# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

25 Pages

### suplecture2

Course: ECON ECON 7230, Spring 2010
School: Utah Valley State
Rating:

Word Count: 1306

#### Document Preview

Lecture Supplemental 2 Optimization in One Dimension Differentiability in One Dimension f is differentiable at x D iff f ( x + x) f ( x) L = lim x 0 x exists. If f is differentiable at x, L is the derivative of f at x, denoted f (x) or df(x)/dx. Differentiability implies continuity but not vice versa. II - 2 Mean-Value Theorem Let f : [a, b] R be a differentiable function. Then there must exist c (a,...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Utah >> Utah Valley State >> ECON ECON 7230

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Lecture Supplemental 2 Optimization in One Dimension Differentiability in One Dimension f is differentiable at x D iff f ( x + x) f ( x) L = lim x 0 x exists. If f is differentiable at x, L is the derivative of f at x, denoted f (x) or df(x)/dx. Differentiability implies continuity but not vice versa. II - 2 Mean-Value Theorem Let f : [a, b] R be a differentiable function. Then there must exist c (a, b) such that df (c) f (b) f (a) = . dx ba increasing. f (x) 0 for all x iff f is weakly decreasing. ac b > increasing. If f (x) 0 for all x, f must be strictly < decreasing. II - 3 Global Minima and Maxima Let x* D. x* is a global maximum of f iff, for all x D, f(x) f(x*). Likewise, x* is a global minimum of f iff, for all x D, f(x) f(x*). When optimizing the function f on D, our aim is to find a global maximum (or minimum) of f. II - 4 Local Minima and Maxima Let x* D. x* is a local maximum of f iff there exists a > 0 such that for all x D (x* , x* + ), f(x) f(x*). Likewise, x* is a local minimum of f iff there exists a > 0 such that for all x D (x* , x* + ), f(x) f(x*). II - 5 Open Sets A set S R is open if any point in S is a member of an open interval that is also contained in S. Formally, for any x S, there exists > 0 such that (x , x + ) S. II - 6 First-Derivative Test If f : D R is differentiable, where D is open, and x* D is a local maximum of f then f (x*) = 0. The first step to find a maximum of a differentiable function f is solve for roots of f . This test cannot distinguish between local maxima and local minima. A root of f need not be a minimum or maximum. Some authors refer to a root of f as an extremum. II - 7 Cubic Function For example, let f(x) = x3. The derivative is f (x) = 3x2, which has a root at x = 0. But x = 0 is neither a minimum nor a maximum of f. 1 0.75 0.5 0.25 -1 -0.5 -0.25 -0.5 -0.75 -1 0.5 1 II - 8 Second Derivative Test Let f be twice continuously differentiable. If x* D satisfies f (x*) = 0, then if f (x*) < 0, x* is a local maximum. if f (x*) > 0, x* is a local minimum. If f (x*) = 0, the second derivative test does not tell us what kind of extremum x* is, and you must look at higher-order derivatives. II - 9 Global Optimization The first and second derivative tests provide ways of finding local minima or maxima. There is no simple way to find a global optimum. Generally, you must go through all local optima and see which has the best value of f. However, if f has certain properties, there will be a unique local maximum (or minimum). II - 1 0 Existence of Optima In the following, we will assume that f : [a, b] R. If f is continuous, then there will exist xmin and xmax [a, b] such that for all x [a, b] f ( xmin ) f ( x) f ( xmax ). II - 1 1 Convexity f is convex iff for all x, y [a, b] and for all [0, 1], f (x + (1 ) y ) f ( x) + (1 ) f ( y ). If, in addition, for all (0, 1) x y f (x + (1 ) y ) < f ( x) + (1 ) f ( y ), we say that f is strictly convex. II - 1 2 Concavity Likewise, f is concave iff for all x, y [a, b] and for all [0, 1], f (x + (1 ) y ) f ( x) + (1 ) f ( y ). If, in addition, for all (0, 1) x y f (x + (1 ) y ) > f ( x) + (1 ) f ( y ), we say that f is strictly concave. II - 1 3 Preservation Under Addition Let f and g be concave functions on [a, b]. Let x, y [a, b] and ( 0 , 1 ). f (x + (1 ) y ) f ( x) + (1 ) f ( y ) g (x + (1 ) y ) g ( x) + (1 ) g ( y ) If