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, y i ≡ y + i ? y ? i =net output, y = (y 1 ,y 2 ,...,y n ) is a production plan. Production possibility set: Y = null technologically feasible production plans y ∈ R n null . y ∈ Y is technologically e?cient ifthereisnoy 0 ∈ Y s.t. y 0 >y. Production frontier = null technological e?cient production plans null . y ∈ Y is economically e?cient if it maximizes profit. Proposition 1.1. Economic e?ciency implies technological e?ciency.  Consider a single output y ∈ R + . Denote x ∈ R n + as the firm’s inputs and define the production function f : R n + → R + as f(x) ≡ max (y,?x)∈Y y. Proposition 1.2. For y ∈ R + , (y,?x) is technologically e?cient =? y = f(x)  1—1 Isoquant: Q(y) ≡ null x ∈ R n + | y = f(x) null . Marginal rate of transformation: MRT(x) ≡ f x 1 (x) f x 2 (x) . MRT(x) is the slope of the isoquant. Example 1.1. Cobb-Douglas Technology. For 0 ≤ α ≤ 1, consider Y ≡ null (y,?x 1 ,?x 2 ) ∈ R + ×R 2 ? | y ≤ x α 1 x 1?α 2 null .  For production function f : R n + → 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 ∈ R n + ,t>1. Example 1.2. Consider f(x 1 ,x 2 )=Ax a 1 x b 2 .  Elasticity of scale at x : e(x) ≡ dlogf(tx) dlogt null null null null 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. For x ∈ R n , 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 · f 0 (x) f(x) , implying local IRS ?? f 0 (x) > f(x) x local CRS ?? f 0 (x)= f(x) x local DRS ?? f 0 (x) < f(x) x . 3. For x ∈ R n 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 x 1 (w,y) x 2 (w,y) ? log w 1 w 2 . σ isthepercentagechangein x ? 1 x ? 2 for 1% increase in w 1 w 2 . 1—3 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 ∈ R n , the firm’s problem is π ≡ max a R(a) ?C(a). FOC: ?R(a ? ) ?a i = ?C(a ? ) ?a i or MR= MC, ? i. (2.1) Assume competitive firms (price takers) and a single output. Profitfunctionis π(p,w) ≡ max x pf(x) ?w · x. (2.2) Demand function: x ? = x(p,w). Supply function: y(p,w) ≡ f[x(p,w)]. We have FOC : pDf(x ? )=w, SOC : D 2 f(x ? ) ≡ null ? 2 f(x ? ) ?x i ?x j null ≤ 0. Cost function: c(w,y) ≡ min x {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 = λDf(x ? ) or w i w j = f x i (x ? ) f x j (x ? ) , ? i, j. (2.4) The SOC for (2.3) is h 0 D 2 x f(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 : p = ?c(w,y ? ) ?y , SOC : ? 2 c(w,y ? ) ?y 2 ≥ 0. Example 1.3. Consider c(w,y)=min x 1 ,x 2 w 1 x 1 + w 2 x 2 s.t. Ax a 1 x b 2 = y. The solution is x 1 (w 1 ,w 2 ,y)=A ? 1 a+b null aw 2 bw 1 null b a+b y 1 a+b , x 2 (w 1 ,w 2 ,y)=A ? 1 a+b null bw 1 aw 2 null a a+b y 1 a+b . Thus, c(w 1 ,w 2 ,y)=A ? 1 a+b null null a b null b a+b + null a b null ? a a+b null w a a+b 1 w b a+b 2 y 1 a+b . Example 1.4. In Example 1.3, c(w 1 ,w 2 ,y) ≡ c(w 1 ,w 2 )y 1 a+b . Profit maximization: max y py ?c(w 1 ,w 2 )y 1 a+b . Solution: y(p,w 1 ,w 2 )= null p a + b c(w 1 ,w 2 ) null a+b 1?a?b , if a + b null=1. Then, π(p,w 1 ,w 2 )= null 1 a + b ?1 null (a + b) 1 1?a?b p 1 1?a?b c(w 1 ,w 2 ) ? a+b 1?a?b . If a + b =1, profit maximization: max y [p?c(w 1 ,w 2 )]y. 1—5 Solution: y s = ? ? ? ? ? ? ? ? ? ? ? ∞ if p>c(w 1 ,w 2 ), [0, ∞ ]ifp = c(w 1 ,w 2 ), 0 if p<c(w 1 ,w 2 ). Example 1.5. CES production function: y = f(x 1 ,x 2 )=(a 1 x ρ 1 + a 2 x ρ 2 ) 1 ρ , ρ ∈ [?∞ , 1]. We find σ ≡? ?(x 1 (w,y)/x 2 (w,y)) ?(w 1 /w 2 ) (w 1 /w 2 ) (x 1 /x 2 ) = 1 1 ?ρ . If ρ =1 or σ = ∞ , linear production function: y = a 1 x 1 + a 2 x 2 . If ρ =0 or σ =1, assume a 1 + a 2 =1. We have y = lim ρ→0 (a 1 x ρ 1 + a 2 x ρ 2 ) 1 ρ = x a 1 1 x a 2 2 , which is the Cobb-Douglas Production Function. If ρ = ?∞ or σ =0, assuming a 1 = a 2 null=0, we have y =lim ρ→?∞ (a 1 x ρ 1 + a 2 x ρ 2 ) 1 ρ =min(x 1 ,x 2 ), 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 su?cient 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 x i (p,w) is an interior solution, y(p,w)= ?π(p,w) ?p ,x i (p,w)=? ?π(p,w) ?w i , ? i.  Proposition 1.8. (Shephard’s Lemma). If x i (w,y) is an interior solution, x i (w,y)= ?c(w,y) ?w i , ? i.  Proposition 1.9. (Conditional Demand). If x(w,y) is twice continuously di?eren- tiable, (1) x(w,y) is zero homogeneous in w; (2) substitution matrix D w x(w,y) ≤ 0; (3) symmetric cross-price e?ects: ?x i (w,y) ?w j = ?x j (w,y) ?w i ; (4) x i (w,y) is decreasing in w i .  Proposition 1.10. (Demand and Supply). If x(p,w) and y(p,w) are twice contin- uously di?erentiable, (1) x(p,w) and y(p,w) are zero homogeneous in (p,w); (2) x i (p,w) is decreasing in w i ,y(p,w) is increasing in p; (3) symmetric cross-price e?ects: ?x i (p,w) ?w j = ?x j (p,w) ?w i .  1—7