1. Frank Cowell: Microeconomics Consumer: Aggregation MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Firm: Optimisation Consumption: Basics Almost essential Firm: Optimisation Consumption: Basics Prerequisites October 2006

2. Frank Cowell: Microeconomics Use of consumer models is often simplified... (1) We usually suppose that a many-dimensioned commodity space can be represented appropriately in terms of just a few commodities. (1) We usually suppose that a many-dimensioned commodity space can be represented appropriately in terms of just a few commodities.  Requires aggregation over goods (2) We often assume that there is a “representative consumer.” (2) We often assume that there is a “representative consumer.”  Requires aggregation over consumers We can use economic analysis to see whether and when these two simplifications are appropriate We can use economic analysis to see whether and when these two simplifications are appropriate

3. Frank Cowell: Microeconomics Aggregation over goods: the issue Why n goods? Why n goods? What determines the boundaries between goods? What determines the boundaries between goods? Diagrams all with 2 goods. Diagrams all with 2 goods.  Is this valid?  What assumptions are we making? Is it legitimate to simplify the n-commodity problem to, say, a 2-commodity problem? Is it legitimate to simplify the n-commodity problem to, say, a 2-commodity problem?

4. Frank Cowell: Microeconomics Aggregation over goods: the model Use the standard preference model with n goods. Use the standard preference model with n goods. Find an aggregate  x and a function  U(x 1,  x ) that yield the same behaviour as U(x 1, x 2,  x 3,..., x n ) Find an aggregate  x and a function  U(x 1,  x ) that yield the same behaviour as U(x 1, x 2,  x 3,..., x n ) Then we can say that  U(,) also exactly represents the consumer’s preferences. Then we can say that  U(,) also exactly represents the consumer’s preferences. The aggregation problem is then solved. The aggregation problem is then solved.

5. Frank Cowell: Microeconomics Aggregation over goods: result The “composite commodity” theorem: The “composite commodity” theorem: You can always aggregate over goods 2,3,..,n if relative prices of goods 2,3,..,n stay constant. You can always aggregate over goods 2,3,..,n if relative prices of goods 2,3,..,n stay constant.  U(,, …, ) and  U(,) then represent the same preferences Clearly this can be done for any arbitrary group of commodities. Clearly this can be done for any arbitrary group of commodities.  You just need the condition on relative prices

6. Frank Cowell: Microeconomics Aggregation over consumers We need to model the behaviour of n h consumers. We need to model the behaviour of n h consumers. Consumer h has utility function U h and income y h. Consumer h has utility function U h and income y h. From this get demand for good i in usual way, given prices p. From this get demand for good i in usual way, given prices p.  D hi (p, y h ). If all goods are “private” we can easily get total demand for i. If all goods are “private” we can easily get total demand for i.  Just add up over the D hi Let’s look at the simple mechanics. Let’s look at the simple mechanics. Alf and Bill

7. Frank Cowell: Microeconomics Aggregation of consumer demand Alf Bill The Market p1p1 x1x1 a x1x1 b x1x1 p1p1 p1p1   Alf’s demand curve for good 1.   Bill’s demand curve for good 1.   Pick any price   Sum of consumers’ demand   Repeat to get the market demand curve

8. Frank Cowell: Microeconomics Aggregation over consumers: the issues Demand for good i by each consumer h depends on prices p and income y h. Demand for good i by each consumer h depends on prices p and income y h. Aggregation problems could arise as with firms. Aggregation problems could arise as with firms. But main issue is: will the mass of consumers behave in the same way as a single consumer? But main issue is: will the mass of consumers behave in the same way as a single consumer? In general market demand will depend on the distribution of incomes y h. In general market demand will depend on the distribution of incomes y h. Can we write average demand as  D i (p,  y ), say? Can we write average demand as  D i (p,  y ), say?  For example  y could be average income in the market.  Just take the mean over the consumers We can do this only in special cases… We can do this only in special cases… Graphical illustration

9. Frank Cowell: Microeconomics Aggregable demand functions xihxih aih(p)aih(p) bi(p)bi(p) D hi (p,y h ) yhyh   h’s demand for good i.   Plot demand against income.   Must be linear in income.   “subsistence minimum” demand   Intercept could differ amongst the h.   Relation of demand to income   Slope must be the same for all h.

10. Frank Cowell: Microeconomics Aggregability Aggregable demands require restrictions on income effects. Aggregable demands require restrictions on income effects.  In our case average demand depends on average income.  Must have demand that is linear in income, with the same slope for all. Implies restrictions on preferences Implies restrictions on preferences But what could happen if this condition were not satisfied? But what could happen if this condition were not satisfied? Let's consider an example... Let's consider an example... Alf and Bill again

11. Frank Cowell: Microeconomics A consumer (Alf) x1x1 x2x2   Preferences look like this.   Prices then change thus.   Initial prices determine budget constraint.   Equilibrium moves from A to A' A' A' A A

12. Frank Cowell: Microeconomics B' B' B Another consumer (Bill) x1x1 x2x2   Different preferences, (and resources) but....   Prices then change the same way as for Alf   Faces the same initial prices.   Equilibrium moves from B to B'

13. Frank Cowell: Microeconomics Alf and Bill combined x1x1 x2x2   Initial equilibrium   In the aggregate WARP does not hold (!)   New equilibrium. A'&B' A&B   Each consumer’s behaviour is conventional.   Each individual satisfies WARP.   But the joint behaviour does not satisfy WARP.   This is because Alf and Bill are “too dissimilar” in their preferences.

14. Frank Cowell: Microeconomics What next? Integrate production and consumption decisions. Integrate production and consumption decisions. Examine behaviour in general equilibrium Examine behaviour in general equilibriumgeneral equilibriumgeneral equilibrium