Economics 401. Sample questions 1

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Economics 401. Sample questions 1"

Transcription

1 Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B 3 ) = {c}; (b) X = {a, b, c},b = {B, B 2 }, B = {a, b}, B 2 = {a, b, c}, C (B ) = {a}, C (B 2 ) = {c}; (c) X = {a, b, c, d},b = {B, B 2 }, B = {a, b, c}, B 2 = {a, b, d}, C (B ) = {a, c}, C (B 2 ) = {a, d}; WARP holds in some choice structure (B, C ( )) if: Assume B, B 2 B, {a, b} B B 2 ; a C (B ) and b C (B 2 ) a C (B 2 ). It follows from this definition that WARP has no leverage unless our observations give us at least two budget sets with two elements in common say a and b AND a is amongst the best elements in one budget set and b is amongst the best elements in the other budget set. These assumptions are not met in any of these cases above so WARP is trivially satisfied. 2. MWG,.B.3 Proof: We say that u ( ) represents on choice set X if the following statement is true for all x, y X, x y if and only if u (x) u (y) To complete the proof note that if f ( ) is a strictly increasing function defined on the real numbers then for all x, y X, x y if and only if f (u (x)) f (u (y)) MWG,.B.4

2 Proof: Denote ( ) for all x, y X, x y if and only if u (x) = u (y) ( ) for all x, y X, x y if and only if u (x) > u (y) Given that ( ) and ( ) are true we need to prove the definition in MWG.B.3 holds. Note that x y implies either x y in which case u (x) > u (y) or x y in which case u (x) = u (y); combining the two options then x y implies u (x) u (y). Conversely, u (x) u (y) implies either u (x) > u (y) in which case x y or u (x) = u (y) in which case x y. So combining the two options then u (x) u (y) implies x y. { } {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, 3. Let X = {a, b, c, d}, B = and (B, C ( )) be a {c, d}, {d, b, c}, {a, b, d}, {a, c, d} choice structure. Suppose C ( ) is such that d is the best choice whenever d is available and C ({a, b}) = {a}, C ({b, c}) = {b}, and C ({c, a}) = {c}. (a) Does C ( ) satisfy WARP? (b) Is there a rational preference relation which rationalizes C ( ) on B? (c) If {a, b, c} were another budget set could C ( ) satisfy WARP? Defend your answers to (a), (b) and (c) carefully. (a) WARP holds in some choice structure (B, C ( )) if: B, B B, {x, y} B, {x, y} B : x C (B), y C (B ) x C (B ). The only way to contradict WARP here is for some problem to arise with budget sets that don t contain d. But there is none of these with two or more elements in common, so the supposition in the definition of WARP is not satisfied and thus WARP must be true. Remember, anything is true of the empty set. (b) No, because a b c but c a, so transitivity doesn t hold. (c) Try C ({a, b, c}) = {a} ; but C ({c, a}) = {c} so WARP is violated. Try C ({a, b, c}) = {b} ; but C ({a, b}) = {a} so WARP is violated. Try C ({a, b, c}) = {c} ; but C ({b, c}) = {b} so WARP is violated. So the only way to satisfy WARP is to set C ({a, b, c}) = the empty set, which is not allowed. 4. Answer both parts. (a) What does it mean to say that a utility function, u( ), represents a preference relation on some choice set X? Prove that if u( ) represents preference relation, this preference relation must be complete and transitive. 2