we define (f + g)(z) = f(z) + g(z) for [a, z b], ( f + g )(x + (1 ) y ) ( f + g )( x) + (1 )( f + g )( y ). Convexity, strict convexity, and strict concavity are also preserved. II - 1 4 Convexity of Maximal Set for a Concave Function Let f : [a, b] R be concave and continuous. Let S = {x [a, b] : (y [a, b])[f(x) f(y)]}. S is denoted arg max f. We know S . Then S is a convex set, meaning that if x < y S then, for all z (x, y), z S. Note that f(x) must be the same for all x S. II - 1 5 Uniqueness of Maximum for a Strictly Concave Function If f is strictly concave, S must be a singleton. Suppose x, y S and x < y. Then x + y f ( x) + f ( y ) f = f ( x). > 2 2 But this contradicts x and y both being in S. II - 1 6 First-Derivative Test and Concavity Let f : [a, b] R be concave and differentiable. If x* [a, b] satisfies f (x*) = 0, then x* is a global maximum. Likewise if f is convex and f (x*) = 0, then x* is a global minimum. II - 1 7 Taylors Theorem A function is called Cn if its nth-order derivative exists and is continuous. Let f be Cn+1 for x (x0 , x0 + ) for some x0 R and > 0. For all x (x0 , x0 + ), there exists a z between x and x0 such that n 1 (i ) 1 i f ( x) = f ( x0 )( x x0 ) + f ( n +1) ( z )( x x0 ) n +1. (n + 1)! i = 0 i! II - 1 8 Convexity, Concavity, and The Second Derivative Let f : [a, b] R be twice differentiable. f is everywhere nonnegative iff f is convex. If f is everywhere positive, f is strictly convex. f is everywhere nonpositive iff f is concave. If f is everywhere negative, f is strictly concave. II - 1 9 Strict Concavity And the Second Derivative You might expect that strict concavity would imply a strictly negative second derivative. This is not true. f(x) = 4 is strictly x concave. But f (x) = x2, so 12 f (0) = 0. -1 -0.5 -0.2 -0.4 -0.6 -0.8 -1 0.5 1 II - 2 0 To What Extent Is the Strict Converse True? With more advanced methods, you can show that if f is strictly concave, then {x [a, b] : f ' ' ( x) = 0} is a set of measure zero. That is, f (x) < 0 almost everywhere. Likewise if f is strictly convex. II - 2 1 Optimization on Non-Open Sets If f is differentiable on an open set D, we have seen that, at a local optimum x*, f (x*) = 0. But most optimization is not done on open sets. For example, we usually require consumption of goods to be nonnegative, not strictly positive. We have a theorem that says global optima must exist on closed and bounded sets, but not open sets. If D is not open, we can have f (x*) 0 at a local or global optimum x* D. II - 2 2 A Corner Solution Suppose you are maximizing f(x) = x on [0, 1]. We have f (x) = 1 for all x [0, 1]. There is no point in [0, 1] that satisfies the first-derivative test. 1 0.8 Yet clearly x* = 1 is a global maximum. This is a corner solution. 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 II - 2 3 Constrained Optimization Suppose you are maximizing f : [a, b] R, where f is differentiable. An interior solution x* (a, b) must satisfy f (x*) = 0. There are two possible corner solutions: x* = a must satisfy f (a) 0. x* = b must satisfy f (b) 0. Constrained optimization is more difficult in multiple dimensions. a b II - 2 4 Summary 1. If f is a differentiable function, An interior maximum must satisfy f = 0. A boundary maximum can satisfy weaker conditions. 2. Second-order conditions are needed to verify that an interior extremum is an optimum. 3. The maxima of a concave function will form a convex set, of a strictly concave function a singleton set. II - 2 5
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Utah Valley State - ECON - ECON 7230
Supplemental Lecture 3 Differential EquationsDifferential Equations Let f : Rn R. The equation d n x(t ) dx(t ) d n 1 x(t ) = f x(t ), , , n n 1 dt dt dt is an nth-order differential equation.III - 2Complete Specification To complete the problem, w
Utah Valley State - ECON - ECON 7230
6543Column B Column B210 0 20 40 60 80 100 1200 0 20 40 60 80 100 1201.351.31.251.2 Column C Column C 1.151.11.051 0 20 40 60 80 100 1201 0 20 40 60 80 100 120ALPHA N 0.33G 0.01 0.02 0K/Y 3C/YDELTA K* 0.75 0.06Y* 5.2 1.73C* 1.3T 0
