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Multi-Variable Calculus and Optimization Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus Reading 1 Chapter 7.4-7.6, 8 and 11 (section 6 has lots of economic examples) of CW 2 Chapters 14, 15 and 16 of PR Plan 1 Partial differentiation and the Jacobian matrix 2 The Hessian matrix and concavity and convexity of a function 3 Optimization (profit maximization) 4 Implicit functions (indifference curves and comparative statics) Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 2 / 51 Multi-variable Calculus Functions where the input consists of many variables are common in economics. For example, a consumer's utility is a function of all the goods she consumes. So if there are n goods, then her well-being or utility is a function of the quantities (c1, ... , cn) she consumes of the n goods. We represent this by writing U = u(c1, ... , cn) Another example is when a firm’s production is a function of the quantities of all the inputs it uses. So, if (x1, ... , xn) are the quantities of the inputs used by the firm and y is the level of output produced, then we have y = f (x1, ... , xn). We want to extend the calculus tools studied earlier to such functions. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 3 / 51 Partial Derivatives Consider a function y = f (x1, ... , xn). The partial derivative of f with respect to xi is the derivative of f with respect to xi treating all other variables as constants and is denoted, ¶f or fxi ¶xi Consider the following function: y = f (u, v) = (u + 4)(6u + v). We can apply the usual rules of differentiation, now with two possibilities: ¶f = 1 (6u + v) + 6 (u + 4) = 12u + v + 24 ¶u ¶f = 0 (6u + v) + 1 (u + 4) = u + 4 ¶v Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 4 / 51 Partial Derivatives - Economic Examples Again, suppose y = f (x1, ... , xn). Assume n = 2 and x1 = K and x2 = L; that is, y = f (K, L). Then assume the following functional form: f (K, L) = K 0.25L0.75; i.e., a Cobb-Douglas production function. The partial derivatives are given by, ¶f ¶f = 0.25K −0.75L0.75, = 0.75K 0.25L−0.25 ¶K ¶L a 1−a Alternatively; suppose utility is U = ca cb , where a =apples, b =bananas. We find: 1−a a ¶U cb ¶U ca = a , = (1 − a) ¶ca ca ¶cb cb So, for a given labor input, more capital raises output. For given consumption of apples, consuming more bananas makes you happier. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 5 / 51 Interpretation of Partial Derivatives Mathematically, the partial derivative of f with respect to xi tells us the rate of change when only the variable xi is allowed to change. Economically, the partial derivatives give us useful information. In general terms: 1 With a production function, the partial derivative with respect to the input, xi , tells us the marginal productivity of that factor, or the rate at which additional output can be produced by increasing xi , holding other factors constant. 2 With a utility function, the partial derivative with respect to good ci tells us the rate at which the consumer's well being increases when she consumes additional amounts of ci holding constant her consumption of other goods. I.e., the marginal utility of that good. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 6 / 51 Derivative of a Linear Function As with functions of one variable, to gain some intuition, we start with functions that are linear. We motivated the notion of a derivative by saying that it was the slope of the line which \looked like the function around the point x0." When we have n variables, the natural notion of a \line" is given by y = a0 + a1x1 + ... + anxn When there are two variables, x1 and x2, the function y = a0 + a1x1 + a2x2 is the equation of a plane. In general, the function y = a0 + a1x1 + ... + anxn is referred to simply as the equation of a plane. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 7 / 51 Two Variable Linear Function Consider the plane y = a0 + a1x1 + a2x2. How does the function behave when we change x1 and x2? Clearly, if dx1 and dx2 are the amounts by which we change x1 and x2, we have, dy = a1dx1 + a2dx2 Note furthermore that the partials are, ¶y ¶y = a1 and = a2 ¶x1 ¶x2 We can then write, ¶y ¶y dy = dx1 + dx2 ¶x1 ¶x2 Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 8 / 51 Jacobian Matrix We can take this result (i.e., that dy = ¶y dx + ¶y dx ) and recast it ¶x1 1 ¶x2 2 in matrix (here, vector) form. h ¶y ¶y i dx1 dy = x x ¶ 1 ¶ 2 dx2 | {z } ≡J We call J the Jacobian. h ¶y ¶y i Take our previous example. The vector = a1 a2 is ¶x1 ¶x2 what we can think of as the derivative of the plane. This vector tells us the rates of change in the directions x1 and x2. The total change is therefore computed as a1dx1 + a2dx2. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 9 / 51 General Functions Now consider a more general two variable function, y = f (x1, x2). With a general function y = f (x1, x2), the idea is to find a plane which looks locally like the function around the point (x1, x2). Since the partial derivatives give the rates of change in x1 and x2, it makes sense to pick the appropriate plane which passes through the point (x1, x2) and has slopes ¶f /¶x1 and ¶f /¶x2 in the two directions. The derivative (or the total derivative) of the function f (x1, x2) at h ¶y ¶y i (x1, x2) is simply the vector where the partial derivatives ¶x1 ¶x2 are evaluated at the point (x1, x2). We can interpret the derivative as the slopes in the two directions of the plane which looks \like the function" around the point (x1, x2). Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 10 / 51 An Example When we have a function y = f (x1, ... , xn), the derivative at h ¶y ¶y i (x1, ... , xn) is simply the vector ... ¶x1 ¶xn Take a more concrete example. Suppose, 2 y = 8 + 4x1 + 6x2x3 + x3 The Jacobian must be, h ¶y ¶y ¶y i 8x1 6x3 1 = ¶x1 ¶x2 ¶x2 So we can write, 2 3 dx1 dy = 8x1 6 1 4 dx2 5 dx3 Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 11 / 51 An Economic Example a 1−a Consider the utility function from before; U = ca cb , where a =apples, b =bananas. The Jacobian is a vector of the partials (which we already computed). The change in utility is therefore given by the following. h ¶U ¶U i dca dU = ¶c ¶c a b dcb h 1−a a i dca = a cb (1 − a) ca ca cb dcb The same is true for the Cobb-Douglas production function, f (K, L) = K 0.25L0.75. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 12 / 51 Second-Order Partial Derivatives We now move on to second-order partial derivatives. Our motivation will be economic problems/interpretation. ¶2f Given a function f (x1, ... , xn), the second-order derivative is the ¶xi xj ¶f partial derivative of with respect to xj . ¶xi The above may suggest that the order in which the derivatives are taken matters: in other words, the partial derivative of ¶f with ¶xi ¶f respect to xj is different from the partial derivative of with respect ¶xj to xi . While this can happen, it turns out that if the function f (x1, ... , xn) is well-behaved then the order of differentiation does not matter. This result is called Young's Theorem. We shall only be dealing with well-behaved functions (i.e., Young's Theorem will always hold). Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 13 / 51 Economic Example For the production function y = K 0.25L0.75, we can evaluate the ¶2y second-order partial derivative, ¶K ¶L , in two different ways. ¶y −0.75 0.75 First, since ¶K = 0.25K L , taking the partial derivative of this with respect to L, we get, ¶ ¶y 3 = K −0.75L−0.25 ¶L ¶K 16 ¶f 0.25 −0.25 And since ¶L = 0.75K L , we have, ¶ ¶y 3 = K −0.75L−0.25 ¶K ¶L 16 This illustrates Young's Theorem: no matter in which order we differentiate, we get the same answer. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 14 / 51 Economic Interpretation The second-order partial derivatives can be interpreted economically. Our example production function is y = K 0.25L0.75. Therefore, ¶2y 3 = − K −1.75L0.75 ¶K 2 16 This is negative, so long as K > 0 and L > 0. This tells us that the marginal productivity of capital decreases as we add more capital. Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 15 / 51 Economic Interpretation As another example, consider, ¶2y 3 = K −0.75L−0.25 ¶K ¶L 16 This is positive, so longas K > 0, L > 0. This says that the marginal productivity of labour increases as you add more capital. It also suggests a symmetry. The above can be interpreted as saying that the marginal productivity of capital increases when you add more labor.
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