Ch. 8 Comparative-Static Analysis of General-Function Models 8.1 Differentials 8.2 Total Differentials 8.3 Rules of Differentials (I-VII) 8.4 Total Derivatives 8.5 Derivatives of Implicit Functions 8.6 Comparative Statics of General-Function Models 8.7 Limitations of Comparative Statics

8.1 Differentials 8.1.1 Differentials and derivatives 8.1.2 Differentials and point elasticity

8.1.1 Differentials and derivatives Problem: What if no explicit reduced-form solution exists because of the general form of the model? Example: What is Y / T when Y = C(Y, T0) + I0 + G0 T0 can affect C direct and indirectly thru Y, violating the partial derivative assumption Solution: Find the derivatives directly from the original equations in the model. Take the total differential The partial derivatives become the parameters

Differential: dy & dx as finite changes (p. 180) Mathematics. Being neither infinite nor infinitesimal. Having a positive or negative numerical value; not zero. Possible to reach or exceed by counting. Used of a number. Having a limited number of elements. Used of a set.

Difference Quotient, Derivative & Differential f(x0+x) f(x) f(x0) x0 x0+x y=f(x) x y x f’(x) B x D A f’(x0)x C

Overview of Taxonomy - Equations: forms and functions Primitive Form Function Specific (parameters) General (no parameters) Explicit (causation) y = a+bx y = f(x) Implicit (no causation) y3+x3-2xy = 0 F(y, x) = 0

Overview of Taxonomy – 1st Derivatives & Total Differentials Differentiation Form Function Specific (parameters) General (no parameters) Explicit (causation) Implicit (no causation)

8.1.1 Differentials and derivatives From partial differentiation to total differentiation From partial derivative to total derivative using total differentials Total derivatives measure the total change in y from the direct and indirect affects of a change in xi

8.1.1 Differentials and derivatives The symbols dy and dx are called the differentials of y and x respectively A differential describes the change in y that results for a specific and not necessarily small change in x from any starting value of x in the domain of the function y = f(x). The derivative (dy/dx) is the quotient of two differentials (dy) and (dx) f '(x)dx is a first-order approximation of dy

8.1.1 Differentials and derivatives “differentiation” The process of finding the differential (dy) (dy/dx) is the converter of (dx) into (dy) as dx 0 The process of finding the derivative (dy/dx) or Differentiation with respect to x

8.1.2 Differentials and point elasticity Let Qd = f(P) (explicit-function general-form demand equation) Find the elasticity of demand with respect to price

8.2 Total Differentials Extending the concept of differential to smooth continuous functions w/ two or more variables Let y = f (x1, x2) Find total differential dy

8.2 Total Differentials (revisited) Differentiation of U wrt x1 U/ x1 is the marginal utility of the good x1 dx1 is the change in consumption of good x1

8.2 Total Differentials (revisited) Total Differentiation: Let Utility function U = U (x1, x2, …, xn) To find total derivative divide through by the differential dx1 ( partial total derivative)

8.2 Total Differentials Let Utility function U = U (x1, x2, …, xn) Differentiation of U wrt x1..n U/ xi is the marginal utility of the good xi dxi is the change in consumption of good xi dU equals the sum of the marginal changes in the consumption of each good and service in the consumption function

8.3 Rules of differentials, the straightforward way Find dy given function y=f(x1,x2) Find partial derivatives f1 and f2 of x1 and x2 Substitute f1 and f2 into the equation dy = f1dx1 + f2dx2

8.3 Rules of Differentials (same as rules of derivatives) Let k is a constant function; u = u(x1); v = v(x2) 1.  dk = 0 (constant-function rule) 2. d(cun) = cnun-1du (power-function rule) 3. d(u  v) = du  dv (sum-difference rule) 4. d(uv) = vdu + udv (product rule) 5.  (quotient rule)

8.3 Rules of Differentials (I-VII) 6. 7. d(uvw) = vwdu + uwdv + uvdw

Rules of Derivatives & Differentials for a Function of One Variable

Rules of Derivatives & Differentials for a Function of One Variable

Rules of Derivatives & Differentials for a Function of One Variable

8.3 Example 3, p. 188: Find the total differential (dz) of the function

8.3 Example 3 (revisited using the quotient rule for total differentiation)

8.4 Total Derivatives 8.4.1 Finding the total derivative 8.4.2 A variation on the theme 8.4.3 Another variation on the theme 8.4.4 Some general remarks

8.4.1 Finding the total derivative from the differential

8.4.3 Another variation on the theme

8.4.3 Another variation on the theme

8.5 Derivatives of Implicit Functions 8.5.3 Extension to the simultaneous-equation case

8.5.1 Implicit functions Explicit function: y = f(x)  F(y, x)=0 but reverse may not be true, a relation? Definition of a function: each x  unique y (p. 16) Transform a relation into a function by restricting the range of y0, F(y,x)=y2+x2 -9 =0