Utah Valley State - ECON - ECON 7230
HW #Anvar Suyundikov2. a) OUTPUT Diagram: If theta is less that 5, output returns to zero but takes longer to return as theta increases. If theta is equal to five, it goes horizontal at 2. If theta is ten capital continues increasing and does not return
Utah Valley State - ECON - ECON 7230
Macro Prelim 2010Each question should be answered with a clear and concise explanation. Yes-no questions should be answered with a supportive argument or counterexample. Please write legibly. 1) Consider the following continuous-time dynastic model. Each
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010MacroEcon Theory IIProfessor: James Feigenbaum B620 J.Feigen@aggiemail.usu.edu 4:00 PM 5:15 PM Mondays and Wednesdays B318 1:00 2:00 PM Mondays and Wednesdays, or by appointmentClass Time: Office Hours:Course Description: At l
Utah Valley State - ECON - ECON 7230
SUMMARYOUTPUT RegressionStatistics MultipleR 0.96 RSquare 0.92 AdjustedRSquare 0.92 StandardError 22.06 1 Observations 36 ANOVA Regression Residual Totaldf1 34 35SS MS # # # 14899.11 #F SignificanceF 400.34 0Intercept MCoefficients StandardError tSt
Utah Valley State - ECON - ECON 7230
1.210.80.6 A K C Y R0.40.20-0.2 0 20 40 60 80 100 120ALPHA0.33 G0.02 K/Y3 C/Y0.8 DELTALAMBDA1 Q0 ETACK+ ETACKT 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 A1.04 LAMBDA2 0.07 Q1 -0.48 ETAKK+
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010Homework 1 Solutions1) a) The public knows xt, so the public is minimizing2 L p = E[( xt + ~t wt ) 2 ] = ( xt wt ) 2 + E[ ~t ]. z zThis has the first-order condition wt = xt. The Phillips curve then becomes u t = U n z t . The
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010Homework 2 Solutions1) a) Denote the transition matrix Pr[ t +1 = l | t = l ] Pr[ t +1 = h | t = l ] 1 p p = = . Pr[ t +1 = l | t = h ] Pr[ t +1 = h | t = h ] p 1 p One way to compute the powers of a matrix is to diagonalize it
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010Homework 4 Solutions1) We augment the Consumption RBC Model with Cobb-Douglas production and CRRA utility by including government spending financed by lump-sum taxes. Let Tt be the tax at t with t = ln Tt = t0 + t1. We assume th
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010Homework 6 Solutions1) Let f : R R be a continuously differentiable function with a fixed point x*. Suppose that |f (x*)| &lt; &lt; 1. a) There exists &gt; 0 such that if x R and 0 &lt; |x x*| &lt; then f ( x) f ( x*) f ' ( x*) &lt; f ' ( x*) . x
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010Homework 8 Solutions1) Consider a model where members of the labor force maximize T E t ~t , y t =0 where yt is the income received at t and (0, 1). In each period, labor force members are either employed with wage w or unemploy
Utah Valley State - ECON - ECON 7230
ECN/APEC 7240Spring 2010Homework 9 Solutions1) Consider an economy with a single consumer. There is one good in the economy, which arrives in the form of an exogenous endowment obeying y t +1 = t +1 y t , where yt is the endowment at time t and and t+1
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 1Due 1/20/10 1) Kydland and Prescott (1977). Consider a model of inflation and unemployment where unemployment ut is determined by the Phillips curve u t = U n ( t wt ) , where Un is the natural rate of unemployment, &gt;
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 2Due 1/3/10 Let us now further generalize the CKR model with technology shocks. We introduce a government that purchases goods Gt using a distortionary income tax on both capital and labor income with rate t. Thus we ha
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 3Due 2/10/10 1) We have shown that the Consumption RBC Model with = 1, u(C) = ln C, and F(K, L) = K L1 then C ( K , A) = (1 ) K A1 . Show that our log-linear approximation to the consumption function is exact in this ca
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 4Due 2/24/10 1) We augment the Consumption RBC Model with Cobb-Douglas production and CRRA utility by including government spending financed by lump-sum taxes. Let Tt be the tax at t with t = ln Tt = t0 + t1. We assume
