14 March 2018 Module 1: Marginal analysis and single variable calculus John Riley. ( x, f ( x )) are the convex combinations of these two

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1 4 March 28 Module : Marginal analysis single variable calculus John Riley 4. Concave conve functions A function f( ) is concave if, for any interval [, ], the graph of a function f( ) is above the line joining the end-points, the line segment joining (, f ( )) (, f ( )) (, f ( )). Consider Fig. 4-. Note that (, f ( )) are the conve combinations of these two points. The function is strictly concave if for all such pairs of points, the graph of the function is strictly above the line. A function is concave if the graph of the function is on or above the line. In the eamples below, the first function is strictly concave. The second is concave but not strictly concave since it has a linear segment. Fig 4-: Strictly concave function Fig 4-2: Concave function Section 4, page

2 4 March 28 Module : Marginal analysis single variable calculus John Riley Definition: Concave function f( ) is concave on some interval X if for any in this interval, any conve combination,, t ( t) ( t) t, f t t f tf ( ( )) ( ) ( ) ( ) Definition: Strictly concave function f( ) is strictly concave on some interval X if for any in this interval, any conve combination, ( t) ( t) t t,, f t t f tf ( ( )) ( ) ( ) ( ) In this section we will focus primarily on concave functions. All of the conclusions hold for strictly concave functions ecept that all statements of propositions hold strictly inequalities become strict inequalities. Rules for concave functions. Linear function rule A linear function is concave. 2. Sum rule: The sum of concave functions is concave 3. Composite function rule Define the composite function h( ) g( f ( )) where f( ) g( y ) are both concave. If (i) f( ) is linear or (ii) g( y ) is increasing, then h ( ) is concave. A composite function is a function of a function Section 4, page 2

3 4 March 28 Module : Marginal analysis single variable calculus John Riley Derivation of the Linear function rule This is clear by considering the graph of a linear function, f ( ) a b. All the points on the line joining are on the graph of the function so f ( ( t)) ( t) f ( ) tf ( ). Fig. 4-3: Linear function For completeness, a simple derivation is presented. For the conve combination t (), f ( ( t)) a b( t) a b(( t) t ) a b( t) tb. Also f ( ) a b f ( ) a b. Multiplying the first equation by t, the second by t adding, Therefore ( t) f ( ) tf ( ) ( t)( a b ) t( a b ) a ( t) b tb f ( ( t)) ( t) f ( ) tf ( ). Derivation of the sum rule any To see that the sum rule is true, note that if f( ) g ( ) are concave on X, then for in X, conve combination t () f t t f tf ( ( )) ( ) ( ) ( ) Section 4, page 3

4 4 March 28 Module : Marginal analysis single variable calculus John Riley g( ( t)) ( t) g( ) tg( ). Adding these two inequalities, f t g t t f tf t g tg ( ( )) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ) Rearranging the terms on the right h side, f t g t t f g t f g ( ( )) ( ( )) ( )[ ( ) ( )] [ ( ) ( )] The final step is to define the sum h( ) f ( ) g( ). Substituting this into the above inequality, h( ( t)) ( t) h( ) th( ). Derivation of the Composite function rule (case (i)) The composite function is depicted below for case (i) when f( ) is linear. Fig. 4-3: composite function, case (i) The point Similarly is first mapped to is mapped to t () is mapped to yˆ f ( ( t)). y f ( ) then y this is mapped to Appealing to the Linear function Rule y is mapped to g f ( ( )) yˆ f ( ( t)) ( t) f ( ) tf ( ) ( t) y ty y( t) g y ( ) g( f ( )).. Finally the conve combination. (4-) The point yˆ y() t is then mapped to g( yˆ ) g( y( t)). Since g( y ) is a concave function g yˆ g y t t g y g y ( ) ( ( )) ( ) ( ) ( ) Section 4, page 4

5 4 March 28 Module : Marginal analysis single variable calculus John Riley But g yˆ g f t g y g f ( ) ( ( ( )), ( ) ( ( )) g f t t g f g f ( ( ( )) ( ) ( ( )) ( ( )) g y ( ) g( f ( )). Therefore Derivation of the Composite function rule (case (i)) The composite mapping is depicted below for case (ii) when g ( ) is increasing concave. Fig. 4-4: Composite function, case (ii) Since f( ) is concave, i.e., f t t f tf ( ( )) ( ) ( ) ( ) yˆ ( t) y ty y( t) Since g( y ) is increasing, it follows that g( yˆ ) g( y( t)). Since g( y ) is concave, g( y( t)) ( t) g( y ) tg( y ). Combining these inequalities, as in case (i), g( yˆ ) ( t) g( y ) tg( y ). Section 4, page 5