8.5.1 Implicit functions Implicit function theorem: given F(y, x1 …, xm) = 0 a) if F has continuous partial derivatives Fy, F1, …, Fm and Fy  0 and b) if at point (y0, x10, …, xm0), we can construct a neighborhood (N) of (x1 …, xm), e.g., by limiting the range of y, y = f(x1 …, xm), i.e., each vector of x’s  unique y then i) y is an implicitly defined function y = f(x1 …, xm) and ii) still satisfies F(y, x1 … xm) for every m-tuple in the N such that F  0 (p. 195) dfn: use  when two side of an equation are equal for any values of x and y dfn: use = when two side of an equation are equal for certain values of x and y (p.197)

8.5.1 Implicit functions If the function F(y, x1, x2, . . ., xn) = k is an implicit function of y = f(x1, x2, . . ., xn), then where Fy = F/y; Fx1 = F/x1 Implicit function rule F(y, x) = 0; F(y, x1, x2 … xn) = 0, set dx2 to n = 0

8.5.1 Implicit functions Implicit function rule

8.5.1 Deriving the implicit function rule (p. 197)

8.5.1 Deriving the implicit function rule (p. 197)

Implicit function problem: Exercise 8.5-5a, p. 198 Given the equation F(y, x) = 0 shown below, is it an implicit function y = f(x) defined around the point (y = 3, x = 1)? (see Exercise 8.5-5a on p. 198) x3 – 2x2y + 3xy2 - 22 = 0 If the function F has continuous partial derivatives Fy, F1, …, Fm ∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2

Implicit function problem Exercise 8.5-5a, p. 198 If at a point (y0, x10, …, xm0) satisfying the equation F (y, x1 …, xm) = 0, Fy is nonzero (y = 3, x = 1) This implicit function defines a continuous function f with continuous partial derivatives If your answer is affirmative, find dy/dx by the implicit-function rule, and evaluate it at point (y = 3, x = 1) ∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2 dy/dx = - Fx/Fy =- (3x2-4xy+3y2 )/-2x2+6xy dy/dx = -(3*12-4*1*3+3*32 )/(-2*12+6*1*3)=-18/16=-9/8

8.5.2 Derivatives of implicit functions Example If F(z, x, y) = x2z2 + xy2 - z3 + 4yz = 0, then

8.5 Implicit production function F (Q, K, L) Implicit production function K/L = -(FL/FK) MRTS: Slope of the isoquant Q/L = -(FL/FQ) MPPL Q/K = -(FK/FQ) MPPK (pp. 198-99)

Overview of the Problem – 8.6.1 Market model Assume the demand and supply functions for a commodity are general form explicit functions Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) where Q is quantity, P is price, (endogenous variables) Y0 is income, T0 is the tax (exogenous variables) no parameters, all derivatives are continuous Find P/Y0, P/T0 Q/Y0, Q/T0

Overview of the Procedure - 8.6.1 Market model Given Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) Find P/Y0, P/T0, Q/Y0, Q/T0 Solution: Either take total differential or apply implicit function rule Use the partial derivatives as parameters Set up structural form equations as Ax = d, Invert A matrix or use Cramer’s rule to solve for x/d

8.5.3 Extension to the simultaneous-equation case Find total differential of each implicit function Let all the differentials dxi = 0 except dx1 and divide each term by dx1 (note: dx1 is a choice ) Rewrite the system of partial total derivatives of the implicit functions in matrix notation

8.5.3 Extension to the simultaneous-equation case

8.5.3 Extension to the simultaneous-equation case Rewrite the system of partial total derivatives of the implicit functions in matrix notation (Ax=d)

7.6 Note on Jacobian Determinants Use Jacobian determinants to test the existence of functional dependence between the functions /J/ Not limited to linear functions as /A/ (special case of /J/ If /J/ = 0 then the non-linear or linear functions are dependent and a solution does not exist.

8.5.3 Extension to the simultaneous-equation case Solve the comparative statics of endogenous variables in terms of exogenous variables using Cramer’s rule

8.6 Comparative Statics of General-Function Models 8.6.1 Market model 8.6.2 Simultaneous-equation approach 8.6.3 Use of total derivatives 8.6.4 National income model 8.6.5 Summary of the procedure

Overview of the Problem – 8.6.1 Market model Assume the demand and supply functions for a commodity are general form explicit functions Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) where Q is quantity, P is price, (endogenous variables) Y0 is income, T0 is the tax (exogenous variables) no parameters, all derivatives are continuous Find P/Y0, P/T0 Q/Y0, Q/T0

Overview of the Procedure - 8.6.1 Market model Given Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) Find P/Y0, P/T0, Q/Y0, Q/T0 Solution: Either take total differential or apply implicit function rule Use the partial derivatives as parameters Set up structural form equations as Ax = d, Invert A matrix or use Cramer’s rule to solve for x/d

General Function Comparative Statics: A Market Model (8.6.1)

General Function Comparative Statics: A Market Model

General Function Comparative Statics: A Market Model

General Function Comparative Statics: A Market Model

General Function Comparative Statics: A Market Model

General Function Comparative Statics: A Market Model

General Function Comparative Statics: A Market Model

Market model comparative static solutions by Cramer’s rule

Market model comparative static solutions by matrix inversion

8.7 Limitations of Comparative Statics Comparative statics answers the question: how does the equilibrium change w/ a change in a parameter. The adjustment process is ignored New equilibrium may be unstable Before dynamic, optimization