM.A. Keyzer Presentation available: www.sow.vu.nl/downloadables.htm www.ccap.org.cn

2. Overview of the lecture 1. Importance of the subject and approach 2. Comparable utilities, exact aggregation 3. Noncomparable utilities, optimal aggregation 4. Spatial aggregation over markets 5. Aggregation over commodities 6. Conclusions

3. 1. Subject and approach Question: How to represent the behavior of many individuals by a tractable number of agents and markets? Principles: 1. Socio-economic environment of individuals is described by finite number of fixed (spatial and social) characteristics that follow a smooth joint distribution. 2. Individuals choose optimally from options.

4. 1. Subject and approach (end) Why special attention to aggregation issues? (a) Aggregation errors are of particular importance in China (b) Current developments require disaggregated approach 1. Decentralisation and liberalization 2. Role of state to achieve basic economic targets 3. Increased diversity of lifestyles (migration) 4. Spatially explicit policies in agriculture and environment 5. Increased product heterogeneity

5. 2. Case 1: comparable utilities or payoffs Exact aggregation is possible and representative agents exist. Moreover, their response to changes in the fixed characteristics of their environment is smooth. Two examples: 2.1 Farmland allocation 2.2 Transportation

6. 2.1 Example: Farmland allocation Index s for farmland sites Index j for crops grown on site s : acreage of crop j at site s (decision variables) : revenue per acre of crop j at site s : cost per acre of crop j at site s : rental price of land at site s : characteristics of farming at site s : variation of productivity of farmers growing j at site s : marginal density of , from joint density

7. 2.1 Farmland allocation (2) Allocation by individual farmers (profitabilities ): Allocation by all farmers jointly:

8. 2.1 Farmland allocation (3) After adjusting the rental price to satisfy land balance : This allocation by all farmers jointly coincides with the allocation by a representative farmer:

9. 2.1 Farmland allocation (end) The function has all the properties common in microeconomics: it is strictly convex, non-decreasing, differentiable, and homogeneous of degree one. Hence, it has major practical advantages: - exact aggregation implementable at various scales - continuous responses to price reforms - estimation can be based on data generated by the underlying density.

10. 2.2 Example: Transportation The farmland allocation model has many features that can be used to model the transportation of goods to the most rewarding markets and along the cheapest route. In example 2.1, the representative agent was a farmer at site s allocating a given piece of land over various crops j . Now the representative agent will be a transport firm at site s transporting a given quantity procured from within the continuum of the site to some discrete destination j .

11. 2.2 Transportation (2) Index s for the origin (say, production sites) Index j for the destination (say, markets at district centers) : quantity shipped from site s to market j : net revenue per ton produced at site s : opportunity cost of shipping one ton away from s : characteristics of trading at site s : cost per ton-kilometer at site s : distance between a site in s and the center of j : joint density

12. 2.2 Transportation (end) The same construct as for farmland allocation applies and leads to the following representative ‘transporter’ model.

13. 3 Case 2: Noncomparable utilities Overview: 3.1 Exact aggregation is no longer possible, unless certain strict conditions are met. 3.2 Optimal aggregation is an attractive way out. Methodologies originally developed in mathematical statistics are increasingly made available to economic analysis. Promising research.

14. 3.1 Exact aggregation Suppose individuals in the smooth continuum maximize utility from a discrete set of options, and subject to a budget constraint, with given prices and income . A representative agent construct holds if : a) consumers have common income characteristics, b) the economy has a fixed income distribution, and, c) consumers spend their last penny on a common priced good.

15. 3.2 Optimal aggregation Index i for consumers, i = 1, 2, ..., I Index h for commodities, h = 1, 2, ..., H : price of commodity h : characteristics of consumer i : individual demand by consumer i : aggregate demand Aggregate demand must equal sum of individual demands:

16. 3.2 Optimal aggregation (2) Questions for optimal aggregation: (a) How many income groups would be needed to represent the underlying individual demand functions ? (b) How should the corresponding population weights be determined ? (c) How should the corresponding group demand functions be specified ?

17. 3.2 Optimal aggregation (3) Answer to question (c): The model for a group should simply be the model of one individual of that group. This is required for an analysis of welfare responses to policy reforms. Questions (a) and (b): boil down to an investigation into a choice of weights other than with less than I groups. For this we use kernel learning techniques from the vector-support regression literature.

