Comparative Statics. Autumn 2018

Transcription

1 Comparative Statics Autumn 2018

2 What is comparative statics? Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total derivative Implicit differentiation 3 Summary Comparative Statics Autumn / 41

3 What is comparative statics? Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total derivative Implicit differentiation 3 Summary Comparative Statics Autumn / 41

4 What is comparative statics? Comparative statics Static equilibrium analysis solves for equilibrium, i.e. values of endogenous variables when the values of exogenous variables and parameters are given. Comparative statics compares equilibria across different values of exogenous variables. Comparative statics does not address how the new equilibrium is reached. Time dimension is still missing. The trajectories from one equilibrium to another are a subject of dynamic analysis. Comparative Statics Autumn / 41

5 What is comparative statics? Comparative statics A typical problem of comparative statics is to explore how the values of endogenous variables change when one of the exogenous variables or parameters changes. The equilibria can be compared qualitatively (does the value increase or decrease) or quantitatively (how much the value changes). The most important tool of comparative statics is differential calculus, in particular the concept of derivative. Comparative Statics Autumn / 41

6 What is comparative statics? Partial equilibrium Example Let us consider a market which is described by the following equations: q d = a bp, (a, b > 0) q s = c + dp, (c, d > 0) q d = q s. The equilibrium is { p = a+c b+d q = ad bc b+d We can easily see that when parameter a increases (demand curve shifts up), the equilibrium price (p ) and quantity (q ) increase. If parameter b increases the equilibrium price and quantity decrease. Comparative Statics Autumn / 41

7 Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total derivative Implicit differentiation 3 Summary Comparative Statics Autumn / 41

8 Derivative Definition The derivative of a function f : R R at x = x 0 is the limit of the difference quotient: f f (x 0 + h) f (x 0 ) (x 0 ) = lim. h 0 h The derivative of a function f : R R is the function which returns the derivative at an arbitrary x. The derivative of a function y = f (x) is denoted by f (x), y, dy dx, df dx (x), or Df (x). The derivative function is defined point-wise. The derivative tells the rate of growth of the function at a given point. The rate of growth is given by how much the value y would change along the tangent line if x would change by one unit. Comparative Statics Autumn / 41

9 Geometric interpretation of the derivative The definition can be interpreted as drawing secants across points (x, f (x)) and (x + h, f (x + h)) with successively smaller h. The derivative at x 0 is the slope of the tangent drawn through point (x 0, f (x 0 )). The function can be approximated linearly around x 0 by ˆf (x) = f (x 0 ) + f (x 0 )(x x 0 ). Comparative Statics Autumn / 41

10 Existence of derivative Function f has a derivative at a point x 0 if the limit in the above definition exists (i.e. limits from both side are equal and the function is continuous at x 0 ) Basically function has to be smooth in such a way that an unambiguous tangent line (linear approximation) can be drawn. Function f is differentiable at an open interval (a, b) if the derivative f (x 0 ) exists at all points x 0 (a, b). Only a continuous function can be differentiable, but not all continuous functions are differentiable (e.g. f (x) = x is not differentiable at x = 0 though it is continuous) If the derivative f is continuous, function f is continuously differentiable. (We sometimes denote this by f C 1.) Comparative Statics Autumn / 41

11 Taylor series Infinitely differentiable functions can be approximated around x 0 by the Taylor series: f (x) f (x 0 )+f (x 0 )(x x 0 )+ f (x 0 ) 2! (x x 0 ) 2 + f (x 0 ) (x x 0 ) ! More terms generally result in a better approximation. There are some weird functions for which this does not hold, but these are rarely met in economics. In many applications higher order terms can be omitted without significant loss of accuracy The approximation is better close to the expansion point x 0. In many applications linear approximation is enough. Comparative Statics Autumn / 41

12 Partial equilibrium Example We return to the previous market equilibrium example. The equilibrium price was given by p = a + c b + d = 1 b + d a + c b + d = p (a). This can be thought of as a linear function of the parameter a. If we want to know how the equilibrium price changes when parameter a changes, we differentiate function p (a) with respect to a dp da (a) = 1 b + d > 0. Because the derivative is positive, the equilibrium price increases when parameter a increases. Because the derivative is constant, the rate of growth is independent of a. Comparative Statics Autumn / 41

