Cost_minimization2

# Cost_minimization2 - Econ 301 Microeconomic Theory 2...

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Unformatted text preview: Econ 301 - Microeconomic Theory 2 Lecture Note Chapter 20 – Cost Minimization Department of Economics University of Waterloo Fall 2011 Preview • In this chapter, we will break up the profit maximization problem into two pieces. • First, we will look at the problem of how to minimize the costs of producing any given level of output, and then we will look at how to choose the most profitable level of output. Economic Profits • Total costs for the firm are given by total costs = C = w l + vk • Total revenue for the firm is given by total revenue = pq = pf ( k , l ) • Economic profits () are equal to = total revenue - total cost = pq- w l- vk = pf ( k , l ) - w l- vk Economic Profits • Economic profits are a function of the amount of k and l employed – we could examine how a firm would choose k and l to maximize profit • “derived demand” theory of labor and capital inputs – for now, we will assume that the firm has already chosen its output level ( q 0) and wants to minimize its costs. Cost-Minimizing Input Choices • Minimum cost occurs where the RTS is equal to w / v – the rate at which k can be traded for l in the production process = the rate at which they can be traded in the marketplace Cost-Minimizing Input Choices • We seek to minimize total costs given q = f ( k , l ) = q • Setting up the Lagrangian: ℒ = w l + vk + [ q 0 - f ( k , l )] • FOCs are ℒ / l = w- ( f/ l ) = 0 ℒ / k = v- ( f/ k ) = 0 ℒ / = q 0 - f ( k , l ) = 0 Cost-Minimizing Input Choices • Dividing the first two conditions we get ) for ( / / k RTS k f f v w l l = ∂ ∂ ∂ ∂ = • The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices. Cost-Minimizing Input Choices • Cross-multiplying, we get w f v f k l = • For costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs. Cost-Minimizing Input Choices • The inverse of this equation is also of interest λ = = k f v f w l • The Lagrangian multiplier shows how the extra costs that would be incurred by increasing the output constraint slightly. q0 Given output q0, we wish to find the least costly point on the isoquant C1 C2 C3 Costs are represented by parallel lines with a slope of -w/v Cost-Minimizing Input Choices l per period k per period C1 < C2 < C3 C1 C2 C3 q0 The minimum cost of producing q0 is C2 Cost-Minimizing Input Choices l per period k per period k* l* The optimal choice is l*, k* This occurs at the tangency between the isoquant and the total cost curve Contingent Demand for Inputs • In Chapter 5, we considered an individual’s expenditure-minimization problem – to develop the compensated demand for a good....
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Cost_minimization2 - Econ 301 Microeconomic Theory 2...

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