18. 3.2 Optimal aggregation (4) The idea for obtaining weights for optimal aggregation is to minimize the sum of squared weights, subject to the aggregation constraint, applied for all possible prices . For given aggregate function and given feature functions the optimal aggregation problem reads

19. 3.2 Optimal aggregation (5) The integral that appears in the constraint makes a direct solution to this optimal aggregation problem impossible. Therefore approximation is required. For a series of randomly sampled prices , and a regularization term that accounts for the fact that aggregation cannot be exact, we obtain an optimal aggregation model.

20. 3.2 Optimal aggregation (6) We now write this model in matrix-vector notation. This will clarify that it is a quadratic program that possesses a particularly practical dual formulation. The optimal aggregation model is rewritten as: for , non-negative -matrix and -vectors and with unit elements.

21. 3.2 Optimal aggregation (end) The key feature of this problem is its coincidence with the dual formulation. For the positive semidefinite -matrix defined as , optimal aggregation weights can also be identified as after solving the dual model. In the Chinese context with very large numbers of households this pre-aggregation of information in the matrix would seem necessary to find an optimal number of groups.

22. 4 Spatial aggregation over markets The full representation of transportation economy is possible in a single-commodity welfare model. In multi-commodity welfare model either the number of feasible flows has to be restricted drastically, or, spatial aggregation is required.

23. 4.1 Transportation in welfare model The representation of transport follows the described under exact aggregation Indices (s, r) for the sites (say, cells on a grid) : quantity shipped from site s to site r : price on the market at site s : total availability of the good at site s : cost associated with flow from shipments

24. 4.1 Transportation in welfare model (end) The single-commodity representative trader model for site s. At given production and money-metric utility one can define the corresponding welfare model.

25. 4.2 Spatial aggregation for transition from partial to general equilibrium Spatial aggregation means application of the welfare program at a larger scale, say counties that will be indexed . Now we can move from a single-commodity, partial equilibrium to a multi-commodity general equilibrium framework.

26. 4.2 Spatial aggregation (2) However, the general equilibrium framework with spatial aggregation of markets is to some extent inconsistent. Within a location it abstracts from price variation and allows supply to meet demand along the cheapest route. This problem can partly be overcome by assuming fixed price differentials within a region: Immediate extension of the framework is to allow for production employing endowments as well as current inputs .

27. 4.2 Spatial aggregation (3) After incorporation of production and price variation within regions, the general equilibrium welfare model reads:

28. 4.2 Spatial aggregation (end) This spatially aggregated general equilibrium model forms the basis for the CHINAGRO welfare model. It has many features to capture the response of Chinese agriculture to changing prices and policy reforms. It also has its shortcomings. It rules out changes in routing, while the price band may act as a price distortion rather than reflect true cost. Therefore, in parallel with the general equilibrium model, a set of partial commodity-specific equilibrium models is developed that do not require spatial aggregation.

29. 5 Aggregation over commodities Spatial aggregation is a special case of aggregation over commodities. It aggregates over commodities that only differ with respect to location and can be converted in one another through transportation. Aggregation over commodities requires some sort of nested hierarchy on the supply side (technology), on the demand side (utility), or on both sides. Little can be said in general about such aggregation.

30. 6 Conclusions I. On the representative agent (comparable utilities / payoffs) 1) Nano-foundation of micro assumptions: Discrete choice combined with smooth densities leads to strictly concave and differentiable production and utility functions. 2) Profit maximizing farmers can be represented in a spatial and social continuum, and yet their behavior follows relatively standard micro models of production. 3) Likewise, the approach can deal with transportation from a continuum to a finite number of market places. 4) Risk aversion behavior follows through aggregation, even though the underlying choices are risk neutral.

31. 6 Conclusions (2) II. On the representation of consumers: 1) Representative consumers are selected individuals, not average individuals. 2) Exact aggregation is difficult under individual budget constraints. 3) Kernel learning techniques can be used to determine optimal level of aggregation. 4) It is “safer” to work with aggregate consumers with utilities express in money metric and with an exogenous marginal utility of income, as is done in welfare programs.

32. 6 Conclusions (3) III. On spatial aggregation over markets There is no clean solution. Hence, we operate two models in parallel: (a) a general equilibrium welfare model, in which intra-regional trade is subject to fixed transportation costs for all net purchase and net sales of the county. (b) a set of single commodity partial equilibrium models on a 10 by 10 kilometer grid. 

33. 6 Conclusions (end) IV On aggregation over commodities Aggregation over commodities requires assuming constant returns and a nested hierarchy in production, in utility, or in both. Whether this is warranted depends on the application at hand.