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 5Due 3/1/10 1) Let x, y Rn. We define the d1, d2, and d metrics such that d 1 ( x, y ) = x i y ii =1 nn d 2 ( x, y ) = ( x i y i ) 2 i =1 d ( x, y ) = max i xi y i .1/ 2Note that the d2 metric corresponds to our usu
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 6Due 3/31/10 1) Let f : R R be a continuously differentiable function with a fixed point x*. Suppose that |f (x*)| &lt; &lt; 1. a) Use Taylors Theorem to show there exists a neighborhood U of x* such that if we restrict f to
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 7Due 4/7/10 1) Consider the corresponding planners problem associated with the production model, in which, given Y0, the planner maximizest =0tS (Yt , Yt +1 ) ,where S (Y , Y ' ) = A0Y 1 d A1Y 2 (Y Y ' ) 2 . 2 2a)
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Homework 9Due 4/28/10 1) Consider an economy with a single consumer. There is one good in the economy, which arrives in the form of an exogenous endowment obeying y t +1 = t +1 y t , where yt is the endowment at time t and and t
Utah Valley State - ECON - ECON 7140
Lecture 1 Introduction to Modern MacroeconomicsPer Capita GNPECN/APEC 7240I-2Income-Expenditure Identity Y = C + I + G = zF(K, N) C = C(K, Y) K = I + (1 )K = zF ( K , N ) C ( K , zF ( K , N ) G + (1 ) K Key question: What are F and C? F is determine
Utah Valley State - ECON - ECON 7140
Lecture 2 Technology Shocks and the Business CycleBusiness Cycle Regularities Real GDP fluctuates irregularly. But other economic variables have similar fluctuationsthey comove with GDP. With respect to qualitative behavior of comovements among series,
Utah Valley State - ECON - ECON 7140
Lecture 3 Technology Shocks and ConsumptionThe Household There is a representative household that lives forever. The household maximizes t E u (Ct ). t =0 Utility is CRRA with risk aversion . Discount factor (0, 1). Since future income is uncertain,
Utah Valley State - ECON - ECON 7140
Lecture 4 Log-Linearizing the Consumption RBC ModelOur Approach There is no exact analytic solution to the Consumption RBC Model. We can solve for the balanced growth path the model converges to in the absence of technology shocks. We can also solve for
Utah Valley State - ECON - ECON 7140
Lecture 5 Endogenizing LaborEndogenizing Labor Thus far we have always assumed that households supply labor inelastically. One unit of time corresponds to a forty-hour work week. We endogenize the labor choice by assuming leisure is also a good that h
Utah Valley State - ECON - ECON 7140
Lecture 6 Metric SpacesMetric Spaces A metric space is a space X endowed with a metric d. A function d : X X R+ is a metric iff it satisfies the following properties: For all x, y X, d(x, y) = 0 iff x = y. (positive definiteness) For all x, y X, d(x, y
Utah Valley State - ECON - ECON 7140
Lecture 7 Contraction Mapping TheoremContraction Mappings Let (X, d) be a metric space and T : X X. We say that T is a contraction mapping iff for some (0, 1), for all x, y X,d (T ( x), T ( y ) d ( x, y ). We call the modulus of T.ECN/APEC 7240VII -
Utah Valley State - ECON - ECON 7140
Lecture 8 Recursive Competitive EquilibriumLinear Regulator Problem Let the state space X = Rn and suppose the choice space is U = Rk. Given x0, the problem istT T max [ xt Rxt + ut Qut ] t =0 subject to xt+1 = Axt + But. R is n n and positive semide
Utah Valley State - ECON - ECON 7140
Lecture 10 Search ModelsFinding a Trading Partner An important real-world friction is that buyers and sellers often have to find each other before they can trade. Job seekers have to search for a good match among job opportunities. Employers have to se
Utah Valley State - ECON - ECON 7140
Lecture 10 Complete MarketsStates of the World Consider an infinite-horizon model where in each period t the state of the world is described by a stochastic variable st S. Let st denote the history (s0, . . . , st). The probability of realizing st is t(
Utah Valley State - ECON - ECON 7140