3 (b) Now suppose X = R 2 + and (x, x 2) (x 2, x 2 2) when x > x 2, or x = x 2 and x 2 > x 2 2. Is this preference relation complete, transitive and continuous? Defend your answers. if (a) A utility function u : X R represents a preference relation on the choice set X x, y X, x y u(x) u(y). Completeness: x, y X, u(x) and u(y) are real numbers so either u(x) u(y) in which case x y or u(y) u(x) in which case y x. Transitivity: Suppose x, y, z X, and x y and y z. We need to show x z. Since u represents, u(x) u(y) and u(y) u(z). Since these are real numbers u(x) u(z) x z. (b) Lexicographic preferences are complete and transitive but not continuous. Completeness: Consider any two distinct points in R 2 + (x, x 2) (point a) and (x 2, x 2 2) (point b). If x > x 2, a b. If x 2 > x, b a. If x = x 2, since a and b are distinct, it must be that either x 2 > x 2 2 in which case a b or x 2 2 > x 2 in which case b a. Transitivity: Let a, b, c R 2 + where a = (x, x 2), b = (x 2, x 2 2) and c = (x 3, x 3 2). Suppose a b and b c. We want to prove a c. a b either x > x 2 (call this ) or x = x 2 and x 2 > x 2 2 (call this 2). b c either x 2 > x 3 (call this 3) or x 2 = x 3 and x 2 2 > x 3 2 (call this 4). If and 3 are true then x > x 3 in which case a c. If and 4 are true then x > x 3 in which case a c. If 2 and 3 are true then x > x 3 in which case a c. If 2 and 4 are true then x = x 3 and x 2 > x 3 2 in which case a c. Thus in every possible situation a c, which is what we wanted to prove. Continuity: Suppose and {x n } R 2 + and lim n x n = x {y n } R 2 + and lim n y n = y and x n y n, n. Then if were continuous we would be able to deduce that x y. Let x n = (/n, 0) and y n = (0, ). Then x = (0, 0), y = (0, ), x n y n n but y x. So lexicographic preferences are complete and transitive but they violate continuity. 5. You are given the following information about a consumer s purchases. Goods and 2 are the only goods consumed. Year Year 2 Quantity Price Quantity Price Good Good ? 80 3

4 For what values of good 2 consumed in year 2 would you conclude: (a) that the consumer s consumption bundle in year is revealed preferred to that in year 2? (b) that the consumer s consumption bundle in year 2 is revealed preferred to that in year? (c) that her/his behaviour contradicts the weak axiom? (d) that good is an inferior good somewhere for this consumer (assume WARP holds)? (e) that good 2 is an inferior good somewhere for this consumer (assume WARP holds)? (a) Denote the consumption of good 2 in year 2 by x 2 2. If the consumer s consumption bundle in year is revealed preferred to that in year 2 then the year 2 bundle must be affordable with year prices and income. So (00) (00) + (00) (00) (00) (20) + (00) x 2 2 or x (b) If the consumer s consumption bundle in year 2 is revealed preferred to that in year then the year bundle must be affordable with year 2 prices and income. So (00) (20) + 80x 2 2 (00) (00) + (80) (00) or x (c) This requires that the inequalities in (a) and (b) both hold, so 75 x (d) x 2 2 < 75. (e) 80 < x Consider a price-taking consumer in a two-good world. Let w denote money wealth, (, ) prices, x j (,, w) Marshallian demands, h j (,, u) Hicksian demands, V (,, w) the indirect utility function and e (,, u) the expenditure function. Fill in the following tables, and verify the Slutsky equation for the effect of changing the price of good on the demand for good 2, for someone whose preferences are represented by: u = x x 2. 4

5 x (,, w) x 2 (,, w) V (,, w) Range h (,, u) h 2 (,, u) e (,, u) Range x (,, w) x 2 (,, w) V (,, w) Range w w w , > 0, w 0 h (,, u) h 2 (,, u) e (,, u) Range p /2 /2 2 u /2 /2 p /2 2 u /2 2/2 /2 2 u /2, > 0, u 0 Slutsky equation: from the duality between the UMP and the EMP we know x 2 (,, e (,, u)) = h 2 (,, u). Differentiate this with respect to and use the envelope theorem to obtain x 2 x 2 + x w = h 2 LHS = 0 + w 2 2 But w = e (,, u) = 2/2 /2 2 u /2, so LHS = 2 p /2 p /2 2 u /2 = RHS 7. Consider a price-taking consumer in a two-good world. Let w denote money wealth, (, ) prices, x j (,, w) Marshallian demands, h j (,, u) Hicksian demands, V (,, w) the indirect utility function and e (,, u) the expenditure function. Fill in the following tables for someone whose preferences are represented by: u = min (x, 2x 2 ). x (,, w) x 2 (,, w) V (,, w) Range h (,, u) h 2 (,, u) e (,, u) Range 5

