The Space of Money Economies

We are now able to define a space of money economies at period t. First, for each u h   ∈ u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math176.tif"/> and γ h   ∈   Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math177.tif"/> we have an expected utility function v^h defined by (2).Hence the function spaces Γ and u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math178.tif"/> are important ingredients in the definition of the space of money economies. Moreover, money and commodity endowments in period t are also allowed to vary in the space P × R₊ in addition to varying the γ^h’s and u^h’s for every agent h. In particular, we do not restrict our analysis to a fixed amount of money supply in the model. Therefore, at time t, all economic characteristics of the model are completely specified by the product space ( Γ × u × P × R + ) n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math179.tif"/> . Denote the space of money economies at time t by ℰ   =   ( Γ × u × P × R + ) n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math180.tif"/> , and a money economy E   =   ( γ , u , x ¯ , m ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math181.tif"/> is an element of ℰ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math182.tif"/> where γ = (γ¹,…,γⁿ), u = (u^l,…,uⁿ), x ¯   =   ( x ¯ 1 , ... , x ¯ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math183.tif"/> and m ¯   =   ( m ¯ 1 , ... , m ¯ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math184.tif"/> . In other words, E   ∈   ℰ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math185.tif"/> is a list of expectations, direct utility functions, commodity and money endowments at time t for all agents in the model. In particular, ℰ 0   =   ( Γ × u 0 × P × R + ) n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math186.tif"/> is a space of classical money economies. Clearly, ℰ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math187.tif"/> and ℰ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math188.tif"/> are infinite-dimensional spaces, since for each agent h the 31utility function u^h and expectation function γ^h are allowed to vary in u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180588/ce7d9624-5292-47bf-b72f-b96b28b34859/content/inline-math189.tif"/> and Γ, respectively. Hence, in addition to allowing commodity and money endowments to vary, we also allow for changes of tastes and beliefs. This extends Debreu’s [9] case (in the general Walrasian equilibrium model) where the space of economies is naturally a finite dimensional Euclidean space, since only initial endowments are allowed to vary. ⁸