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Unformatted text preview: Theory of the Firm Moshe BenAkiva 1.201 / 11.545 / ESD.210 Transportation Systems Analysis: Demand & Economics Fall 2008 Outline ● Basic Concepts ● Production functions ● Profit maximization and cost minimization ● Average and marginal costs 2 Basic Concepts ● Describe behavior of a firm ● Objective: maximize profit − ( ) max π = R a ( ) C a s t . . a ≥ 0 – R, C, a – revenue, cost, and activities, respectively ● Decisions: amount & price of inputs to buy amount & price of outputs to produce ● Constraints: technology constraints market constraints 3 Production Function ● Technology: method for turning inputs (including raw materials, labor, capital, such as vehicles, drivers, terminals) into outputs (such as trips) ● Production function: description of the technology of the firm. Maximum output produced from given inputs. q = q ( X ) – q – vector of outputs – X – vector of inputs (capital, labor, raw material) 4 Using a Production Function ● The production function predicts what resources are needed to provide different levels of output ● Given prices of the inputs, we can find the most efficient (i.e. minimum cost) way to produce a given level of output 5 Isoquant ● For twoinput production: Labor (L) Capital (K) q 1 K’ L’ q=q(K,L) q 2 q 3 6 Production Function: Examples ● CobbDouglas : ● InputOutput: q =α x 1 a x 2 b q = min( ax 1 , bx 2 ) x 2 q 1 x 1 Isoquants q 2 q 2 q 1 x 1 x 2 7 Rate of Technical Substitution (RTS) ● Substitution rates for inputs – Replace a unit of input 1 with RTS units of input 2 keeping the same level of production x 2 ∂ x 2 ∂ q ∂ x 1 RTS = = − ∂ x ∂ q ∂ x 2 1 q ' x 2 q slope=RTS ' x 1 x 1 8 RTS: An Example ● CobbDouglas Technology q =α x 1 a x 2 b ∂ q =α ax 1 a − 1 x 2 b ∂ x 1 ∂ ∂ x q 2 =α bx 1 a x 2 b − 1 ∂ x RTS = 2 = − a x 2 ∂ x b x 1 1 q 9 Elasticity of Substitution ● The elasticity of substitution measures the percentage change in factor proportion due to 1 % change in marginal rate of technical substitution ∂ ( x 2 / x 1 ) RTS s = ∂ [ RTS ] ( x 2 / x 1 ) ∂ ln( x 2 / x 1 ) s = ∂ ln( RTS ) ● For CobbDouglas: ∂ ( x 2 / x 1 ) = ∂ [ RTS ] 1 s = ( − b / a ) ( − a / b ) ( x 2 / x 1 ) = 1 ∂ [ RTS ] 1 ∂ ( x 2 / x 1 ) − a b ( x 2 / x 1 ) = = − b a 10 Other Production Functions...
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 Fall '08
 MosheBenAkiva
 Economics, transportation systems, marginal costs

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