6 x (,, w) x 2 (,, w) V (,, w) Range 2w 2 +, > 0, w 0 2w 2 + w 2 + h (,, u) h 2 (,, u) e (,, u) Range u u/2 u ( + /2), > 0, u 0 The Slutsky substitution matrix is S = = [ h h h 2 h 2 [ ]. ] 8. Susan is a price taker and lives in a three-good world. Denote her money wealth by w, prices by (,, ) and her consumption bundle by (x, x 2, x 3 ). Find her Marshallian demands if her preferences can be represented by u = ln x + 2 ln x 2 + x 3. x (,, w) x 2 (,, w) x 3 (,, w) Range 2 w 3 w 3 w 2 w w < 3 9. Suppose a consumer s preferences can be represented by u = 2x /2 + 2x /2 2 + x 3. Find Marshallian demands and the indirect utility function. Find Hicksian demands and the expenditure function. Be sure to specify the range over which each function is valid. Verify the Slutsky equation for the effects of a change in the price of good 2 on the demand for good 3. The Marshallian demands are: 6

7 x x 2 x 3 V (,, w) Range 3 w ( + ) 3 2 w w ( + ) 0 2 w + + w > p2 3 + p2 3 p ( ) 2 /2 w(p + ) w p2 3 + p2 3 The Hicksian demands are: ( h h 2 h ( 3 ) e(,, u) Range ( ) 3 3 u 2p u p2 3 p2 3 u > 2 + p ) 2 2 ( 2 ( ) u p 2( + )) 0 u 2 4( + u 2p ) 3 + u 2( + ) The Slutsky equation we want is Verifying we have h 3 = x 3 x 3 + x 2 w Range LHS RHS Range u > 2 ( + p2 ) p w > p2 3 + p2 3 u 2 ( + p2 ) 0 0 w > p2 3 + p Consider the utility function Assume x, x 2 > 0. u = x a x b 2, a > 0, b > 0. (a) Under what conditions is this function strictly concave? (b) Under what conditions is this function strictly quasi-concave? (c) Use your answers to (a) and (b) to discuss the relationship between concavity and quasi-concavity. The function f (x, x 2 ) = x a x b 2, a > 0, b > 0 is strictly concave only if f < 0 and f f 22 f2 2 > 0. And the function is strictly quasi-concave only if f 2 f 22 2f f 2 f 2 +f2 2 f < 0. We have 7

8 u = au x u 2 = bu x 2 a(a )u u = x 2 u 2 = abu x x 2 b(b )u u 22 = x 2 2 Thus u u 22 u 2 2 = ( ab(a )(b ) a 2 b 2) u 2 = ab( a b) u2 x 2 x 2 2 x 2 x 2 2 and = u 2 u 22 2u u 2 u 2 + u 2 2u ( ) 2 ( ) ( ) ( ) 2 au b(b )u au bu abu bu a(a )u 2 + x x 2 x x 2 2 x x 2 = abu3 (a(b ) 2ab + b(a )) x 2 x 2 2 = abu3 (a + b) x 2 x 2 2 So strict concavity holds only if a+b < but strict quasi-concavity holds for any a, b > 0. Thus strict quasiconcavity is a weaker condition than strict concavity.. Suppose a consumer s preferences can be represented by u = x + 2x /2 2. Find Marshallian demands and the indirect utility function. Find Hicksian demands and the expenditure function. Be sure to specify the range over which each function is valid. Verify the Slutsky equation for the effects of a change in the price of good 2 on the demand for good. The Marshallian demands are: 8 x 2 x 2