13 Rules of differentiation 1 Dk = 0, where k is constant 2 Dkf (x) = kdf (x) = kf (x) 3 D[f (x) + g(x)] = f (x) + g (x) 4 Df (x)g(x) = f (x)g(x) + g (x)f (x) 5 D f (x) g(x) = f (x)g(x) g (x)f (x) [g(x)] 2 6 D(f g)(x) = Df (g(x)) = f (g(x))g (x) 7 Df (x) n = nf (x) n 1 f (x), especially Dx n = nx n 1 8 De f (x) = e f (x) f (x), especially De x = e x 9 D ln f (x) = f (x) f (x), especially D ln x = 1 x 10 Da x = a x ln a Comparative Statics Autumn / 41

14 Differentiation Example Let f (x) = x 2 and g(x) = ln x. The composite function (f g)(x) is given by (f g)(x) = f (g(x)) = (ln x) 2. The derivative of this function is f (g(x))g (x) = 2(ln x) 1 x = 2 ln x x. The composite function with the reverse order is (g f )(x) = g(f (x)) = ln(x 2 ). The derivative of this function is g (f (x))f (x) = 1 2x = 2 x 2 x. Example Let f (x) = x 2 ln x. The derivative is given by f (x) = 2x ln x + 1 x x 2 = 2x ln x + x. Example Let f (x) = x2 ln x. Then f (x) = 2x ln x 1 x x2 (ln x) 2 = 2x ln x x (ln x) 2. Comparative Statics Autumn / 41

15 Cost function Example 1/3 Assume that the cost function of a firm is C(q) where q is the quantity produced and the value of C gives the costs. We only know that C(0) > 0, C (q) > 0, C (q) > 0 i.e. 1) there are some costs even if nothing is produced, 2) costs are increasing with respect to the quantity produced and 3) the growth in costs per unit produced increses with production. The marginal cost function MC(q) is the derivative of the cost function: MC(q) = C (q). It tells how much additional costs incur from producing an additional unit. The average cost function AC(q) = C(q) q tells the average costs per unit produced. Comparative Statics Autumn / 41

16 Cost function Example 2/3 The derivative of the average cost function is AC (q) = C (q)q C(q) q 2 = C (q) q C(q) q = MC(q) AC(q). q The average costs are minimized at the zero of the derivative, which means that MC(q) = AC(q). Therefore the marginal cost function MC(q) crosses the average cost function AC(q) (from below) in the point where average costs are minimized. Comparative Statics Autumn / 41

17 Cost function Example 3/3 Comparative Statics Autumn / 41

18 Multi-variable differentiation With multiple variables one can choose with respect which variable to differentiate. The concept that is used is partial derivative. Definition The i:th partial derivative of a function f : R n R at a point (x 0 1,..., x 0 n) is the limit f (x1 0,..., x 0 f (x1 0 x n) = lim,..., x i 0 + h,..., xn) 0 f (x1 0,..., x n) 0. i h 0 h The i:th partial derivative of a function y = f (x 1,..., x n ) can be denoted by f x i, y x i, f xi, f i, i f or D i f. Comparative Statics Autumn / 41

19 Partial derivative When computing the partial derivative, the other variables are treated as constants. Partial derivative with respect to x i tells the rate of growth of function when x i increases and the other variables remain constant. Comparative Statics Autumn / 41

20 Partial derivative Example Assume function f (x, y, z) = 4x + xy 3 + y 4 + y 4 z 2 + xyz. The partial derivatives of the function are f x (x, y, z) = 4 + y 3 + yz f y (x, y, z) = 3xy 2 + 4y 3 + 4z 2 y 3 + xz f z (x, y, z) = 2y 4 z + xy. Comparative Statics Autumn / 41

21 Cobb-Douglas Example Let us examine the Cobb-Douglas production function F (K, L) = AK α L 1 α, where K is the amount of capital and L is the amount of labor. Here, A and α are (positive) parameters. The production function gives the quantity produced with the given amounts of factors of production K and L. We want to know how much production changes when either of the inputs increases. For that purpose, we compute the partial derivatives: ( ) F K α 1 K (K, L) = αak α 1 L 1 α = αa L F L (K, L) = (1 α)ak α L α = (1 α)a ( ) K α. L Comparative Statics Autumn / 41

22 Gradient The gradient vector of a function f is the (column) vector of partial derivatives: grad(f ) = f = f x 1. f x n The gradient points to the direction of fastest increase of the function. The length of the gradient is proportionate to the rate of growth. We will later encounter another way to arrange the partial derivatives. In Jacobian the partial derivatives form a row vector. Comparative Statics Autumn / 41

