Advanced icroeconomic (lecture 2: production theory II Ye Jianliang
Advanced Microeconomics (lecture 2: production theory II) Ye Jianliang
profit maximization o Content, Some definition Isoprofit curve Demand function and it's properties supply function and It's properties Profit function and it's properties lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College profit maximization • Content: – Some definition – Isoprofit curve – Demand function and it’s properties – supply function and it’s properties – Profit function and it’s properties
1. Some definition Basic assumption y is changeless, no technical improvement The firm is the price taker. o Profit function m(p)=max{py:y∈Y} When it is one production I(p=I(p, w)=max pf(x)-wX lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 1.Some definition • Basic assumption: – Y is changeless, no technical improvement. – The firm is the price taker. • Profit function: – When it is one production: ( ) max { : } p p = y y Y ( ) ( , ) max ( ) = = p p f x p w x - wx
2. Isoprofit curve o The profit p·f(x) 丌(p,W)=p·f(x)-W.X=元 slope=w/ We then got the q=f(x) isoprofit curve Then Vf(x)=wlp and D2f(x')is negative semi-definite lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 2. Isoprofit curve • The profit : • We then got the isoprpfit curve . • Then and is negative semi-definite. ( , ) ( ) p p f w x w x = − q x p f − ( ) x w x = / p slope / = w p q f = ( ) x x * f p ( ) = x w/ 2 D f ( ) x
2. Isoprofit curve Weak Axiom of Profit Maximization(WAPM) if ys yt are in Y, and choice by firm under price ps and pt. then pys>py.we can get:(p3-p2)(y-y)≥0Or△p△y≥0 lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 2.Isoprofit curve • Weak Axiom of Profit Maximization (WAPM) if y s , y t are in Y, and choice by firm under price p s and p t . then . we can get: or s s s t p y p y ) 0 s t s t (p -p )(y - y p y 0
2. Isoprofit curve q q 1 lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 2. Isoprofit curve y 1 y 2 x q y 1 y 2 x q
2. Isoprofit curve YI q YO q X lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 2. Isoprofit curve y 1 y 2 x q y 1 y 2 x YI q YO
3. Demand function factor demand function(set): x=x(p, w) X=XEV(q: wx=pg-I(p, w)) propositions: it's homogenous of degree zero lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 3.Demand function • factor demand function (set): • proposition5: it’s homogenous of degree zero. X { ( ) : ( , )} = = − x wx w V q pq p x = x w ( , ) p
3. Demand function One production: regular p as 1. let x(w) be the profit maximization choice(function means x is single point under w) of input factor vector under factor price w. then must hold n(=0→Vf(x)=w Ox( w) D f(x) is symmetric negative definite lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 3.Demand function • One production: regular p as 1. let x(,w) be the profit maximization choice (function means x is single point under w) of input factor vector under factor price w. then must hold: (.) 0 ( ) (, ) f = → = x w x w 2 2 (.) 0 ( ) is symmetric negative definite. D f → 2 x x
3. Demand function When Vf(x( w))=w, differentiate with respect to w we get D f(x( w). Dx(w)= or Dx( w)=Df(x( wI- Substitution matrix Dx(w) is a symmetric negative matrIx 1. Ox, /Ow=ax, /Ow -2.dwdk= dwDX( w)lw≤0 lecture 2 for Chu Kechen Honors College
lecture 2 for Chu Kechen Honors College 3.Demand function • When , differentiate with respect to w, we get: or • Substitution matrix is a symmetric negative matrix. – 1. – 2. = f ( (, )) x w w 2 D f D I ( ) x(,w) x(,w) = 2 1 D D f [ ( )]− x(,w) x(,w) = Dx(,w) 0 T d d d D d w x w x(,w) w = / / i j j i = x w x w