9 The Hicksian demands are: x x 2 V (,, w) Range w/ / / 2 w/ + / w / 0 w/ 2(w/ ) /2 w < / The Slutsky equation we want is h h 2 e(,, u) Range u 2 / / 2 u / u 2 / 0 u 2 /4 u 2 /4 u < 2 / Verifying we have h = x x + x 2 w Range LHS RHS Range u 2 / 2 / 2 / 2 + ( / 2)(/ ) w / u < 2 / 0 0 w < / 2. Answer both parts of this question. (a) Prove that the expenditure function, e(,, u) is concave in prices (, ). (b) Consider a three-good demand system where x = a + b / + c / + d / 3 x 2 = e + f / + g / + h / 3. where the demand function for the third good follows from the person s budget constraint. If this system is derived from a well-behaved utility-maximizing problem what are the restrictions on a, b, c, d, e, f, g and h? (a) Consider two sets of prices (, 2) and (, 2) and the convex combination of them (p t, p t 2) = t(, 2) + ( t)(, 2) for 0 t. Then e(,, u) is concave in prices (, ) if e(p t, p t 2, u) te(, 2, u) + ( t)e(, 2, u). Let (h, h 2) be the expenditure-minimizing bundle for prices (p t, p t 2) and utility level u. Then 9

10 e(p t, p t 2, u) = p t h + p t 2h 2 = (t + ( t) )h + (t 2 + ( t) 2)h 2 = t( h + 2h 2) + ( t)( h + 2h 2) te(, 2, u) + ( t)e(, 2, u). (b) If this system is derived from a well-behaved utility-maximizing problem then it must satisfy the budget constraint, the demands must be homogeneous of degree zero in prices and money income and the Slutsky substitution matrix must be symmetric and nsd. The demand for the third good is defined so that the budget constraint holds. Inspection of the equations for x and x 2 shows that scaling prices and income by the same positive number has no effect on the demands, so homogeneity is satisfied. Inspection of the demands for the first two goods also shows that they are independent of income so not only are they Marshallian demands but they are also Hicksian demands. Thus S = = h h S 3 h 2 h 2 S 23 S 3 S 32 S 33 b/ + d / 3 c/ + d / 3 S 3 f/ + h / 3 g/ + h / 3 S 23 S 3 S 32 S 33 Note that the budget constraint and homogeneity imply the rows and columns of S are linearly dependent so we need check only the upper 2 X 2 submatrix of S. For symmetry we must have c/ + d / 3 = f/ + h / 3,,, > 0. This means that d = h = 0 and c = f. Then S 0 means b 0 and S 22 0 means g 0. S S 22 S implies bg c 2 0. Finally, if we want the demands to be positive for all prices then a, e Consider a life-cycle consumer whose lifetime is T years and whose utility function is ln(c ) + + ρ ln(c 2) ( + ρ) ln(c T T ), where c i, i =,..., T, is real consumption in period i and ρ > 0 is the utility discount rate. Suppose the real interest rate is r. Under what conditions will her consumption-age profile (annual consumption graphed against age) be upward-sloping? Justify your answer. In an optimal consumption plan 0

11 MRS ctc t+ = + r so (+ρ) t c t (+ρ) t c t+ = + r, t =,..., T. Rewriting we have c t+ c t = + r + ρ. Thus if r > ρ, c t+ > c t and the consumption-age profile slopes upward throughout the person s life cycle.

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price

More information

CITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2

CITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2 CITY AND REGIONAL PLANNING 7230 Consumer Behavior Philip A. Viton March 4, 2015 Contents 1 Introduction 2 2 Foundations 2 2.1 Consumption bundles........................ 2 2.2 Preference relations.........................

