z = f (x; y) = x 3 3x 2 y x 2 3

Transcription

1 BEE Mathematics for Economists Week, ecture Thursday..7 Functions in two variables Dieter Balkenborg Department of Economics University of Exeter Objective This lecture has the purpose to make you familiar with functions in two variables. Partial di erentiation is the basic technique to analyze such functions. We will illustrate such functions using computer graphics. Of particular importance are level curves, which occur as indi erence curves or isoquants in economics. Marginal rates of substitution correspond to the slopes of tangents to such curves. The lecture should enable you to calculate partial derivatives and marginal rates of substitution. Functions in two variables A function z = f (x; y) or simply z (x; y) in two independent variables with one dependent variable assigns to each pair (x; y) of (decimal) numbers from a certain domain D in the two-dimensional plane a number z = f (x; y). x and y are hereby the independent variables and z is the dependent variable. The graph of f is the surface in -dimensional space consisting of all points (x; y; f (x; y)) with (x; y) in D. Example The cubic polynomial z = f (x; y) = x x y is de ned on the entire plane. It has the graph: x y 8 The vertical line is here the z-axis, the x- and the y- axis span a horizontal plane. Thus the height of the graph indicates the values of the function. ighter colours correspond to higher values of the function. For instance, f (; ) =, so the origin of the coordinate system (; ; ) belongs to the graph and is a peak. The function has a saddle point at (x; y) = (; ) where z = f (x; y) = 8 =. The two-dimensional plane consisting of all pairs of numbers (x; y) is usually denoted by IR.

2 Exercise Evaluate Could the information t with the graph? Example A production function like z = f (; ) z = f (; ) z = f (; ) z = f (; ) Q = p p = of a rm describes for each combination of capital ( ) and labour ( ) the total amount Q of output which the rm can produce with these inputs. The speci ed function has the graph: 8 Example Consider a rm with the above production function and assume that the rm is a price taker in the product market and in both factor markets. If P is the price of the commodity produced, r the interest rate (= the price of capital) and w the wage rate (= the price of labour) then the pro t of this rm is (; ) = P Q r w = P r w if it employs units of capital and units of labour. For the prices P =, r =, w = we obtain the function (; ) =

3 with the graph: One can show that pro ts is maximized for = = 8. The level curve of the function z = f (x; y) for the level c is the solution set to the equation f (x; y) = c where c is a given constant. Geometrically, a level curve is obtained by intersecting the graph of f with a horizontal plane z = c and then projecting into the (x; y)-plane. This is illustrated here for the cubic polynomial discussed above: y x y x In the case of a production function the level curves are called isoquants. An isoquant shows for a given output level capital-labour combinations which yield the same output. 8 Finally, the linear function z = x + y

4 has the graph and the level curves: y x y x The level curves of a linear function form a family of parallel lines. This can be seen algebraically as follows: For a given level c the level curve is given by c = x + y y = c x y = c x The level curves are therefore all lines with slope. They di er only in the intercept c. Hence they are all parallel lines. Exercise Describe the isoquant of the production function for the quantity Q = : Q = Exercise Describe the isoquant of the production function for the quantity Q = : Q = p Remark The exercises illustrate the following general principle: If h (z) is an increasing (or decreasing) function in one variable, then the composite function h (f (x; y)) has the same level curves as the given function f (x; y) : However, they correspond to di erent levels. Q = Q = p

5 Partial derivatives Exercise What is the derivative of z (x) = a x with respect to x when a is a given constant? Exercise What is the derivative of z (y) = y b with respect to y when b is a given constant? Consider now a function in two variables z = f (x; y). If we x y at a value y = y and vary only x we obtain a function in one variable z = F (x) = f (x; y ). The derivative of this function F (x) at x = x is called the partial derivative of f with respect to x and denoted jx=x ;y=y = df dx jx=x = df dx (x ). The partial jx=x ;y=y is de ned symmetrically. (Notation: d = dee, = = del ) We do not have to specify y numerically. To partially di erentiate in the direction of x it su ces to think of y and all expressions containing only y as exogenously xed constants. We can then use the familiar rules for di erentiating functions in one variable in order to Other common notations for partial or f x, f y. Example et = y x z (x; y) = y = y x Example et z = x + x y + y : Setting e.g. y = we obtain z = x + x + and jx=;y= = x + x = For xed, but arbitrary, y we obtain From now on I will often write dz dx jx=x special value of = x + xy instead of dz dx (x ) to denote the derivative evaluated at a

