《成本最小化》PPT课件

1、Chapter Twenty,Cost Minimization 成本最小化,Cost Minimization (成本最小化),Cost-minimization: How to choose the optimal input bundle that minimizes the cost of producing a given amount of output. Profit-maximization requires that the cost be minimized. How to understand it?,Cost Minimization,A firm is a cost-

2、minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) denotes 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) and planned

3、 product y, the total cost function will be written asc(w1,wn,y).,The Cost-Minimization Problem,Consider a firm using two inputs to make one output. The production function isy = 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 w1

4、x1 + w2x2.,The Cost-Minimization Problem,For given w1, w2 and y, the firms cost-minimization problem is to solve,subject 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 (smal

5、lest possible) total cost for producing y output units is therefore,Conditional Input Demands,Given w1, w2 and y, how is 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 a

6、mount is an iso-cost curve. E.g., given w1 and w2, the $100 iso-cost line has the equation,Iso-cost Lines （等成本线）,Generally, given w1 and w2, the equation of the $c iso-cost line isi.e. Slope is - w1/w2.,Iso-cost Lines （等成本线）,c w1x1+w2x2,c” w1x1+w2x2,c c”,x1,x2,Slopes = -w1/w2.,The y-Output Unit Isoq

7、uant,x1,x2,All input bundles yielding y unitsof output. Which is the cheapest?,f(x1,x2) y,The Cost-Minimization Problem,x1,x2,All input bundles yielding y unitsof output. Which is the cheapest?,f(x1,x2) y,The Cost-Minimization Problem,x1,x2,All input bundles yielding y unitsof output. Which is the c

8、heapest?,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) 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 t

9、he firms conditional input demand functions?,A Cobb-Douglas Example of Cost Minimization,At the input bundle (x1*,x2*) which minimizesthe 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 condition

10、aldemand for input 1.,A Cobb-Douglas Example of Cost Minimization,is the firms conditional demand for input 2.,Since,and,A Cobb-Douglas Example of Cost Minimization,For the production function the cheapest input bundle yielding y output units is,A Cobb-Douglas Example of Cost Minimization,So the fir

11、ms 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 the firms total cost function?,A Perfect Complements Example of Cost Minimization,x1,x2,x

12、1* = 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 input demands are,and,So the firms total cost function is,Average Total Production Costs (平均总生

13、产成本),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 change with output level. Our firm is presently producing y output units. How does t

14、he 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.

15、 Average production cost does not change.,Decreasing Returns-to-Scale and Average Total Costs,If a firms technology exhibits decreasing returns-to-scale then doubling its output level from y to 2y requires more than doubling all input levels. Total production cost more than doubles. Average producti

16、on cost increases.,Increasing Returns-to-Scale and Average Total Costs,If a firms technology exhibits increasing returns-to-scale then doubling its output level from y to 2y requires less than doubling all input levels. Total production cost less than doubles. Average production cost decreases.,Retu

17、rns-to-Scale and Av. Total Costs,y,$/output unit,constant r.t.s.,decreasing r.t.s.,increasing r.t.s.,AC(y),Returns-to-Scale and Total Costs,What does this imply for the shapes of total cost functions?,Returns-to-Scale and Total Costs,y,$,c(y),y,2y,c(y),c(2y),Slope = c(2y)/2y = AC(2y).,Slope = c(y)/y

18、 = AC(y).,Av. cost increases with y if the firmstechnology exhibits decreasing r.t.s.,Returns-to-Scale and Total Costs,y,$,c(y),y,2y,c(y),c(2y),Slope = c(2y)/2y = AC(2y).,Slope = c(y)/y = AC(y).,Av. cost decreases with y if the firmstechnology exhibits increasing r.t.s.,Returns-to-Scale and Total Co

19、sts,y,$,c(y),y,2y,c(y),c(2y) =2c(y),Slope = c(2y)/2y = 2c(y)/2y = c(y)/y so AC(y) = AC(2y).,Av. cost is constant when the firmstechnology exhibits constant r.t.s.,Short-Run & Long-Run Total Costs,In the long-run a firm can vary all of its input levels. Consider a firm that cannot change its input 2

20、level from x2 units. How does the short-run total cost of producing y output units compare to the long-run total cost of producing y units of output?,Short-Run & Long-Run Total Costs,The long-run cost-minimization problem is The short-run cost-minimization problem is,subject to,subject to,Short-Run

21、& Long-Run Total Costs,x1,x2,Consider three output levels.,Short-Run & Long-Run Total Costs,x1,x2,In the long-run when the firmis free to choose both x1 andx2, the least-costly inputbundles are .,Short-Run & Long-Run Total Costs,x1,x2,Long-runoutputexpansionpath,Long-run costs are:,长期产量扩展线,Short-Run

22、 & Long-Run Total Costs,Now suppose the firm becomes subject to the short-run constraint that x1 = x1”.,Short-Run & Long-Run Total Costs,x1,x2,Short-runoutputexpansionpath,Long-run costs are:,Short-Run & Long-Run Total Costs,x1,x2,Short-runoutputexpansionpath,Long-run costs are:,Short-run costs are:

23、,Short-Run & Long-Run Total Costs,x1,x2,Short-runoutputexpansionpath,Long-run costs are:,Short-run costs are:,Short-Run & Long-Run Total Costs,x1,x2,Short-runoutputexpansionpath,Long-run costs are:,Short-run costs are:,one point in common,Short-Run & Long-Run Total Costs,This says that the long-run

24、total cost curve always has one point in common with any particular short-run total cost curve. Short-run total cost exceeds long-run total cost except for the output level where the short-run input level restriction is the long-run input level choice.,Short-Run & Long-Run Total Costs,y,$,c(y),cs(y)

25、,A short-run total cost curve always hasone point in common with the long-runtotal cost curve, and is elsewhere higherthan the long-run total cost curve.,Fixed, variable and sunk costs,Fixed costs （不变成本）are associated with fixed factors. They are independent of the level of output and must be paid e

26、ven if the firm produces zero output. Variable costs（可变成本）only need to be paid if the firm produce a positive amount of output.,Sunk costs (沉没成本),Sunk cost: an expenditure that has been made and cannot be recovered. Zero opportunity cost. After sunk cost has been occurred, it should not affect a fir

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