Chapter 20经济学 资料

The relation between the Long-run and Short-run costs




Suppose the long-run factor demands are X1=X1(W1,W2,Y)=X1(Y); X2=X2(W1,W2,Y)=X2(Y) Then the long-run cost function can be written as C（Y）= Cs（Y,X2(Y)）=Cs（Y,X2） The equation says that the minimum costs when all factors are variable is just the minimum cost when factor 2 is fixed at the level that minimize long-run cost. So, X1(W1,W2,Y)= X1s(W1,W2,X2(Y),Y) The cost minimizing amount of the variable factor in the long run is that amount that the firm would choose in the short run 在长期内，使成本最小化的可变要素的使用量就是厂商在短 期内所选择的使用量。
AC(Y)= C(W1,W2,Y)/Y =C(W1,W2,1)Y/Y= C(W1,W2,1) The average cost will be constant no matter what
level of output the firm wants to produce.


(2) In the case of increasing returns to scale, the minimum cost would be less than c(w1,w2,y) ≤c(w1,w2,1)y, then the costs increase less than linearly in output. The average cost will diclining in output. (3) In the case of decreasing returns to scale, the minimum cost would be less than c(w1,w2,y) ≥c(w1,w2,1)y, then the costs increase more than linearly in output. The average cost will rise as output increases
Model: min X1+X2 S.T (X11/2+3X21/2)2=16 … ⑴ (1)Substitution method：solving the constraint for X2 as a function of X1,then substituting the function into the objective function to get the unconstrainted minimization problem. (2)Lagrangian method：
Chapter 22, 24, 27
Firm Supply, monopoly, and oligopoly
In these chapters, we will look at the problem of how to choose the most profitable level of output.
x2
Optimal choice
Isocost lines slope= – w 1 / w 2 Isoquant f (x1 , x2 ) = y
x2*
.
x 1*
x1
Isocost lines: p351



It give us all the combinations of inputs that have some given level of cost, C. we can write this as w1 x1 + w2 x2 =C Which can be rearranged to give x2= C/w2 – w1x1/w2 Every point at this line has the same cost, and higher isocost lines are associated with higher costs.
Key points


1. Market environments 2. The supply decision of a competitive firm 3. The supply decision of a monopoly 4. the strategic interations between two firms in a situation of duopoly.
AC
LAC
increasing returns decreasing returns
.
y
3. Long-run and Short-run costs



(1) The short-run cost function is defined by Cs（Y,X2）=min X1W1+X2W2 x1 Such that f(X1,X2)=Y s Which gives X1=X1 (W1,W2,X2,Y); X2=X2 (2) The long-run cost function is defined by C（Y）=min X1W1+X2W2 x1,x2 Such that f(X1,X2)=Y Which gives X1=X1 (W1,W2, Y); X2=X2 (W1,W2, Y);
Minimizing costs for specific thechnologies
Minimizing costs for y = min{ax1 , bx2};完全互补 y = ax1 + bx2; 完全替代 and y = x1a x2b. Cobb-Douglas Refer to P352-353
1. Market environments


The firm faces two sorts of constraints on its actions. First, it faces the technological constraints summarized by the production function. Second, it faces the market constraint. A firm can only sell as much as people are willing to buy. So we should investigate the demand curve facing the firm（厂商面临的需求曲线）, which describes the relationship between the price a firm sets and the amount that it sells.
A tangency condition


If the isoquant is a nice smooth curve, and the optimal solution involves using some of each factor, then the cost-minimizing point will be characterized by a tangency condition: the slope of the isoquant must be equal to the slope of the isocost curve. That is the technical rate of subsitution must equal to the factor price ratio: – MP1 (x1 , x2 ) / MP2 (x1 , x2 ) = TRS(x1 , x2 ) = – w1 / w2
For example

Suppose the prodcution function of the firm is f(X1,X2)=(X11/2+3X21/2)2,and the prices of factor 1 and factor 2 are 2，what is the minimum cost of producing 16 units of output?
Chapter 20
Cost Minimization
the profit–maxmization problem



We will break up the profit– maxmization problem into two pieces. (1)how to minnimize the costs of producing any given level of output. (chapter 20) (2)how to choose the most profitable level of output. (chapter 22)
2. Retuns to scales and the cost function


Suppose the minimum cost of producing 1 unit of output is c(w1,w2,1). Now what is the cheapest way to produce y units of output? (1)In the case of constant returns to scale, the minimum cost would be c(w1,w2,y)=c(w1,w2,1)y, which means the cost function is linear in output.
Key points


1. Cost minimization 2. Retuns to scales and the cost function 3. Long-run and Short-run costs
1. Cost minimization



Assumptions: Two factors: x1 ,x2 , with the prices w1 ,w2 . Prodcution function: f (x1 , x2 ) Basic model: min x1, x2 w1 x1 + w2 x2 subject to f (x1 , x2 ) = y Which gives: x1 ( w1 , w2 , y ) x 2 ( w 1 , w2 , y ) —— conditional factor demand function or derived factor demands and c ( w1 , w2 , y ) —— cost function

1 1/ 2 1/ 2 3x2 ) x1 2 3 1/ 2 1/ 2 1/ 2 1 2( x1 3x2 ) x2 2 1/ 2 1/ 2 ( x1 3x2 ) 16 1 2( x1
1/ 2
X2=9X1=144/100，C=160/100
Cost minimization on diagram
Байду номын сангаас Attention!


If we have a boundary solution where one of the two factors isn’t used, this tangency condition need not be met. If the production function has ―kinks‖,the tangency condition has no meaning.