23 Gradient Example Cobb-Douglas production function Y = F (K, L) = K 0.4 L 0.6. Gradient is orthogonal to the isoquant. Comparative Statics Autumn / 41

24 Functions Real-valued one variable function f : R R. E.g. y = f (x) = x 2 Derivative Real-valued multiple variable function f : R n R. E.g. y = f (x 1, x 2 ) = x1 2 + x 2 2 Partial derivatives, gradient Vector-valued multiple variable function f : R n R m E.g. y = [ y1 y 2 Jacobian matrix ] = [ f 1 (x 1, x 2 ) f 2 (x 1, x 2 ) ] [ ] x1 + x = 2 x1 2 + x 2 2 Comparative Statics Autumn / 41

25 Vector-valued function Let us examine a vector-valued function of the form y 1 f 1 (x 1,..., x n ) y 2 f 2 (x 1,..., x n ). y m =. f m (x 1,..., x n ). Component functions f 1,..., f m can be linear or non-linear. We assume them to be differentiable. The generalization of the derivative to vector-valued functions involves the Jacobian matrix. Comparative Statics Autumn / 41

26 Jacobian The Jacobian is formed in the following way f 1 f 1 f x 1 x 1 x n f 1 f 2 f 2 f J = x 1 x 2 2 x n =. f m f x 2 m x n f m x 1 Sometimes also notation J = (f 1,f 2,...,f m ) (x 1,x 2,...,x n) 1 f2 1 fn 1 f1 2 f2 2 fn f1 m f2 m fn m or J = Df is used.. For a real-valued function (m = 1) the Jacobian is the transpose of the gradient. Note that the i th row is the transpose of the gradient of f i. For a linear function f (x) = Ax the Jacobian is A. Comparative Statics Autumn / 41

27 Jacobian Example Let us examine the vector-valued function f : R 2 R 2 that is given by the equations { y1 = 2x 1 + 3x 2 The Jacobian matrix of function f is [ ] y1 y 1 [ J = x 1 x 2 = y 2 x 1 y 2 x 2 y 2 = 4x x 1x 2 + 9x x x 2 12x x 2 The value of the Jacobian depends on the values of x 1 and x 2. ]. Comparative Statics Autumn / 41

28 Jacobian determinant The determinant of the Jacobian, J, is called the Jacobian determinant. The value of the Jacobian determinant tells about the mutual dependency of the component functions. Note that the determinant exists only if n = m. Theorem Assume a differentiable function f : R n R n. The Jacobian determinant J = 0 x if and only if the functions f 1,..., f n are functionally dependent. Functions are functionally dependent if there exists a continuously differentiable function F for which F (f 1,..., f n ) = 0 Comparative Statics Autumn / 41

29 Jacobian determinant Example In the previous example the Jacobian determinant is J = 2 3 8x x 2 12x x 2 = 2(12x x 2 ) 3(8x x 2 ) = 24x x 2 24x 1 36x 2 = 0 x 1, x 2 R Functions f 1 = 2x 1 + 3x 2 and f 2 = 4x x 1x 2 + 9x2 2 functionally dependent. To be precise f 2 = (f 1 ) 2. are therefore If the functions are linear (i.e. f (x) = Ax), the value of the Jacobian determinant is simply the determinant of A. Comparative Statics Autumn / 41

30 dx and dy In differential calculus we often use the notation dx and dy. These are called differentials. Let us examine the definition of the derivative: f (x) = y lim x 0 x = dy dx We can interpret dy and dx as infinitesimal changes in x and y. Derivative is the change in the value of the function divided by the change in the variable, when both are infinitely small. The terms dx and dy can be read as infinitely small real numbers, with meaningful interpretation only for the quotient. Comparative Statics Autumn / 41

31 Approximation of change of function value Change in function value ( y) can be approximated for a given change of x ( x) using the derivative: y f (x) x. Smaller x leads to more accurate approximation. With infinitely small x there is an equivalence dy = f (x) dx. This equation defines the differential of the function f. Comparative Statics Autumn / 41

32 Differential Originally Leibniz defined the derivative as the quotient of infinitesimal changes dy dx. This is known as the Leibniz notation. Later derivative was defined as the limit of difference quotient and differential was defined using the derivative. Definition Let y = f (x). The differential dy of function f is the following function of variables x and dx: df (x) = dy = f (x)dx. Example Differential of a function tells the change of the value of the function at the given point when the change of x is infinitely small. Let y = f (x) = x 2. The differential of f is dy = 2xdx. Comparative Statics Autumn / 41

