Micro Theory, 2005 Chapter 1 Neoclassical Economics 1. Producer Theory 1. Technology yi =input of good i, y =output of good i, i= yi-yi=net output, y yn) is a production plan Production possibility set Y=technologically feasible production plans yE Rn) y E Y is technologically efficient if there is no yE Y s.t. y>y Production frontier=(technological efficient production plans) y E Y is economically efficient if it maximizes profit Proposition 1.1. Economic efficiency implies technological efficiency. Consider a single output y E R+. Denote E Rn as the firms inputs and define the production function f: Rn -R+ as f(x)≡,max.y ,-x)∈Y Proposition1.2.Pory∈R+,(3,-x) is technologically efficient→y=f(x)■
Chapter 1 Neoclassical Economics Micro Theory, 2005 1. Producer Theory 1.1. Technology y− i = input of good i, y+ i = output of good i, yi ≡ y+ i − y− i = net output, y = (y1, y2,...,yn) is a production plan. Production possibility set: Y = technologically feasible production plans y ∈ Rn . y ∈ Y is technologically efficient if there is no y0 ∈ Y s.t. y0 > y. Production frontier = technological efficient production plans . y ∈ Y is economically efficient if it maximizes profit. Proposition 1.1. Economic efficiency implies technological efficiency. Consider a single output y ∈ R+. Denote x ∈ Rn + as the firm’s inputs and define the production function f : Rn + → R+ as f(x) ≡ max (y,−x)∈Y y. Proposition 1.2. For y ∈ R+, (y, −x) is technologically efficient =⇒ y = f(x) 1—1
Isoquant Q()={x∈R+|y=f(x)} Marginal rate of transformation MRT()= /zi(a) f2(x) MRT() is the slope of the isoquant Example11.Cobb- Douglas Technology.For0≤a≤1, consider y≡ {(v,-1,-2)∈R+×R2|y≤a}.■ For production function f: R+-R+, it exhibits global constant returns to scale(CRS) if f(t r)=tf( global increasing returns to scale(Irs) if f(tr)>tf() lobal decreasing returns to scale(drs if f(t <tf(a) x∈R,t>1 Example 1.2. Consider f(a1, T2)=Azqr2 Elasticity of scale at a e(x)≡ dlog f(ta) e(a)=percentage increase in output for 1% increase in scale At a, we say that f exhibits local constant returns to scale(CRS) if e(r )=1 local increasing returns to scale(IRs) if e(a)>1 local decreasing returns to scale(DRS) if e()<1
Isoquant: Q(y) ≡ x ∈ Rn + | y = f(x) . Marginal rate of transformation: MRT(x) ≡ fx1 (x) fx2 (x) . MRT(x) is the slope of the isoquant. Example 1.1. Cobb-Douglas Technology. For 0 ≤ α ≤ 1, consider Y ≡ (y, −x1, −x2) ∈ R+ × R2 − | y ≤ xα 1x1−α 2 . For production function f : Rn + → R+, it exhibits global constant returns to scale (CRS) if f(tx) = tf(x); global increasing returns to scale (IRS) if f(tx) > tf(x); global decreasing returns to scale (DRS) if f(tx) < tf(x), ∀ x ∈ Rn +, t> 1. Example 1.2. Consider f(x1, x2) = Axa 1xb 2. Elasticity of scale at x : e(x) ≡ d log f(tx) d log t t=1 . e(x) = percentage increase in output for 1% increase in scale. At x, we say that f exhibits local constant returns to scale (CRS) if e(x) = 1; local increasing returns to scale (IRS) if e(x) > 1; local decreasing returns to scale (DRS) if e(x) < 1. 1—2
Proposition 1.3.(Returns to Scale) 1.Forx∈Rn, we have global IRS local IRS or CRS, Va global CRS local CRS,va global DRS => local DRS or CRS, Vr 2.Forx∈R+, we have e(a)=xf(a) f(ar) implying local IRS→f"(x) local crs→f(x)=e2 local Drs→f()< 3. For a E Rn and y= f(a), we have Ac (y imply local Irs←→AC>MC, local crs←AC=MC local DRS→→AC<MC Elasticity of substitution alos alog y o is the percentage change in = for 1% increase in w1
