Overview Definitions Mathematical Properties Properties of Economic Functions Exam Tips. Midterm 1 Review. ECON 100A - Fall Vincent Leah-Martin

Transcription

1 ECON 100A - Fall UCSD October 20, vleahmar@uscd.edu

2 Preferences We started with a bundle of commodities: (x 1, x 2, x 3,...) (apples, bannanas, beer,...)

3 Preferences We started with a bundle of commodities: (x 1, x 2, x 3,...) (apples, bannanas, beer,...) We then suppose a consumer can rank every bundle: (2, 2, 2,...) (1, 1, 1,...) (1, 1, 0,...)

4 Utility Representation Now we assign a number to each bundle such that for any bundles A and B: A B u(a) > u(b)

5 Utility Representation Now we assign a number to each bundle such that for any bundles A and B: A B u(a) > u(b) This yields a utility function u(x 1, x 2,...). The function is ordinal because the it outputs doesn t matter as long as the order of ranking bundles is the same.

6 The Consumer Problem Given this utility function, we want to model how a consumer behaves. We say that a consumer wants to maximize their utility - which just means they want whatever bundle they most prefer: max x1,x 2,...u(x 1, x 2,...)

7 The Consumer Problem Given this utility function, we want to model how a consumer behaves. We say that a consumer wants to maximize their utility - which just means they want whatever bundle they most prefer: max x1,x 2,...u(x 1, x 2,...) However, they are restricted by the price of each commodity and their income: p 1 x 1 + p 2 x = I

8 First Order Conditions Solving this problem yields that for any goods i and j: MRS i,j = MU i MU j = p i p j

9 First Order Conditions Solving this problem yields that for any goods i and j: MRS i,j = MU i MU j = p i p j MU 1 p 1 = MU 2 p 2 or =...for all goods

10 First Order Conditions Solving this problem yields that for any goods i and j: MRS i,j = MU i MU j = p i p j or MU 1 = MU 2 =...for all goods p 1 p 2 If these conditions are not satisfied then the consumer can do better by buying more of whatever good gives him more marginal utility per dollar.

11 Demand Functions With some algebra we can use the first order conditions and the budget constraint to solve for the optimal value of each good as a function of prices and income: x (p 1, p 2,...I )

12 Indirect Utility The last thing we did was find a function for the most utility the consumer can get given prices and income:

13 Indirect Utility The last thing we did was find a function for the most utility the consumer can get given prices and income: u(x 1 (p 1, p 2,...I ), x 2 (p 1, p 2,...I ),...) V (p 1, p 2,...I )

14 Elasticity Definition Elasticity a measure how a percent change in an independent variable affects a dependent variable in percentage terms.

15 Elasticity Definition Elasticity a measure how a percent change in an independent variable affects a dependent variable in percentage terms. Formula ɛ y,x = y x x y

16 Indifference Curves Definition Indifference curves are a plot of all the bundles which the consumer is indifferent between.

17 Indifference Curves Definition Indifference curves are a plot of all the bundles which the consumer is indifferent between. Mathematically They are a collection of points (x 1, x 2 ) for which u(x 1, x 2 ) = c for any given constant c.

18 Marginal Rate of Substitution What it is... The MRS is the slope of the level curve of a function at a given point (x 1, x 2 ).

19 Marginal Rate of Substitution What it is... The MRS is the slope of the level curve of a function at a given point (x 1, x 2 ). Formula MRS 1,2 = MU 1 MU 2 = u(x 1,x 2 ) x 1 u(x 1,x 2 ) x 2

20 Demand Functions Mathematically The demand function is the optimal value of a choice variable as a function of parameters.

21 Demand Functions Mathematically The demand function is the optimal value of a choice variable as a function of parameters. In Economics xi (p 1, p 2,..., I ) is the amount of good i I consume when faced with these prices and income.

22 Demand Functions Mathematically The demand function is the optimal value of a choice variable as a function of parameters. In Economics xi (p 1, p 2,..., I ) is the amount of good i I consume when faced with these prices and income. Note: Hold all p j for j i and I constant, then you can plot the demand function with x i on the x-axis and p i on the y-axis. This is the normal demand curve from ECON 1.

23 Engel Curve Definition The Engle curve for good x i is the plot of income and x i with x i on the x-axis and I on the y-axis.

24 Engel Curve Definition The Engle curve for good x i is the plot of income and x i with x i on the x-axis and I on the y-axis. This tells us if you buy more or less of a good when income increases.

25 Engel Curve Definition The Engle curve for good x i is the plot of income and x i with x i on the x-axis and I on the y-axis. This tells us if you buy more or less of a good when income increases. Note: You can also plot an Engel curve with the axes switched. On the exam if it is not explicitly stated, ask which variable is on which axis.

26 Indirect Utility The question in words... What is the most utility a consumer can get give prices and income?

27 Indirect Utility The question in words... What is the most utility a consumer can get give prices and income? The answer in math... V (p 1, p 2,..., I ) u(x 1 (p 1, p 2,...I ), x 2 (p 1, p 2,...I ),...)