Lecture 11 Asset PricingAssets and Contingent Claims A claim on consumption at time t conditional on history st is known as a contingent claim. An asset can be viewed as a bundle of contingent claims. If markets are complete, the price of all these cont
Utah Valley State - ECON - ECON 7140
Lecture 12 Equity-Premium PuzzlePreferences Consider a representative-agent model where the consumer maximizes t ~ Et u (ct + j ). j =0 The period utility function u has the usual probabilities and (0, 1). Et is the expectation operator conditional o
Utah Valley State - ECON - ECON 7140
ECN/APEC 7240Spring 2010Practice FinalEach question should be answered with a clear and concise explanation or proof. You may assume any theorems or results that have been discussed in class unless the question specifically asks you to prove that resul
Utah Valley State - ECON - ECON 7140
Cobb-Douglas Example for Market Equilibrium and Social OptimumI. Input\$Ontext Cobb-Douglas Example for 7130 \$Offtext \$Offsymxref Variables x1c, x2c, xj, y1, y2, lam, mu; Equations foc1, foc2, foc3, foc4, foc5, mc1, mc2; foc1 . 0.5*Sqrt(y1/x1c)-mu =e= 0;
Utah Valley State - ECON - ECON 7140
Cobb-Douglas Example for Pareto Efficiency (A)I. InputVariables x1c, x2c, xj, y1, y2, lam1, lam2, lam3; Equations foc1, foc2, foc3, foc4, foc5, mc1, mc2, mc3; foc1 . 0.5*Sqrt(y1/x1c)-lam3 =e= 0; foc2 . -0.5*lam1*Sqrt(y2/x2c)-lam3 =e= 0; foc3 . 0.5*Sqrt(
Utah Valley State - ECON - ECON 7140
Cobb-Douglas Example for Pareto Efficiency (B)I. InputVariables x1c, x2c, xj, y1, y2, lam1, lam2, lam3; Equations foc1, foc2, foc3, foc4, foc5, mc1, mc2, mc3; foc1 . 0.5*Sqrt(y1/x1c)-lam3 =e= 0; foc2 . -0.5*lam1*Sqrt(y2/x2c)-lam3 =e= 0; foc3 . 0.5*Sqrt(
Utah Valley State - ECON - ECON 7140
Economics 7130Microeconomic Theory IAutumn Semester, 2009 Course Description: This course covers the same areas in microeconomics as in your introductory principles and intermediate theory courses, but at a more rigorous graduate level. We begin by exam
Utah Valley State - ECON - ECON 7140
Homework Assignment 1Econ 7130Instructions: Please provide your answers to the following questions in the spaces beneath each question. Make sure to clearly show your work and explain your answers for full credit. There are a total of 50 points possible
Utah Valley State - ECON - ECON 7140
Homework Assignment 2Econ 7130Instructions: Please provide your answers to the following questions in the spaces beneath each question. Make sure to clearly show your work and explain your answers for full credit. There are a total of 50 points possible
Utah Valley State - ECON - ECON 7140
Quasi-Linear Example for Pareto Efficiency (A)I. Input\$Ontext Quasi-linear Example for 7130 \$Offtext Variables x1c, x2c, xj, y1, y2, lam1, lam2, lam3; Equations foc1, foc2, foc3, foc4, foc5, foc6, foc7, foc8; foc1 . 1/x1c =e= lam3; foc2 . 1 =e= lam2; fo
Utah Valley State - ECON - ECON 7140
Quasi-Linear Example for Pareto Efficiency (B)I. Input\$Ontext Quasi-linear Example for 7130 \$Offtext Variables x1c, x2c, xj, y1, y2, lam1, lam2, lam3; Equations foc1, foc2, foc3, foc4, foc5, foc6, foc7, foc8; foc1 . 1/x1c =e= lam3; foc2 . 1 =e= lam2; fo
Utah Valley State - ECON - ECON 7140
640ORDINARY DIFFERENTIAL EQUATIONS: SCALAR EQUATIONS[24]=To find (10) as the solution, write the differential equation as y - a(t)y b(t) and multiply every term by exp(- J&quot;' 0 ) a(s) 5 :Since the left-hand side of (11) is precisely the derivative of
Utah Valley State - ECON - ECON 7140
Utah Valley State - ECON - ECON 7140
LyonWheat Storage Home-Work Problem This is a wheat storage problem. Even though, it is incomplete, it is complicated enough for a home-work problem. We are working with a closes economy with perfect forsight. We start the problem at t = 0 with the harve
Utah Valley State - ECON - ECON 7140
Lyon, January, 2000 1.Econ 701Will a shift from a wage system which pays only for hours worked in a mine lead to a different earning that one based upon portal-to-portal pay? A company cannot have a monopoly if its shareholders receive only the normal r
Utah Valley State - ECON - ECON 7140