Proposition 1.3. (Returns to Scale). 1. For x ∈ Rn, we have global IRS =⇒ local IRS or CRS, ∀ x global CRS =⇒ local CRS, ∀ x global DRS =⇒ local DRS or CRS, ∀ x 2. For x ∈ R+, we have e(x) = x · f0 (x) f(x) , implying local IRS ⇐⇒ f0 (x) > f(x) x local CRS ⇐⇒ f0 (x) = f(x) x local DRS ⇐⇒ f0 (x) < f(x) x . 3. For x ∈ Rn and y = f(x), we have e(x) = AC(y) MC(y) , implying local IRS ⇐⇒ AC > MC, local CRS ⇐⇒ AC = MC, local DRS ⇐⇒ AC < MC. Elasticity of substitution: σ ≡ − ∂ log x1(w,y) x2(w,y) ∂ log w1 w2 . σ is the percentage change in x∗ 1 x∗ 2 for 1% increase in w1 w2 . 1—3
1.2. The firm's Problem The firm maximizes its profit or expected profit rofit= total revenue- total cost The cost is the economic cost or opportunity cost. The revenue is the money received from sales For n actions aERn, the firms problem is R(a-C(a) FOC aR(a*) aC(a2 or MR=MC, V i da Assume competitive firms(price takers)and a single output. Profit function 丌(D,)≡ max pf(x)-t·x Demand function: =c(p, w). Supply function: y(p, w)= fa(p, w. We have FOC Df(a D2f(r") a)2f(x”) <0 ac: a Cost function c(u,9)≡min{·x|y≤f(x)} Conditional demand function: a'=z(w, y). Lagrange function is C(a, A)=w.x+ №y-f(x).Then Df(a") fr , The SOC for(2.3) hD2f(x”)h≤0, for all h satisfying Df(x)·h=0 An equivalent problem of (2. 2)is py-c, y
1.2. The Firm’s Problem The firm maximizes its profit or expected profit. profit = total revenue − total cost. The cost is the economic cost or opportunity cost. The revenue is the money received from sales. For n actions a ∈ Rn, the firm’s problem is π ≡ maxa R(a) − C(a). FOC: ∂R(a∗) ∂ai = ∂C(a∗) ∂ai or MR = MC, ∀ i. (2.1) Assume competitive firms (price takers) and a single output. Profit function is π(p, w) ≡ maxx pf(x) − w · x. (2.2) Demand function: x∗ = x(p, w). Supply function: y(p, w) ≡ f[x(p, w)]. We have FOC : p Df(x∗ ) = w, SOC : D2 f(x∗ ) ≡ ∂2f(x∗) ∂xi∂xj ≤ 0. Cost function: c(w, y) ≡ minx {w · x | y ≤ f(x)}. (2.3) Conditional demand function: x∗ = x(w, y). Lagrange function is L(x, λ) = w · x + λ[y − f(x)]. Then, FOC: w = λ Df(x∗ ) or wi wj = fxi (x∗) fxj (x∗) , ∀ i, j. (2.4) The SOC for (2.3) is h0 D2 xf(x∗ )h ≤ 0, for all h satisfying Df(x∗ ) · h = 0. An equivalent problem of (2.2) is max y py − c(w, y). (2.5) 1—4
Then FOC. n dc(w, y") Example 1.3. Consider c(w, y)= min w1. 1+W2.C2 t. Ara The solution is r1(m2,m2,y)=4h/m2)命 1\a 2(U1,2,y)=Aa+ Thus xample 1.4. In e c(t1,2,y)=c(1,2)y+ Profit maximization max py-c(w1, w2)ya+6 Solution: +b v(P,t1,u2) c(1,2) ifa+b≠1.Then, P a+b 1)(a+b If a+b=l, profit maximization Inax p-c(n,2)]y
Then, FOC : p = ∂c(w, y∗) ∂y , SOC : ∂2c(w, y∗) ∂y2 ≥ 0. Example 1.3. Consider c(w, y) = min x1, x2 w1x1 + w2x2 s.t. Axa 1xb 2 = y. The solution is x1(w1, w2, y) = A− 1 a+b aw2 bw1 b a+b y 1 a+b , x2(w1, w2, y) = A− 1 a+b bw1 aw2 a a+b y 1 a+b . Thus, c(w1, w2, y) = A− 1 a+b a b b a+b + a b − a a+b w a a+b 1 w b a+b 2 y 1 a+b . Example 1.4. In Example 1.3, c(w1, w2, y) ≡ c(w1, w2)y 1 a+b . Profit maximization: max y py − c(w1, w2)y 1 a+b . Solution: y(p, w1, w2) = p a + b c(w1, w2) a+b 1−a−b , if a + b 9= 1. Then, π(p, w1, w2) = 1 a + b − 1 (a + b) 1 1−a−b p 1 1−a−b c(w1, w2) − a+b 1−a−b . If a + b = 1, profit maximization: max y [p − c(w1, w2)]y. 1—5