More information

Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17

Revealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17 Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t

More information

Lecture 2: Consumer Theory

Lecture 2: Consumer Theory Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which

More information

Problem Set I: Preferences, W.A.R.P., consumer choice

Problem Set I: Preferences, W.A.R.P., consumer choice Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y

More information

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes

More information

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}. Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

More information

Indifference Curves and the Marginal Rate of Substitution

Indifference Curves and the Marginal Rate of Substitution Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and

More information

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing... Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima

More information

Advanced Microeconomics

Advanced Microeconomics Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

More information

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

More information

G021 Microeconomics Lecture notes Ian Preston

G021 Microeconomics Lecture notes Ian Preston G021 Microeconomics Lecture notes Ian Preston 1 Consumption set and budget set The consumption set X is the set of all conceivable consumption bundles q, usually identified with R n + The budget set B

More information

Economic Principles Solutions to Problem Set 1

Economic Principles Solutions to Problem Set 1 Economic Principles Solutions to Problem Set 1 Question 1. Let < be represented b u : R n +! R. Prove that u (x) is strictl quasiconcave if and onl if < is strictl convex. If part: ( strict convexit of

More information

First Welfare Theorem

First Welfare Theorem First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Demand. Lecture 3. August 2015. Reading: Perlo Chapter 4 1 / 58

Demand. Lecture 3. August 2015. Reading: Perlo Chapter 4 1 / 58 Demand Lecture 3 Reading: Perlo Chapter 4 August 2015 1 / 58 Introduction We saw the demand curve in chapter 2. We learned about consumer decision making in chapter 3. Now we bridge the gap between the

More information

Envelope Theorem. Kevin Wainwright. Mar 22, 2004

Envelope Theorem. Kevin Wainwright. Mar 22, 2004 Envelope Theorem Kevin Wainwright Mar 22, 2004 1 Maximum Value Functions A maximum (or minimum) value function is an objective function where the choice variables have been assigned their optimal values.

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4 Economics 00a / HBS 4010 / HKS API-111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with

More information

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

More information

Multi-variable Calculus and Optimization

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

More information

Chapter 21: The Discounted Utility Model

Chapter 21: The Discounted Utility Model Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal

More information

Consumer Theory: The Mathematical Core

Consumer Theory: The Mathematical Core Consumer Theory: The Mathematical Core Dan McFadden, C13 Suppose an individual has a utility function U(x) which is a function of non-negative commodity vectors x = (x 1,x,...,x N ), and seeks to maximize

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Section 3 Sequences and Limits

Section 3 Sequences and Limits Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

More information

Consumer Theory. The consumer s problem

Consumer Theory. The consumer s problem Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).

More information

Chapter 3: The effect of taxation on behaviour. Alain Trannoy AMSE & EHESS

Chapter 3: The effect of taxation on behaviour. Alain Trannoy AMSE & EHESS Chapter 3: The effect of taxation on behaviour Alain Trannoy AMSE & EHESS Introduction The most important empirical question for economics: the behavorial response to taxes Calibration of macro models

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where. Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

More information

Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11

Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11 Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11 LECTURE 1: BUDGETS AND REVEALED PREFERENCE 1.1. Quantity Discounts and the Budget Constraint Suppose that a consumer

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

More information

Deriving Demand Functions - Examples 1

Deriving Demand Functions - Examples 1 Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x

More information

Chapter 12: Cost Curves

Chapter 12: Cost Curves Chapter 12: Cost Curves 12.1: Introduction In chapter 11 we found how to minimise the cost of producing any given level of output. This enables us to find the cheapest cost of producing any given level

More information

Methods for Finding Bases

Methods for Finding Bases Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

More information

PART II THEORY OF CONSUMER BEHAVIOR AND DEMAND

PART II THEORY OF CONSUMER BEHAVIOR AND DEMAND 1 PART II THEORY OF CONSUMER BEHAVIOR AND DEMAND 2 CHAPTER 5 MARSHALL S ANALYSIS OF DEMAND Initially Alfred Marshall initially worked with objective demand curves. However by working backwards, he developed

