2020/6/21,中级微观经济学,Chapter Twenty,Cost Minimization,2020/6/21,中级微观经济学,Cost Minimization,A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) will denote the firms smallest possible total cost for producing y units of output. c(y) is the firms total cost function.,2020/6/21,中级微观经济学,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).,2020/6/21,中级微观经济学,The Cost-Minimization Problem,Consider a firm using two inputs to make one output. The firms production function is y = f(x1,x2). Take the output level y 0 as given. Take the input prices w1 and w2 as given. Then the cost of an input bundle (x1,x2) is w1x1 + w2x2.,2020/6/21,中级微观经济学,The Cost-Minimization Problem,For given w1, w2 and y, the firms cost-minimization problem is to solve,subject to,2020/6/21,中级微观经济学,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. The firms total (smallest possible) cost for producing y output units is therefore,2020/6/21,中级微观经济学,Conditional Input Demands,For given w1, w2 and y, how are the levels of inputs 1 and 2 in the least costly input bundle located? And how is the firms total cost function computed?,2020/6/21,中级微观经济学,Iso-cost Lines,A curve which contains all of the input bundles which cost the same amount is an iso-cost curve. E.g., given w1 and w2, the $100 iso-cost line has the equation,2020/6/21,中级微观经济学,Iso-cost Lines,In general, given w1 and w2, the equation of the $c iso-cost line is which rearranges to The slope is - w1/w2.,2020/6/21,中级微观经济学,Iso-cost Lines,c w1x1+w2x2,c” w1x1+w2x2,c c”,x1,x2,2020/6/21,中级微观经济学,Iso-cost Lines,c w1x1+w2x2,c” w1x1+w2x2,c c”,x1,x2,Slopes = -w1/w2.,2020/6/21,中级微观经济学,The y-Output Unit Isoquant,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,2020/6/21,中级微观经济学,The Cost-Minimization Problem,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,2020/6/21,中级微观经济学,The Cost-Minimization Problem,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,2020/6/21,中级微观经济学,The Cost-Minimization Problem,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,2020/6/21,中级微观经济学,The Cost-Minimization Problem,x1,x2,All input bundles yielding y units of output. Which is the cheapest?,f(x1,x2) y,x1*,x2*,2020/6/21,中级微观经济学,The Cost-Minimization Problem,x1,x2,f(x1,x2) y,x1*,x2*,At an interior cost-min input bundle: (a),2020/6/21,中级微观经济学,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,2020/6/21,中级微观经济学,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.,2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,A firm has a Cobb-Douglas technology, its production function being The firm faces given input prices w1 and w2. What are the firms conditional input demand functions?,2020/6/21,中级微观经济学,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,2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,(a),(b),2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,(a),(b),From (b),2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,(a),(b),From (b),Now substitute into (a) to get,2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,(a),(b),From (b),Now substitute into (a) to get,2020/6/21,中级微观经济学,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.,2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,is the firms conditional demand for input 2.,Since,and,2020/6/21,中级微观经济学,A Cobb-Douglas Example of Cost Minimization,So the answer to the question “Which input bundle yielding y output units is cheapest when input prices are w1 and w2?” is,2020/6/21,中级微观经济学,Fixed w1 and w2.,Conditional Input Demand Curves,2020/6/21,中级微观经济学,Fixed w1 and w2.,Conditional Input Demand Curves,2020/6/21,中级微观经济学,Fixed w1 and w2.,Conditional Input Demand Curves,2020/6/21,中级微观经济学,Fixed w1 and w2.,Conditional Input Demand Curves,2020/6/21,中级微观经济学,Fixed w1 and w2.,Conditional Input Demand Curves,output expansion path,2020/6/21,中级微观经济学,Fixed w1 and w2.,Conditional Input Demand Curves,output expansion path,Cond. demand for input 2,Cond. demand for input 1,2020/6/21,中级微观经