6 as follows: We can di erentiate the sum x + x y + y with respect to x term-by-term. Di erentiating x yields x, di erentiating x y yields xy because we think now of y as a constant and d(ax ) = ax holds for any constant a. Finally, the derivative of any dx constant term is zero, so the derivative of y with respect to x is zero. Similarly considering x as xed and y variable we = x y + x Example The and of a production function q = f are called the marginal product of capital and (respectively) labour. They describe approximately by how much output increases if the input of capital (respectively labour) is increased by a small unit. If one xes for the production function in Example capital input at = one obtains q = = which has, as a function in the one variable, the graph Q This graph is obtained from the graph of the function in two variables by intersecting the latter with a vertical plane parallel to -q-axes. 8 thus the and describe geometrically the slope of in the - and, respectively, the - direction. Notice that for any given level of capital there is diminishing productivity of labour: The more labour is used, the less is the increase in output when one more unit of labour is employed. Algebraically this can be veri ed = p = p > ;

7 the marginal product of labour is always positive. Di erentiating a second time we = p p < ; so the marginal product of labour is decreasing. Verify similarly that there is diminishing productivity of capital! Exercise Find the partial derivatives of z = x + x y y + x + y The tangent plane For a function y = f (x) in one variable the derivative df dx jx=x tangent to the graph of f through the point (x ; f (x )). at x is the slope of the x The tangent of f (x) = x at x =. This tangent is the graph of the linear function y = l (x) = f (x ) + df dx jx=x (x x ). This is so because l (x ) = f (x ) ; so the graph of l goes through the point (x ; f (x )). Moreover, df y = l (x) = f (x ) x + df x; dx jx=x dx jx=x so l (x) is indeed a linear function in x with intercept f (x ) df dx jx=x x and the right slope df dx jx=x. The graph of a function in one variable is a curve and its tangent is a line. By analogy, the graph of a function z = f (x; y) in two variables is a surface and its tangent at a point 7

8 (x ; y ) is a plane. Non-vertical planes are the graphs of linear functions z = ax + by + c. The tangent plane of the function z = f (x; y) in (x ; y ) is, correspondingly, the graph of the linear function z = l (x; y) = f (x ; y jx=x ;y=y (x x jx=x ;y=y (y y ) () since the graph of this linear function contains the point (x ; y ; f (x ; y )) and has the right slopes in the x- and y-directions. Example 7 This is the example shown in the above graph. The tangent plane of z = f (x; y) = p x + p y at the point (; ) is obtained as follows: We = p x = p y and jx=;y= = jx=;y= = : Since f (; ) = the tangent plane is the graph of the linear function z = l (x; y) = + (x ) + (y ) = + x + y Using the abbreviations dx = x x, dy = y y and dz = z z the formula for the tangent can be neatly rewritten as an expression which is called the total di erential. dz = dx + 8

9 The slope of level curves Suppose (x ; y ) is a point on the level curve f (x; y) = c. Then the slope of tangent to the level curve in this point in the x-y-plane is dy dx jx=x ;y=y = jx=x jx=x ;y=y provided =. This is known as the rule for implicit di erentiation: If jx=x ;y=y function y = y (x) is implicitly de ned by the equation f (x; y (x)) = c, then its derivative dy dx is given by the above formula. Example 8 A linear function has the partial curves have the slope = a z = ax + by + = b. According to the above formula its = a b. Indeed, its level curves are the solutions to the equations c = ax + by + c with some constant c. Solving the equation for y we obtain which is a linear function with slope y = a b x + c c b a. b Example 9 An isoquant of the production function Q = has according to the above formula the slope d = = = : in the point (; ). Indeed, an isoquant is the set of solutions to the equation q = Solving for we see that the isoquant is the graph of the function = q = q. for xed q. Di erentiation yields d d = q 9

10 and since (; ) is on the isoquant d d = = : d Economically, is the marginal rate of substitution of labour for capital: If labour d is reduced by one unit, how much more capital is (approximately) needed to produce the same output? Just reducing labour by one unit reduces output by the marginal product Each additional unit of capital produces the marginal capital more units. Therefore, if now capital input is changes @ = : Remark A quick way to memorize the formula () is to use the total di erential as follows: In order to stay on the level curve one must have dz = z z =, so hence or = dz = dy dy = dy @y : Remark The implicit function theorem states that if = for some jx=x ;y=y (x ; y ) on a level curve f (x; y) = c, then one can nd a function y = g (x) de ned near x = x such that the level curve is locally the graph of this function, i.e. one has f (x; g (x)) = c for x near x. Exercise 7 For the production function Q = p calculate the marginal rate of substitution for = and =. The chain rule Suppose that z = f (x; y) is a function in two variables x, y. Suppose that x depends on the variable t via x = g (t) and that y depends on the variable t via y = h (t). Then we can de ne the composite function in one variable z = F (t) = f (x (t) ; y (t)) which has t as the independent and z as the dependent variable. The derivative of this function, if it exists, is calculated as dz dt = dt + dt : Notice that this is just the total di erential divided by dt.

11 Example Suppose z = xy, x = t + t, y = t t: Then, according to the chain rule, dz dt = dt + dt = y t + + x (t ) = t t t + + (t ) t + t Since z = xy = (t + t) (t t) we get the same result using the product rule: dz dt = t + t t + t + t (t ).