Homework 6: Present Value

We sought out a soothsayer, who did sayeth some sooth. She stirred her cauldron and foresaw that terrible things would happen to Evanston. 100 years from this very day, the crimes of John Evans will come back to punish the residents of this town, causing $300 million dollars of damages. However, we can avert this terrible fate at the low, low cost of just $6 million dollars today (paid to descendants of those Evans wronged). That’s right, for just $6 million dollars now, we can avert $300 million dollars of damage to future Evanston residents! You can’t beat this deal!

1. What is the most we would be willing to pay to avert this future harm if our discount rate is 1.4% per year?

2. What is the most we would be willing to pay to avert this future harm if our discount rate is 4% per year?

3. What is the most we would be willing to pay to avert this future harm if our discount rate is 10% per year?

Suppose that we could buy a bit of Evanston lakefront for $130 million and build a lovely public beach that would deliver social benefits of $5 million dollars per year forever, starting one year from now.

4. What is the most we would be willing to pay to build this park if our discount rate is 1.4% per year?

5. What is the most we would be willing to pay to build this park if our discount rate is 4% per year?

6. What is the most we would be willing to pay to build this park if our discount rate is 10% per year?

7. Think of the basic Pigouvian Externality situation.

Private Marginal Benefit = 600 - 2*Q

Private Marginal Cost = 30 + Q

Marginal Damage = 90

Private market equilibrium quantity = Q_P = (600-30)/(2+1) = 190

What is the optimal Pigouvian tax and socially optimal quantity?

8. Same setup as in the previous problem, except that the Marginal Damage doesn’t occur now, but will actually happen in 10 years. Let the discount rate be 3%.

What is the optimal Pigouvian tax and socially optimal quantity today?

9. Same setup as in the previous problem, except we just had an election, and so now the discount rate is 7%. What is the optimal Pigouvian tax now? What is the optimal social quantity today?

10.1Document for Analysis: Poor Persuasive Request Inviting Speaker to Discuss Seven Cardinal Sins in Food Service

(L.O. 1–3)

The following letter from a program chair strives to persuade a well-known chef to make a presentation before a local restaurant association. But the letter is not very persuasive. How could this message be more persuasive? What reader benefits could it offer? What arguments could be made to overcome resistance? How should a persuasive message conclude?

Your Task Analyze the following invitation and list its weaknesses, and write a revision.

Current date

Ms. Danielle Watkins

The Beverly Hills Hotel

9641 Sunset Boulevard

Beverly Hills, CA 90210

Dear Ms. Watkins:

We know you are a very busy hospitality professional as chef at the Beverly Hills Hotel, but we would like you to make a presentation to the San Francisco chapter of the National Restaurant Association. I was asked to write you since I am program chair.

I heard that you made a really good presentation at your local chapter in Los Angeles recently. I think you gave a talk called “Avoiding the Seven Cardinal Sins in Food Service” or something like that. Whatever it was, I'm sure we would like to hear the same or a similar presentation. All restaurant operators are interested in doing what we can to avoid potential problems involving discrimination, safety at work, how we hire people, etc. As you well know, operating a fast-paced restaurant is frustrating—even on a good day. We are all in a gigantic rush from opening the door early in the morning to shutting it again after the last customer has gone. It's a rat race and easy to fall into the trap with food service faults that push a big operation into trouble.

Enclosed please find a list of questions that our members listed. We would like you to talk in the neighborhood of 45 minutes. Our June 10 meeting will be in the Oak Room of the Westin St. Francis Hotel in San Francisco and dinner begins at 7 p.m.

How can we get you to come to San Francisco? We can only offer you an honorarium of $200, but we would pay for any travel expenses. You can expect a large crowd of restaurateurs who are known for hooting and hollering when they hear good stuff! As you can see, we are a rather informal group. Hope you can join us!

Sincerely,

R-Code

3. A rival music streaming company wishes to make inference for the proportion of individuals in the United States who subscribe to Spotify. They plan to take a survey. Let S1, . . . , Sn be the yet-to-be observed survey responses from n individuals, where the event Si = 1 corresponds to the ith individual subscribing to Spotify and the event Si = 0 corresponds to the ith individual does not subscribe to Spotify (i = 1, . . . , n). Assume that S1, . . . , Sn are i.i.d. Bernoulli(π).

a) What distribution does the random variable S = sum of Si from i = 1 to n have? Compute E(S) and var(S). The formulas should involve π and n.

(b) Suppose that n = 30 and π = 0.2. Run a Monte Carlo simulation with m = 10000 replications to verify the formulas for E(S) and var(S) from the previous question. That is, simulate 10000 i.i.d. copies of S and compare the observed average of these to the true mean, and the observed (sample) variance to the true variance. Comment.

(c) Let S¯ = S(n ^−1) = (n ^−1)*sum of Si from i=1 to n. What is the mean and variance of S¯?

(d) Verify your answers to the previous question by a Monte Carlo simulation with m = 10000 replications.

