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1、Chapter Twenty,Cost Minimization 成本最小化,Structure,The cost minimization problem Average costs Returns to scale and total and average costs Short run and long run costs,Cost Minimization,A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) denotes
2、the firms smallest possible total cost for producing y units of output. c(y) is the firms total cost function （总成本函数）.,Cost Minimization,When the firm faces given input prices w = (w1,w2,wn) the total cost function will be written as c(w1,wn,y).,The Cost-Minimization Problem,Consider a firm using tw
3、o inputs to make one output. The production function is y = f(x1,x2). Take the output level y 0 as given. Given the input prices w1 and w2, the cost of an input bundle (x1,x2) is w1x1 + w2x2.,The Cost-Minimization Problem,For given w1, w2 and y, the firms cost-minimization problem is to solve,subjec
4、t to,The Cost-Minimization Problem,The levels x1*(w1,w2,y) and x1*(w1,w2,y) in the least-costly input bundle are the firms conditional demands for inputs 1 and 2 （条件要素需求）. The (smallest possible) total cost for producing y output units is therefore,Conditional Input Demands,Given w1, w2 and y, how i
5、s the least costly input bundle located? And how is the total cost function （成本函数）computed?,Iso-cost Lines （等成本线）,A curve that contains all of the input bundles that cost the same amount is an iso-cost curve. E.g., given w1 and w2, the $100 iso-cost line has the equation,Iso-cost Lines,Generally, gi
6、ven w1 and w2, the equation of the $c iso-cost line is i.e. Slope is - w1/w2.,Iso-cost Lines,c w1x1+w2x2,c” w1x1+w2x2,c c”,x1,x2,Slopes = -w1/w2.,The y-Output Unit Isoquant,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,The Cost-Minimization Problem,x1,x2,All i
7、nput bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,The Cost-Minimization Problem,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,x1*,x2*,The Cost-Minimization Problem,x1,x2,f(x1,x2) y,x1*,x2*,At an interior cost-min input bundle: (a),The
8、Cost-Minimization Problem,x1,x2,f(x1,x2) y,x1*,x2*,At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant; i.e.,A Cobb-Douglas Example of Cost Minimization,A firms Cobb-Douglas production function is Input prices are w1 and w2. What are the firms conditional input dem
9、and functions?,A Cobb-Douglas Example of Cost Minimization,At the input bundle (x1*,x2*) which minimizes the cost of producing y output units: (a) (b),and,A Cobb-Douglas Example of Cost Minimization,(a),(b),From (b),Now substitute into (a) to get,So,is the firms conditional demand for input 1.,A Cob
10、b-Douglas Example of Cost Minimization,is the firms conditional demand for input 2.,Since,and,A Cobb-Douglas Example of Cost Minimization,So the cheapest input bundle yielding y output units is,Fixed w1 and w2.,Conditional Input Demand Curves,Fixed w1 and w2.,Conditional Input Demand Curves,Fixed w1
11、 and w2.,Conditional Input Demand Curves,Fixed w1 and w2.,Conditional Input Demand Curves,Fixed w1 and w2.,Conditional Input Demand Curves,output expansion path,Cond. demand for input 2,Cond. demand for input 1,A Cobb-Douglas Example of Cost Minimization,For the production function the cheapest inpu
12、t bundle yielding y output units is,A Cobb-Douglas Example of Cost Minimization,So the firms total cost function is,A Perfect Complements Example of Cost Minimization,The firms production function is Input prices w1 and w2 are given. What are the firms conditional demands for inputs 1 and 2? What is
13、 the firms total cost function?,A Perfect Complements Example of Cost Minimization,x1,x2,x1* = y/4,x2* = y,4x1 = x2,min4x1,x2 y,Where is the least costly input bundle yielding y output units?,A Perfect Complements Example of Cost Minimization,The firms production function is,and the conditional inpu
14、t demands are,and,So the firms total cost function is,Average Total Production Costs,For positive output levels y, a firms average total cost of producing y units is,Returns-to-Scale and Av. Total Costs,The returns-to-scale properties of a firms technology determine how average production costs chan
15、ge with output level. Our firm is presently producing y output units. How does the firms average production cost change if it instead produces 2y units of output?,Constant Returns-to-Scale and Average Total Costs,If a firms technology exhibits constant returns-to-scale then doubling its output level from y to 2y requires doubling all input levels. Total production cost doubles. Average production cost does not change.,Decreasing Returns-to-Scale and Average Total Costs,If a firms technology exhibits decreasing re
