香港科技大学：《微观经济学》（英文版） Chapter 1 Neoclassical Economics

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 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) 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()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) 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

oo if p>c1, w2), 0,]ifp=c(1,2), 0 if p<c(wi, w2) Example 1.5. CES production function f(x1,x2)=(a1 We find (x1(v,y)/r2(0,y)(n/u2) 0(01/12)(x1/12)1 If p=l or a =oo, linear production function If p=0 or g= l, assume a1+a2=1. We have (a1xf+a2x2)2=1 which is the Cobb-Douglas Production Function. Ifp=-∞ora=0, assuming (1=a2≠0, we have y= lim(ai+a2 2)P=min(a1, T2) hich is the leontief Production Function 1.3. Properties Proposition 1. 4. If the production function is homogenous of degree a, c(w, y) c(,1) Proposition 1.5.(Cost Function). c(w, y)is (1) increasing in w (2)linearly homogeneous in w (3) concave in w

Solution: ys = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ ∞ if p>c(w1, w2), [0, ∞] if p = c(w1, w2), 0 if p<c(w1, w2). Example 1.5. CES production function: y = f(x1, x2)=(a1xρ 1 + a2xρ 2) 1 ρ , ρ ∈ [−∞, 1]. We find σ ≡ − ∂(x1(w, y)/x2(w, y)) ∂(w1/w2) (w1/w2) (x1/x2) = 1 1 − ρ . If ρ = 1 or σ = ∞, linear production function: y = a1x1 + a2x2. If ρ = 0 or σ = 1, assume a1 + a2 = 1. We have y = lim ρ→0 (a1xρ 1 + a2xρ 2) 1 ρ = xa1 1 xa2 2 , which is the Cobb-Douglas Production Function. If ρ = −∞ or σ = 0, assuming a1 = a2 9= 0, we have y = lim ρ→−∞(a1xρ 1 + a2xρ 2) 1 ρ = min(x1, x2), which is the Leontief Production Function.  1.3. Properties Proposition 1.4. If the production function is homogenous of degree α, c(w, y) = y 1 α c(w, 1).  Proposition 1.5. (Cost Function). c(w, y) is (1) increasing in w. (2) linearly homogeneous in w. (3) concave in w. 1—6

And, if c(w, y) is continuous, the three conditions are sufficient for cw, y) to be a cost function.■ What causes concavity in cos Proposition 1.6.(Profit Function). T(p, w)is (1) increasing in p, decreasing in w (2) linearly homogeneous in (p, w) (3) convex in(p, w) Proposition 1.7.(Hotelling s Lemma). If Ti(p, w) is an interior solution y(p, w) P Proposition 1.8.(Shephard's Lemma). If i(w, y) is an interior solution, i(, y) Proposition 1.9.(Conditional Demand ). If (w, y) is twice continuously differen tiable (1)aw, y) is zero homogeneous in w; (2) substitution matrix Duc(w, )<0 (3)symmetric cross-price effects: Br u n2=Dao (4)Ti(w, y) is decreasing Proposition 1.10.(Demand and Supply). If a(p, w) and y(p, w) are twice contin- uously differentiable (1)a(p, w) and y(p, w) are zero homogeneous in(p, w) (2)a(p, w) is decreasing in wi, y(p, w) is increasing in p (3)symmetric cross-price effects: Dxp, o = B(p u).D

And, if c(w, y) is continuous, the three conditions are sufficient for c(w, y) to be a cost function.  What causes concavity in cost? Proposition 1.6. (Profit Function). π(p, w) is (1) increasing in p, decreasing in w; (2) linearly homogeneous in (p, w); (3) convex in (p, w).  Proposition 1.7. (Hotelling’s Lemma). If xi(p, w) is an interior solution, y(p, w) = ∂π(p, w) ∂p , xi(p, w) = −∂π(p, w) ∂wi , ∀ i.  Proposition 1.8. (Shephard’s Lemma). If xi(w, y) is an interior solution, xi(w, y) = ∂c(w, y) ∂wi , ∀ i.  Proposition 1.9. (Conditional Demand). If x(w, y) is twice continuously differen￾tiable, (1) x(w, y) is zero homogeneous in w; (2) substitution matrix Dwx(w, y) ≤ 0; (3) symmetric cross-price effects: ∂xi(w,y) ∂wj = ∂xj (w,y) ∂wi ; (4) xi(w, y) is decreasing in wi.  Proposition 1.10. (Demand and Supply). If x(p, w) and y(p, w) are twice contin￾uously differentiable, (1) x(p, w) and y(p, w) are zero homogeneous in (p, w); (2) xi(p, w) is decreasing in wi, y(p, w) is increasing in p; (3) symmetric cross-price effects: ∂xi(p,w) ∂wj = ∂xj (p,w) ∂wi .  1—7