33 Point elasticity Example The price elasticity of demand ɛ is obtained by dividing the relative change of demand q by the corresponding relative change of price p: ɛ = q q. p p The value of the price elasticity at a given point (point elasticity) is obtained when changes are infinitely small. The point elasticity formula is achieved by replacing changes with differentials and forming a derivative: ɛ p = q q p p = dq q p dp = dq dp p q = q (p) p q. Comparative Statics Autumn / 41

34 Point elasticity Example Suppose that the demand function is given by q(p) = 5 2p. The point elasticity at p = 2, q = 1 is = 4. Example Suppose that the demand function is given by q(p) = p ɛ. The constant point elasticity is ɛp ɛ 1 = ɛ. p p ɛ Comparative Statics Autumn / 41

35 Total differential The concept of differential generalizes to multiple variable functions. Assume a two-variable function z = f (x, y). The partial differential of function f with respect to the variable x gives the change of the function value when x changes by dx. This is obtained as the product of the partial derivative and differential z x dx. The total differential of a function is the sum of all the partial differentials. It tells how much function s value changes when all the variables change infinitesimally. Definition Assume a differentiable function f : R n R, y = f (x 1,..., x n ). The total differential of f is dy = y x 1 dx 1 + y x 2 dx y x n dx n Comparative Statics Autumn / 41

36 Total differential Example Assume a utility function of the form U(x, y) = ax + by. The utility U depends on the quantity of good 1, x, and the quantity of good 2, y. An indifference curve is a curve on which the utility U stays constant, that is, all the points (x, y) which satisfy U(x, y) = c for some c R. The following equation holds on an indifference curve: Therefore: du = U U dx + x y dy = U xdx + U y dy = adx + bdy = 0. U y dy = U x dx dy dx = U x U y = a b. The slope of an indifference curve equals the quotient of marginal utilities. In this example indifference curves are lines y = a b x + k, k R and the slope is always constant a b. Comparative Statics Autumn / 41

37 Dependence between variables Example There may be dependencies between variables of a function E.g. assume a function y = f (x, w) where the value of y depends on values of x and w. Assume also that the value of x depends on w according to x = g(w). Then w affects y through two channels: Directly through second argument of f (w changes y changes). Indirectly by first affecting x through g and then affecting y through first argument of f (w changes x changes y changes). Function g can be substituted into f to end up with only one variable. Let y = f (x, w) = 3x w 2, where x = g(w) = 2w 2 + w + 4. Plugging x = g(w) into f we get y = f (g(w), w) = 3(2w 2 + w + 4) w 2 = 5w 2 + 3w + 12 = h(w). Comparative Statics Autumn / 41

38 Total derivative Derivative tells how much the value of a function changes when the value of the variable changes (by a small amount). If the variables are mutually dependent, one must take into account both direct and indirect effects. The resulting derivative is called the total derivative There is nothing fundamentally new about the total derivative. Same result is achieved by substituting the corresponding functions and differentiating normally. Sometimes it may be more convenient to use the formula for total derivative Comparative Statics Autumn / 41

39 Total derivative Let y = f (x, w) and x = g(w). We want to differentiate f with respect to w. The direct effect of w is obtained from the partial derivative y w. The indirect effect is obtained from product y dx x dw. The total derivative is the sum: dy dw = y dx x dw + y w. The same formula can be obtained from the definition of the total differential: dy = y y dx + x w dw : dw Comparative Statics Autumn / 41

40 Total derivative Example Let y = f (x, w) = 3x w 2 where x = g(w) = 2w 2 + w + 4. The total derivative of f with respect to w is dy dw = y dx x dw + y w = (4w + 1) 3 + ( 2w) = 10w + 3. By substitution we obtain the function y = h(w) = 5w 2 + 3w Differentiation of h gives the same result h (w) = dy dw = 10w + 3. Comparative Statics Autumn / 41

41 Total derivative and the chain rule Note the difference between the partial derivative y w and the total derivative dy dw. If there would be no dependencies between the variables partial derivative and total derivative would be equal. With functions y = f (x) and x = g(w) the total derivative formula reduces to the rule of differentiation of a composite function. There may be several chains. Let y = f (x 1, x 2, u, v) where x 1 = g(u, v) and x 2 = h(u, v). The total derivative of f with respect to variable u is dy du = y dx 1 x 1 du + y dx 2 x 2 du + y u. The total derivative formula is often called the chain rule of differentiation. Comparative Statics Autumn / 41