More information

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

More information

Math 312 Homework 1 Solutions

Math 312 Homework 1 Solutions Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

More information

More is Better. an investigation of monotonicity assumption in economics. Joohyun Shon. August 2008

More is Better. an investigation of monotonicity assumption in economics. Joohyun Shon. August 2008 More is Better an investigation of monotonicity assumption in economics Joohyun Shon August 2008 Abstract Monotonicity of preference is assumed in the conventional economic theory to arrive at important

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero

More information

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0. Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

More information

Lecture 6. Inverse of Matrix

Lecture 6. Inverse of Matrix Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

More information

Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan

Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian

More information

Substitution and Income Effect, Individual and Market Demand, Consumer Surplus. 1 Substitution Effect, Income Effect, Giffen Goods

Substitution and Income Effect, Individual and Market Demand, Consumer Surplus. 1 Substitution Effect, Income Effect, Giffen Goods Substitution Effect, Income Effect, Giffen Goods 4.0 Principles of Microeconomics, Fall 007 Chia-Hui Chen September 9, 007 Lecture 7 Substitution and Income Effect, Individual and Market Demand, Consumer

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

Practice Problems Solutions

Practice Problems Solutions Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

More information

THE SECOND DERIVATIVE

THE SECOND DERIVATIVE THE SECOND DERIVATIVE Summary 1. Curvature Concavity and convexity... 2 2. Determining the nature of a static point using the second derivative... 6 3. Absolute Optima... 8 The previous section allowed

More information

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima. Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

More information

Lecture 5 Principal Minors and the Hessian

Lecture 5 Principal Minors and the Hessian Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and

More information

Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )

Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd ) (Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion

More information

Logic and Incidence Geometry

Logic and Incidence Geometry Logic and Incidence Geometry February 27, 2013 1 Informal Logic Logic Rule 0. No unstated assumption may be used in a proof. 2 Theorems and Proofs If [hypothesis] then [conclusion]. LOGIC RULE 1. The following

More information

Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10

Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary

More information

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem

More information

E(p,.Pr,v)=U*D. v =#6 SOLUTIONS. op, ap", Amherst College Department of Economics Economics 58 Fall2009. First Hour Test

E(p,.Pr,v)=U*D. v =#6 SOLUTIONS. op, ap, Amherst College Department of Economics Economics 58 Fall2009. First Hour Test Amherst College Department of Economics Economics 58 Fall2009 Name First Hour Test SOLUTIONS There are three questions on this 80 minute examination. Each is of equal weight in grading. 1. Suppose that

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

More information

2 The Euclidean algorithm

2 The Euclidean algorithm 2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

Vector and Matrix Norms

Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

Solutions to Assignment 4

Solutions to Assignment 4 Solutions to Assignment 4 Math 412, Winter 2003 3.1.18 Define a new addition and multiplication on Z y a a + 1 and a a + a, where the operations on the right-hand side off the equal signs are ordinary

More information

Consumer Theory. Jonathan Levin and Paul Milgrom. October 2004

Consumer Theory. Jonathan Levin and Paul Milgrom. October 2004 Consumer Theory Jonathan Levin and Paul Milgrom October 2004 1 The Consumer Problem Consumer theory is concerned with how a rational consumer would make consumption decisions. What makes this problem worthy

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

More information

1 Polyhedra and Linear Programming

1 Polyhedra and Linear Programming CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

More information

Lecture 2 Dynamic Equilibrium Models : Finite Periods

Lecture 2 Dynamic Equilibrium Models : Finite Periods Lecture 2 Dynamic Equilibrium Models : Finite Periods 1. Introduction In macroeconomics, we study the behavior of economy-wide aggregates e.g. GDP, savings, investment, employment and so on - and their

More information

U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off?

