Economic growth and economic development 244

Introduction to Modern Economic Growth Theorem 5.4 (Representative Firm Theorem) Consider a competitive production economy with N ∈ N∪ {+∞} commodities and a countable set F of firms, each with a convex production possibilities set Y f ⊂ RN Let p ∈ RN + be the price vector in this economy and denote the set of profit maximizing net supplies of firm f ∈ F by Yˆ f (p) ⊂ Y f (so that for any yˆf ∈ Yˆ f (p), we have p · yˆf ≥ p · y f for all y f ∈ Y f ) Then there exists a representative firm with production possibilities set Y ⊂ RN and set of profit maximizing net supplies Yˆ (p) such that ˆ ∈ Yˆ (p) if and only if yˆ (p) = for any p ∈ RN +, y X f ∈F yˆf for some yˆf ∈ Yˆ f (p) for each f ∈ F Proof Let Y be defined as follows: ( ) X Y = y f : y f ∈ Y f for each f ∈ F f ∈F P ˆ = f ∈F yˆf for To prove the “if” part of the theorem, fix p ∈ RN + and construct y / Yˆ (p), some yˆf ∈ Yˆ f (p) for each f ∈ F Suppose, to obtain a contradiction, that yˆ ∈ so that there exists y such that p · y > p · yˆ By definition of the set Y , this implies â ê that there exists y f f ∈F with y f ∈ Y f such that p· Ã X f ∈F X f ∈F y f ! > p· p · yf > Ã X X f ∈F f yˆ f ∈F ! p · yˆf , so that there exists at least one f ∈ F such that 0 p · y f > p · yˆf , which contradicts the hypothesis that yˆf ∈ Yˆ f (p) for each f ∈ F and completes this part of the proof To prove the “only if” part of the theorem, let yˆ ∈ Yˆ (p) be a profit maximizing choice for the representative firm Then, since Yˆ (p) ⊂ Y , we have that yˆ = X f ∈F 230 yf