Ch 21 Cost Minimization
What are the 2 steps to minimize cost
1. Choose x1 and x2 to minimize cost for a given output level y 2. Choose Output Level y to maximize profit
Cost Minimization for Imperfect Substitutes : Cobb-Douglas
Cost minimizing choice is at the tangency of isoquant and isocost line
Revealed Cost Minimization
If a firm is observed to produce the same quantity of output in each period over a number of periods, it should be using the cost-minimizing choice each period The isocost lines give the lowest bound of the isoquant
How an you find a x1 or x2?
Rearrange the isocost lines function to solve for x x2 = c/w2 - (w1/w2)*x1
What is the constraint in Cost Minimization
The constraint f(x1, x2) = y is illustrated by an Isoquant c = w1x1 + w2x2 which is illustrated by isocost lines
Cost Minimization for Perfect Compliments
The cost minimizing choices of x1 and x2 are at the kink of the isoquant. Let w1 = 2, w2 = 1, and y = 10 c = (2)(10) + (1)(10) = 30
Cost Minimizing for Perfect Substitutes
The cost-minimizing choices of x1 and x2 are in general at either end of the isoquant Let w1= 2, w2 = 1, and y = 30 f(x1, x2) = 4x1 + 3x2 You can produce 7.5 units of x1 (30/4) or 10 units of x2 (30/10) (7.5, 0) costs (2)(7.5) = $15 (0, 10) costs (1)(10) = $10
Exponent Rule
The exponent rule emerges from setting the slope of the isocost equal to the slope of the isoquant
WACM (Weak Axium Cost Minimization)
change in w1 and change in x1 + change in w2 change in x2 = 0 In other words the firm should choose the combos that are price max price minimizing
Violation of WACM
is Change (w1x1 + w2x2) less than or equal to 0 then it violates In the example, the total costs are the same, and thus it violates WACM
What is factor demand?
is a function of the price of both inputs (the factor prices) and the desired level of output
What is Cost Minimization?
min w1x1 + w2x2 s.t. f(x1, x2) = y Choose the lowest cost way of producing a desired level of output given the technology available to the firm
