Comparative Statics Overview: I

Transcription

1 Comparative Statics Overview: I 1 Implicit function theorem: used to compute relationship between endogenous and exogenous variables. 1 Context: First order conditions of an optimization problem. Examples e.g., demands as a function of income e.g., input demands as a function of output 2 Context: Some kind of system of equilibrium equations. Examples: market equilibrium prices as a function of economy endowments Cournot quantities as a function of demand parameter, # of firms () November 13, / 13

2 Comparative Statics Overview: II 2 Envelope theorem: simplifies computation of change in value function (optimized level of objective function) with parameters 1 Some function is being optimized 1 function depends on some parameters 2 how does optimum value change with these parameters? 2 Context: First order conditions of an optimization problem Examples: 1 optimized utility as a function of income 2 maximized profit as a function of price Two kinds of optimization: 1 unconstrained optimization (intuition is fairly straightforward) 2 constrained optimization (intuition is much more elusive) () November 13, / 13

3 Comparative Statics Overview: III 3 Envelope theorem (bonus): in very special cases you can short-circuit the implicit function theorem, and use the envelope theorem to get the same information that the implicit function theorem would give you, but much more easily Examples: 1 input demand functions (Hotelling s Lemma) 2 conditional input demand functions (Shephard s Lemma) These results are very special cases: 1 not what the envelope theorem was meant for 2 happens to work because some part of the function being optimized is linear () November 13, / 13

4 Envelope Theorem: unconstrained version I if x (α) solves max x f(x;α), then df (x (α);α) db k = f (x (α);α). i.e., total derivative of f w.r.t. b k equals partial derivative of f w.r.t. b k. Math derivation is straightforward: x f(x (α);α) = 0 necessarily df(x (α);α) db k = f(x (α);α) + = f(x (α);α) n i=1 f(x (α);α) dx i (α) } x {{ i } =0 db k () November 13, / 13

5 Envelope Theorem: unconstrained version II () November 13, / 13

6 Envelope Theorem: tangent planes df/db(x*(3.3),3.3) db 0 0 dx db 0 0 dx () November 13, / 13

7 Unconstrained Envelope Theorem Example E.g. of unconstrained envelope theorem (Hotelling s lemma): Let π (p,w) = pf(x ) w x be the maximized value of profits given output price p and input price vector w. Then the i th input demand function is x i ( ) = π (, ) w i. Note that this is a bonus result: we re not interested in the question how does profit change when w i changes. Again math is straightforward: by the unconstrained envelope theorem, dπ (p,w) = π (p,w) = x i dw i w i () November 13, / 13

8 Envelope Theorem: constrained version If (x (α),λ (α)) solves max x f(x;α) s.t. h j (x;α) 0, j = 1,...m, df(x (α);α) db k = f(x (α);α) + m j=1 λ j (α) hj (x (α);α) total derivative of f w.r.t. b k equals partial derivative of f w.r.t. b k plus λ -weighted sum of partial derivative of h j s w.r.t. b k. Proof: apply unconstrained Env thm to Lagrangian; (Let g j = h j ; h j 0 b j g j 0, where b j = 0, so (b j g j ) h j.) m j=1 L(x,λ;α) = f(x;α) + λ j h j (x;α) L(x,λ ;α) f(x,α) df(x,α) dl(x,λ ;α) = L(x,λ ;α) db k db k b }{{ k } by the unconstrained envelope theorem = f(α,x (α)) + m j=1 λ j (α) hj (α,x (α)) () November 13, / 13

9 Constrained Envelope Theorem Example E.g. of constrained envelope theorem (Shephard s lemma): Let ĉ(w, q) = w x be the minimized level of costs given input prices w and output q (i.e., x is optimal input mix given params). Then the i th conditional input demand function is x i ( ) = ĉ(, ) w i. Again, not asking: how does minimized cost change when w i changes. Proof: min c(x;w, q) x s.t. f(x) q 0 max c(x;w, q) x s.t. q f(x) 0 L(x,λ;w, q) = w x + λ( q f(x)) L(x,λ ;w, q) = w x =: c(x ;w, q) dl(x,λ ;w, q) = L(x,λ ;w, q) = c(x ;w, q) = x i (w) dw i w i w i () November 13, / 13

10 The mother of all envelope theorems The LRATC curve (all inputs variable) is the outer envelope of the SRATC curves (some inputs fixed); when fixed inputs are at their optimal levels, the slopes of LRATC and SRATC curves are equal. Average Cost LRATC( ) SRATC(, k) q q Output () November 13, / 13 1

11 The mother of all envelope theorems: the math Let x,y denote per unit input levels, i.e., l = qx, k = qy, q = f(l,k). The long-run problem: ac LR (x,y; q) = wx + ry min x,y aclr (x,y; q) s.t. f( qx, qy) q 0 L(x,y,λ LR ; q) = (wx + ry) + λ LR ( q f( qx, qy)) dac LR (x ( q),y ( q); q) dq = λ LR ( q) = (wx ( q) + ry ( q)) evaluated anywhere but at optimal mix, daclr (.) would depend on direction dq The short-run problem: ac LR (x; q, k) = wx + r k/ q min ac SR (x; q, k) s.t. f( qx, k) q 0 x L(x,λ LR ; q, k) = (wx + r k/ q) + λ SR ( q f( qx, k)) dac SR (x ( q); q, qy ( q)) dq = λ SR ( q) = (wx ( q) + ry ( q)) () November 13, / 13

12 Cost Increases starting from optimal input mix l Isocost line at q + dq (new optimal input mix) Isocost line at q + dq (increase only l) Isoquant for q + dq increase only l optimal input adjustment (k, l) Isoquant for q Isocost line at q k k () November 13, / 13

13 Cost Increases starting from suboptimal Input Mix l Isocost line at q + dq (new optimal input mix) Isocost line at q + dq (increase only l) increase only l (k, l ) Isoquant for q + dq ( k, l) Isoquant for q optimal input adjustment Isocost line at q k k () November 13, / 13