28 Economic Bads Definition We say a commodity x i is an economic bad if MU i < 0.

29 Economic Bads Definition We say a commodity x i is an economic bad if MU i < 0. Intuition The more the consumer consumes of x i the less happy he is.

30 Economic Bads Definition We say a commodity x i is an economic bad if MU i < 0. Intuition The more the consumer consumes of x i the less happy he is. Implication If a good is an economic bad, the consumer will try to consume as little of it as possible (typically 0).

31 Monotonicity Monotonically Increasing A function is monotonically increasing in x if for every x > x, f (x ) f (x).

32 Monotonicity Monotonically Increasing A function is monotonically increasing in x if for every x > x, f (x ) f (x). Monotonically Decreasing A function is monotonically decreasing in x if for every x > x, f (x ) f (x).

33 Monotonicity Monotonically Increasing A function is monotonically increasing in x if for every x > x, f (x ) f (x). Monotonically Decreasing A function is monotonically decreasing in x if for every x > x, f (x ) f (x). How does this relate to the derivative?

34 Homogeneity (Scale Properties) Homogeneous of Degree 0 A function is homoegeneous of degree 0 if for every λ R: f (λx) = f (x)

35 Homogeneity (Scale Properties) Homogeneous of Degree 0 A function is homoegeneous of degree 0 if for every λ R: f (λx) = f (x) Homogeneous of Degree 1 A function is homoegeneous of degree 1 if for every λ R: f (λx) = λf (x)

36 Homogeneity (Scale Properties) Homogeneous of Degree 0 A function is homoegeneous of degree 0 if for every λ R: f (λx) = f (x) Homogeneous of Degree 1 A function is homoegeneous of degree 1 if for every λ R: f (λx) = λf (x) Homogeneous of Degree k A function is homoegeneous of degree k if for every λ R: f (λx) = λ k f (x)

37 Strictly Diminishing MRS Intuitive Definition The slope of the indifference curve gets flatter as you move down the indifference curve (increasing x 1 and decreasing x 2 )

38 Strictly Diminishing MRS Intuitive Definition The slope of the indifference curve gets flatter as you move down the indifference curve (increasing x 1 and decreasing x 2 ) Implication Take any two points on the indifference curve and the consumer will prefer any linear combination of those two points (see picture).

39 Cobb-Douglas Utility u(x 1, x 2 ) = Ax α 1 x β 2 Must have x 1 > 0 and x 2 > 0 Satisfies diminishing MRS One good could be inferior (not both)

40 Leontief Utility { x1 u(x 1, x 2 ) = min A, x } 2 B Must have x 1 > 0 and x 2 > 0 Positive marginal utility for x i only if consuming too much of x j Also known as perfect complements Always optimize at kinks x 1 A = x 2 B Slope of optimization line (line going through kinks) is A B

41 Linear Utility u(x 1, x 2 ) = αx 1 + βx 2 Unless certain conditions hold, will consume none of one good (corner solution) Constant marginal utility for both goods Also known as perfect substitutes

42 Demand Functions x i (p 1, p 2, I ) Must be homogeneous of degree 0 (pure inflation has no effect on demand) x i p i 0 in most cases Income Properties xi I xi I 0 Normal or superior good < 0 Inferior good Income Elasticity Properties ɛ xi,i > 1 Superior good ɛ xi,i [0, 1] Normal good ɛ xi,i < 0 Inferior good

43 Indirect Utility Functions V (p 1, p 2, I ) Must be homogeneous of degree 0 V p i 0 V 0 I

44 Recommended Problems At this point you should have done a substantial number of problems. Here are a handful I think might be useful to make sure you understand: Elasticity (pg 26-29): 2, 3, 5, 6, 8, 9, 14, 21, 32 Level Curves (pg 29): 1, 3, 4 Constrained Optimization (pg 29-32): 1, 3, 6, 8, 10, 15 Comparative Statics of Solutions Functions (pg 32-34): 6 Consumer Preferences (pg 35-38): 1, 2, 4, 7, 15, 28, 32 Utility Maximization and Demand Functions (pg 38-43): 1, 3, 5, 13, 14, 16, 22, 24, 35 Comparative Statics of Demand (pg 44-53): Don t worry about this section.

45 Last Minute Check Do you have a solid understanding of how optimization with two goods works graphically? Do you know what happens graphically when I change a parameter in the consumer s problem? Can you verbally explain all of the definitions? Can you verbally explain the first order conditions? Can you explain the economic intuition for every mathematical concept?

46 Advice Final Office Hours Today, ECON 300, 2pm-4pm - I will go over any questions you have from the packet or on the concepts.

47 Advice Final Office Hours Today, ECON 300, 2pm-4pm - I will go over any questions you have from the packet or on the concepts. Monday Night Once you feel like there is nothing more you can do, stop worrying about the exam and do something relaxing.

48 Advice Final Office Hours Today, ECON 300, 2pm-4pm - I will go over any questions you have from the packet or on the concepts. Monday Night Once you feel like there is nothing more you can do, stop worrying about the exam and do something relaxing. During the Exam Relax, look at the problem. First identify what the problem is talking about then apply what you know about that topic. Nothing will require a very long answer so if you re writing a lot or doing very messy math you might be doing something wrong.

49 Good luck! Don t panic.