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19_CostFunctions_ToPost-1

# 19_CostFunctions_ToPost-1 - , run HenryDavidThoreau ECON...

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“The cost of a thing is the amount of what I will call life which  is required to be exchanged for it, immediately or in the long  run.” –Henry David Thoreau ECON 410 Cost Functions

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2 Class 19 - Cost Functions
3 Class 19 - Cost Functions

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4 Mathematical Analogy 1 2 ( , ) f x x Of the bundles that can produce q, which is cost minimizing? 1 1 2 2 1 2 ( , ) ( ) w x w x q f x x λ = + + - L 2 1 1 2 f f w w = Class 19 - Cost Functions
5 Class 19 - Cost Functions

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6 Class 19 - Cost Functions Individual Demand Curve An Individual Demand Curve is the graphical relationship between a Good’s price and the amount of the Good an individual will select when optimizing over her changing budget set.
7 Class 19 - Cost Functions Factor Demand Function A Factor Demand Function specifies the relationship between the prices of input goods, the quantity of output produced, and the amount of an input good a firm will select when minimizing its costs.

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8 Class 19 - Cost Functions Steps to Analytically Solve for a Firm’s Factor Demand Functions 1. Create the Lagrange equation with the specified production function, but leave q, w1, and w2 as parameters. 2. Take the FOC’s and solve for x1 and x2. Your results will be as a function of q, w1, and w2 (or some combination of the three). x1 and x2 will be your Factor Demand Functions.
9 Class 19 - Cost Functions Assume a firm has a Cobb- Douglas production function of the form f(x1, x2)=x11/2x21/2. Find the firm’s factor demand functions 1. Create the Lagrange equation with the specified production function, but leave q, w1, and w2 as parameters. 1/ 2 1/ 2 1 2 1 1 2 2 [ ] w x w q x x x λ + + - L 1/ 2 1/ 2 1 2 2 1 2 1 [ ] x w q x x x w λ + + - L Cost Functio n Output Constraint

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10 Class 19 - Cost Functions 2. Take the FOC’s and solve for x1 and x2. Your results will be as a function of q, w1, and w2 (or some combination of the three). x1 and x2 will be your Factor Demand Functions. 1 0 x = L 1/ 2 1/ 2 1 1 2 2 0 1 x w x λ - - = 1/ 2 1/ 2 2 1 2 0 1 2 x x w λ - - = 2 0 x = L 0 λ = L 1/ 2 1/ 2 1 2 0 q x x - = 1/ 2 1/ 2 1 2 1 2 1 2 [ ] x w w q x x x λ + + - L Assume a firm has a Cobb- Douglas production function of the form f(x1, x2)=x11/2x21/2. Find the firm’s factor demand functions
11 Class 19 - Cost Functions 1/ 2 1/ 2 1 1 2 2 0 1 x w x λ - - = 1/ 2 1/ 2 2 1 2 0 1 2 x x w λ - - = 1/ 2 1/ 2 1 2 0 q x x - = 1 1/ 2 1/ 2 1 2 2 x w x λ - = 2 1/ 2 1/ 2 1 2 2 w x x λ - = 2. Take the FOC’s and solve for x1 and x2. Your results will be as a function of q, w1, and w2 (or some combination of the three). x1 and x2 will be your Factor Demand Functions. Assume a firm has a Cobb- Douglas production function of the form f(x1, x2)=x11/2x21/2. Find the firm’s factor demand functions

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12 Class 19 - Cost Functions 2 1 1/ 2 1/ 2 1/ 2 1/ 2 1 2 1 2 2 2 w x x x x w - - = 1/ 2 1/ 2 1 1 2 2 0 1 x w x λ - - = 1/ 2 1/ 2 2 1 2 0 1 2 x x w λ - - = 1/ 2 1/ 2 1 2 0 q x x - = 1 1/ 2 1/ 2 1 2 2 x w x λ - = 2 1/ 2 1/ 2 1 2 2 w x x λ - = 2.
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