Lyon 1.Discussion Questions 1.A In a competitive industry, some firms might be more efficient than others on average, but all firms are equally efficient at the margin. T, F, U. Explain.2.Average cost is rather more popular in economics, and deserves f
Utah Valley State - ECON - ECON 7140
Discussion questions: 1. It has been observed that the best grades of products (oranges, apples and the like) are sent to large cities and are not readily available to consumers in the areas in which they are produced. Explain why. Friedman calls the hori
Utah Valley State - ECON - ECON 7140
LYON Answer True, False, or Uncertain and briefly explain why. 1. The invention of a synthetic substitute for wool will raise the price of mutton. 2. In the United States, real income per capita has risen over time, whereas the number of domestic servants
Utah Valley State - ECON - ECON 7140
Discussion Questions #3 1. The research department of a price searcher firm estimated (assume correctly) that the elasticity of demand for its product is -5, and -2 in markets A and B, respectively. Both are estimated at the price-quantity combination in
Utah Valley State - ECON - ECON 7140
Discussion questions for March 23: 1. Discuss the paradox of voting using the following: Individual 1: cPb, bPa Individual 2: bPa, aPc Individual 3: aPc, cPb 2. Explain what a &quot;Social Welfare Function&quot; is. 3. An acceptable Social Welfare Function must sat
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 First ExamFebruary 2, 20061.a. Show that if y1/L = y0 then the change in income is equal to Slutskys compensating variation of price, where yi is an individuals income in year i, and L is the individuals Laspeyres price index. b. Show th
Utah Valley State - ECON - ECON 7140
Lyon 1.Econ 7140January 31, 2008Using the profit maximization problem for a price-taker firm, where there are n inputs, show (prove) that a.My* Mx* 1 ) = ! ) Mw1 Mpb. CanMy* Mx* 1 ) = ! ) \$ 0 ? Mw1 MpProve yea or nay (yes or no). Explain. c. An inp
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 First ExamFebruary 18, 20101.Let the primal problem be to maximize profits for a price-taker firm. Use the primal-dual problem to show that the solution profit function is convex in w and p. What does this imply about the resource demand
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 Second ExamFebruary 27, 20041.We analyze the welfare loss of placing sales taxes on the commodities x1, and x2. The representative consumer has utility function and transformation function: u(x1, x2) G(x1, x2) = 0 u , C2, u strictly quas
Utah Valley State - ECON - ECON 7140
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 Second ExamMarch 2, 20061. The individuals utility function U ( x1 , x2 , x3 , x4 ) is strictly quasi-concave, where commodities 1 and 2 are food items and 3 and 4 are clothing items. Show how you would go about proving that we can combin
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 Third ExamApril 4, 20061. Individual A has the utility function U ( x ) = ln x for income (the scale of x is \$10, 000). The individual has a job offer that pays \$40,000 with a bonus. The bonus will be 0 or \$50,000 with equal probability.
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 Third ExamApril 3, 20081. For the utility functions U(x ) = ln x and V (x ) = x a. Show they are CRRA utility functions. Explain. b. TFU, At any level of x , U exhibits more risk aversion than does V . Remember that the explanation is the
Utah Valley State - ECON - ECON 7140
LyonEcon 7140 Final ExamApril 29, 20081. Let there be 2 private goods, 2 inputs, 2 strictly concave production functions, fixed endowments of the inputs, no &quot;externalities,&quot; and price-taker individuals and firms. Show (prove) that the market equilibriu
University of Phoenix - BA - 531
1 Alternative Dispute ResolutionAlternateDisputeResolutionClause MariamIbrahim LAW/531 December6,2010 MarleneF.Wilhite2 Alternative Dispute ResolutionTheteamcharteristheguidelinesthatteamsestablish,whichassistsassemblethe rulesstraightforwardforeveryon