The big picture and a math review

Transcription

1 The big picture and a math review Intermediate Micro Topic 1 Mathematical appendix of Varian

2 The big picture Microeconomics Study of decision-making under scarcity Individuals households firms investors Limitations to what they are able to choose

3 Graphs Level sets Solving Calculus Optimization Example problem: setting the alarm clock Deciding when to wake up I Scarcity/limitation I I I 1440 minutes each morning (Midnight - noon) Can work Sleep Multivariate calculus

4 Graphs Level sets Solving Calculus Optimization Example problem: setting the alarm clock Deciding when to wake up I Preferences I I I Want lots of time to sleep and to get ready Some pairs of (Sleep,Work) make you happier than others Choose feasible pair that makes you happiest Multivariate calculus

5 Functions Function f : x y, y = f (x) Rule that takes an input and gives an output Usually inputs and outputs are numbers (or sets of numbers) Given input x, f gives a single output y

6 Functions Function f : x y, y = f (x) One-to-One function: every output y is only reached by one input x Inverse function: the inverse function of one-to-one function y = f (x) is found by solving for x; x = g(y) Independent variable x: the input to a function Dependent variable y: the output of a function Domain The set of values of inputs the function applies to Range The set of values of outputs the function may give

7 Examples of functions Function f : x y, y = f (x) f (x) = 1440 x f (x) = ln(x) f (x) = x 2 3x + 2 f (x, y) = x 2 + y 2 f (x, y) = x y

8 Examples of functions Function f : x y, y = f (x) Linear function: function that can be written as f (x 1,..., x n ) = c 0 + c 1 x c n x n Additively separable function: function that can be written in the form f (x 1,..., x n ) = g 1 (x 1 ) g n (x n )

9 Graphs A graph shows values of independent and dependent variables given by the function y = 1440 x z = x y

10 Graphs Continuous function: The graph does not have gaps or jumps Smooth function: A continuous function that has no corners Piecewise continuous function: The graph is continous over portions of the domain

11 Level sets Graph works well for 1-input functions y = f (x) Not for 2-input functions z = f (x, y) Level set: let z be constant z = c Solve c = f (x, y) for y (if possible) Graph for several values of c c = z = x y y = ( ) c 2 x

12 Level sets of a linear function f (x, y) = 8 + 2x y Level sets are the same color on the 3D graph

13 Solving systems of equations An equation need only hold for some value(s) of the variables (x + 1) 2 = 1 A solution is an x such that an equation is true x = 2 or x = 0 An identity is true for all values of the variables (x + 1) 2 x 2 + 2x + 1

14 Solving systems of equations A system of equations can often be solved (or at least simplified) using algebraic substitution A system of n equations in n unknowns may have 0,1, or many solutions, even

15 Slope between two points Start at some value of x Add x Slope is called rate of change of y wrt x slope = (y+ y) y (x+ x) x slope = f (x+ x) f (x) x y = f (x) slope = rise run = y x

16 Slope at a single point What happens as x shrinks to zero? Get slope at a point, or slope of the tangent line This is equivalent to the derivative of f at x f (x) is smooth slope = f (x+ x) f (x) x

17 Derivatives review Ways of writing the derivative of f wrt x at x df (x) f (x) **Note that x is a variable, a is a constant** f (x) ax x a df (x) a ax a 1 ln(x) 1 e x x e x

18 Derivatives review df (x) = d f (x) = g(x) + h(x) dg(x) (g(x) + h(x)) = + dh(x) df (x) Product rule f (x) = g(x) h(x) = g(x) dh(x) + h(x) dg(x)

19 Derivatives review df (x) = Quotient rule f (x) = g(x) h(x) dg(x) h(x) h(x) 2 g(x) dh(x) df (x) Chain rule f (x) = h(g(x)) = dh(y) dy dg(x), y = g(x)

20 Unconstrained optimization FOC y = f (x), continuous and smooth Optimization: Finding maximum or minimum values of f (x) f (x) increasing when f (x) > 0 f (x) decreasing when f (x) < 0 Maximum must be at a point where x = 0 First order condition for maximizing f (x): f (x ) = 0

21 Unconstrained optimization SOC y = f (x), continuous and smooth If x is a (local) maximum f (x) > 0 for x slightly less than x f (x) < 0 for x slightly greater than x Second order condition for maximizing f (x): f (x ) < 0

22 Unconstrained optimization SOC y = f (x), continuous and smooth When f (x ) < 0, x is a (local) maximum When f (x ) > 0, x is a (local) minimum When f (x ) = 0, we do not know if x is a local maximum without checking nearby values of x

23 Other cases Constrained optimization is optimization with limitations on x, eg x min x x max FOCs give candidate points Check boundary values of x If f (x) is not smooth, check kink points Find maximum among f (x FOC ), f (x max ), f (x min ), f (x kink )

24 Derivatives of functions with multiple inputs f (x 1, x 2 ) Works like derivatives of a single-input function Get effect of small change in x1 Hold x 2 constant Written f (x 1,x 2 ) x 1 = f x1 (x 1, x 2 ) = f 1 (x 1, x 2 ) Example: f (x, y) = 25 x 2 xy + 2x e y f x (x, y) = 2 2x y f y (x, y) = x e y

25 Finding the slope of a level set f (x, y) = c 1. If possible, solve for y = g(x, c) dy = g x(x, c) 2. Otherwise, dy = fx (x,y) f y (x,y) Effect of small changes and dy along level set: f = f x + f y dy Level set means f = 0 Solve for dy

26 Unconstrained optimization with multiple inputs Same idea as with single input Solution is where small change in x or y neither increases or decreases f (x, y) FOC: f x (x, y) = 0, f y (x, y) = 0 SOC math is complicated maxf (x, y), smooth, continuous

27 Constrained optimization with multiple inputs Graphically 1. Find maximal level set of f that does not violate g(x, y) = at some pairs (x, y ) 4. (at one pair (x, y ) where the two graphs are tangent) max {x,y} f (x, y) s.t. g(x, y) = 0

28 Constrained optimization with multiple inputs Algebraically: 1. Find dy in terms of x and y according to the slope of the level sets of f 2. Find dy in terms of x and y according to the slope of constraint g(x, y) = 0 3. Solve out dy from your two equations above 4. Using the above result, and constraint g(x, y) = 0, solve for x, y max {x,y} f (x, y) s.t. g(x, y) = 0

29 Example: Setting the alarm clock Algebraically: 1. dw dw 2. ds max {s,w} s w s.t s w = 0 ds = w 0.5s 1 w = 2w s = w = s w w = 0 5. (s, w ) = (960, 480)