U = x 1 2. 1 x 1 4. 2 x 1 4. What are the equilibrium relative prices of the three goods? traders has members who are best off? Chapter 7 General Equilibrium Exercise 7. Suppose there are 00 traders in a market all of whom behave as price takers. Suppose there are three goods and the traders own initially the following quantities:

More information

A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

A set is a Many that allows itself to be thought of as a One. (Georg Cantor) Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

More information

On Lexicographic (Dictionary) Preference

On Lexicographic (Dictionary) Preference MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

1.3 Induction and Other Proof Techniques

1.3 Induction and Other Proof Techniques 4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

1 Homogenous and Homothetic Functions

1 Homogenous and Homothetic Functions 1 Homogenous and Homothetic Functions Reading: [Simon], Chapter 20, p. 483-504. 1.1 Homogenous Functions Definition 1 A real valued function f(x 1,..., x n ) is homogenous of degree k if for all t > 0

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Definition and Properties of the Production Function: Lecture

Definition and Properties of the Production Function: Lecture Definition and Properties of the Production Function: Lecture II August 25, 2011 Definition and : Lecture A Brief Brush with Duality Cobb-Douglas Cost Minimization Lagrangian for the Cobb-Douglas Solution

More information

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc. 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

More information

Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow.

Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow. Consider a consumer living for two periods, and trying to decide how much money to spend today, and how much to save for tomorrow.. Suppose that the individual has income of >0 today, and will receive

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

More information

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2

MATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2 MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we

More information

In economics, the amount of a good x demanded is a function of a person s wealth and the price of that good. In other words,

In economics, the amount of a good x demanded is a function of a person s wealth and the price of that good. In other words, LABOR NOTES, PART TWO: REVIEW OF MATH 2.1 Univariate calculus Given two sets X andy, a function is a rule that associates each member of X with exactly one member ofy. Intuitively, y is a function of x

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

Choice under Uncertainty

Choice under Uncertainty Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory

More information

Lecture 8: Market Equilibria

Lecture 8: Market Equilibria Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecturer: Bruno Codenotti Lecture 8: Market Equilibria The market setting transcends the scenario of games. The decentralizing effect of

More information

ECON 301: General Equilibrium I (Exchange) 1. Intermediate Microeconomics II, ECON 301. General Equilibrium I: Exchange

ECON 301: General Equilibrium I (Exchange) 1. Intermediate Microeconomics II, ECON 301. General Equilibrium I: Exchange ECON 301: General Equilibrium I (Exchange) 1 Intermediate Microeconomics II, ECON 301 General Equilibrium I: Exchange The equilibrium concepts you have used till now in your Introduction to Economics,

More information

Topic 7 General equilibrium and welfare economics

Topic 7 General equilibrium and welfare economics Topic 7 General equilibrium and welfare economics. The production possibilities frontier is generated using a production Edgeworth box diagram with the input goods on the axes. The following diagram illustrates

More information

Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1 Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

More information

The Market-Clearing Model

The Market-Clearing Model Chapter 5 The Market-Clearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

Mathematical Economics: Lecture 15

Mathematical Economics: Lecture 15 Mathematical Economics: Lecture 15 Yu Ren WISE, Xiamen University November 19, 2012 Outline 1 Chapter 20: Homogeneous and Homothetic Functions New Section Chapter 20: Homogeneous and Homothetic Functions

More information

max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

More information

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint. Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

More information

4.6 Linear Programming duality

4.6 Linear Programming duality 4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

More information

Cost Minimization and the Cost Function

Cost Minimization and the Cost Function Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is

More information

Profit Maximization and the Profit Function

Profit Maximization and the Profit Function Profit Maximization and the Profit Function Juan Manuel Puerta September 30, 2009 Profits: Difference between the revenues a firm receives and the cost it incurs Cost should include all the relevant cost

More information

Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

More information

Tastes and Indifference Curves

Tastes and Indifference Curves C H A P T E R 4 Tastes and Indifference Curves This chapter begins a -chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic

More information

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

More information