(e) Is S¯ a continuous random variable? Explain.

(f) Run a Monte Carlo simulation to estimate the probability P(S¯− 1/ √ n ≤ π ≤ S¯ + 1/ √ n) when π = 0.2 and n = 10, 20, 80, 160. Hint: For every n considered, do the following m = 10000 times: generate a random variable S˜ with the same distribution as S¯ and record whether |S˜−0.2| ≤ 1/ √ n. The Monte Carlo estimate of the desired probability is the number of times this happened divided by the total number of simulations, m = 10000.

y'=y-x^2 ; y(1)= -4

My MATLAB program won't work. I am trying to get the main program to output a plot of the three equations (1 from the main program and two called in the function). The goal is to code a Euler method and a 2nd order Taylor numerical solution for

a. x0= 1.0 , step size h= 0.2, # of steps n=20

b. x0= 1.0 , step size h=0.05 , # of steps n=80 ; write a separate functionn for f(x,y) that is called. Plot the results on the same plot as the exact solution.

I keep getting an error of "Matrix Dimensions must agree ; error in Project_2(my function) with my Tay = ... equation (2nd order taylor equation).

Main Code

t_span = 1:0.2:5;
h=0.2;
y1 = -4;
B=(t_span.^2);
[x,y] = ode45(@(x,y) y-x^2, t_span, y1);
d=[x,y];
project_2(y1,h,d,B)

subplot(4,1,1)
plot(x,y)

xlabel('value of x')
ylabel('value of y(x)')
grid on

t_span = [1:0.05:5];
y1 = -4;
h=0.05;
[x,y]= ode45(@(x,y) y-x^2, t_span, y1);
subplot(4,1,4)
project_2(y1,h,d,B)
plot(x,y)
xlabel('value of x')
ylabel('value of y(x)')
grid on

The Function

function [outputArg,Tay] = project_2(y1,h,d,B)

outputArg = y1 + h*d; %Euler method

Tay= y1 +(h*d)+((1/2)*(h^2))*((y1-2*t_span)+(-B)*d); %2nd order Taylor

subplot(4,1,2)
plot(outputArg)

subplot(4,1,3)
plot(Tay)

end

3. A rival music streaming company wishes to make inference for the proportion of individuals in the United States who subscribe to Spotify. They plan to take a survey. Let S1, . . . , Sn be the yet-to-be observed survey responses from n individuals, where the event Si = 1 corresponds to the ith individual subscribing to Spotify and the event Si = 0 corresponds to the ith individual does not subscribe to Spotify (i = 1, . . . , n). Assume that S1, . . . , Sn are i.i.d. Ber(π).

(a) What distribution does the random variable S = Pn i=1 Si have? Compute E(S) and var(S). The formulas should involve π and n.

(b) Suppose that n = 30 and π = 0.2. Run a Monte Carlo simulation with m = 10000 replications to verify the formulas for E(S) and var(S) from the previous question. That is, simulate 10000 i.i.d. copies of S and compare the observed average of these to the true mean, and the observed (sample) variance to the true variance. Comment. 1

(c) Let S¯ = n ⁻¹S = n ⁻¹ Pn i=1 Si . What is the mean and variance of S¯?

(d) Verify your answers to the previous question by a Monte Carlo simulation with m = 10000 replications.

(e) Is S¯ a continuous random variable? Explain.

(f) Run a Monte Carlo simulation to estimate the probability P(S¯− 1/ √ n ≤ π ≤ S¯ + 1/ √ n) when π = 0.2 and n = 10, 20, 80, 160. Hint: For every n considered, do the following m = 10000 times: generate a random variable S˜ with the same distribution as S¯ and record whether |S˜−0.2| ≤ 1/ √ n. The Monte Carlo estimate of the desired probability is the number of times this happened divided by the total number of simulations, m = 10000.

Consider the population described by the probability distribution shown in the table. The random variable x is observed twice. If these observations are independent, all the different samples of size 2 and their probabilities are shown in the accompanying table. Complete parts a through e below.

a.) Find the sampling distribution of s^2. Type the answers in ascending order for s^2

(type as integers or decimals)

b. Find the population variance:

c.) Find the sampling distribution of the sample standard deviation.

2. A 13.2kg block is on a ramp, with an incline of θ

(a) If the ramp is frictionless what is the magnitude of the acceleration of the block down the ramp?

(b) If the ramp has a coefficient of static friction of µs = 0.3, at what angle θ will the block start to move? Imagine like the friction lab, you slowly increase the incline of the ramp.

(c) Does the angle found in b) depend on mass? (d) What is the work done by the Normal force on the block if it travels 1.00m down the ramp?

How's the economy? A pollster wants to construct a 99.5% confidence interval for the proportion of adults who believe that economic conditions are getting better.

(a) A poll taken in July 2010 estimates this proportion to be 0.3 Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.02?

B. Estimate the sample size needed if no estimate of p is available.