6 4 March 28 Module : Marginal analysis single variable calculus John Riley Differentiable concave functions Very often economists build models assuming functions are differentiable. Then at each point on the graph of the function the slope is well defined. Consider the two tangent lines depicted in Figure 4-5 below. Fig 4-5: Strictly concave function It is intuitively clear that for the graph to lie above every chord, the tangent line at must have a greater slope than at strictly greater if the function is strictly concave. It is much more important to underst this intuitively than to know how to prove equivalence. Thus discussions of how to prove this result are relegated to an appendi at the end of this section. They are not essential reading. In fact the converse is also true. If the slope is everywhere decreasing, then the function is concave. We therefore have the following second definition of a differentiable concave function. Definition: Concave differentiable function The differentiable function f( ) is concave on some interval X if for any in this interval, f ( ) f ( ). Section 4, page 6

7 4 March 28 Module : Marginal analysis single variable calculus John Riley Definition: Strictly concave differentiable function The differentiable function f( ) is strictly concave on some interval X if for any in this interval, f f ( ) ( ) From this definition, the graph of such a function can only have one of three possible shapes. These are shown below. Fig. 4-6: Strictly concave functions For differentiable concave functions there is a third equivalent definition. Consider Figure 4-7 below. Fig 4-7: Tangent line is above the graph of the function Section 4, page 7

8 4 March 28 Module : Marginal analysis single variable calculus John Riley Consider the tangent line through Therefore for the point (, y ) (, y ). The slope of the graph of f( ) is on the tangent line, f ( ). y y y f ( ). It follows that the equation of the tangent line is y y f ( )( ) f ( ) f ( )( ) Since the tangent line has a constant slope a concave function has a decreasing slope it follows that the tangent line must be everywhere above the graph of the function. Definition: Concave differentiable function The differentiable function f( ) is concave on some interval X if for every in this interval, f ( ) f ( ) f ( )( ). Definition: Strictly concave differentiable function The differentiable function f( ) is concave on some interval X if for every in this interval, f ( ) f ( ) f ( )( ). Conve functions A function f( ) is conve if, for any interval [, ], the graph of a function f( ) is below the line joining the end-points, joining (, f ( )) (, f ( )) (, f ( )). Note that the line segment (, f ( )) are the conve combinations of these two points. The function is strictly conve if for all such pairs of points, the line is strictly below the graph of the function. In the eample below, the function is strictly conve. Section 4, page 8

9 4 March 28 Module : Marginal analysis single variable calculus John Riley Fig 4-8: Strictly conve function For a conve function f t t f tf ( ( )) ( ) ( ) ( ) Define g( ) f ( ). Then g t f t t f t f t g tg ( ( )) ( ( )) ( )( ( )) ( ( )) ( ) ( ) ( ) Thus a function, f( ) is conve if only if f( ) is concave. Section 4, page 9

10 4 March 28 Module : Marginal analysis single variable calculus John Riley Appendi: Proofs Proposition: If f( ) is differentiable concave on X [ a, b] then the tangent line at any point in X lies above the graph of f( ). From the first definition, f( ) is concave on some interval X if for any in this interval, any conve combination,, t ( t) ( t) t, f t t f tf ( ( )) ( ) ( ) ( ) Re can rewrite this as follows. f ( ( t)) f ( ()) t( f ( ) f ( )) (4-2) for ( t) t( ), t, (4-3) Note first that (). Net define the function of a function t g( t) f ( ( t)) f ( t( )). The derivative of this function is g ( t) f ( ( t)) ( t) f ( ( t))( ). In particular, g () f ( ())( ) f ( )( ) (4-4) Appealing to (4-2), g( t) g() t( f ( ) f ( )). Therefore g( t) g() t f ( ) f ( ) for all t (,) Section 4, page

11 4 March 28 Module : Marginal analysis single variable calculus John Riley The limit of the left h side is the derivative g (). Appealing to (4-4), g () f ( )( ) f ( ) f ( ). (4-5) QED Proposition: If where f( ), is differentiable concave on f ( ) f ( ). X [ a, b] then for any in X, Proof: If we reverse in (4-5), Therefore ( )( ) ( ) ( ). f f f ( )( ) ( ) ( ). (4-6) f f f The result then follows from (4-5) (4-6). QED Section 4, page