Project description Type your project, including equations, and submit via Blackboard by 5 pm on May 26. The recommended program for doing this is LaTeX (see http: //www.latex-project.org/) together with a graphical user interface such as TeXstudio. There are many instruction manuals and help forums for LaTeX online. Simplify mathematical expressions to be easily readable. 1 Logical analysis of an article from print media [8 marks] 1. Find an article from English-language print media that presents a chain of reasoning (attempts a logical argument). Copy the paragraphs containing the reasoning, distinguishing them from the rest of your answer. 2. Identify the basic statements that are the building blocks of the logical argument. Assign letters to the statements. 3. Identify any implicit assumptions that are important in the reasoning. If the implicit assumptions are statements, assign letters to them. If the implicit assumptions are logical connections between statements that have been assigned letters, express them in the symbolism of logic. 4. Express the entire argument in the symbolism of logic. Then simplify it as much as possible, in simple steps. 5. Say whether the argument is logically valid and why. Print media includes online (and paper) newspapers and magazines, but not blogs, forums, books etc. The media can be from any part of the world, as long as it is in English. Sources must be properly referenced and the reference must include the url if an online source is used. Use the Harvard reference stylesearch online for the correct way to cite according to it. Example texts or arguments given by the instructor cannot be used in the project, e.g. a sample posted on the Blackboard course site analyses the argument (Q P ) (P Q), so analysing an argument of this form is not acceptable in your project. 1 Students should not choose the same article as another student. To achieve this, a table will be posted on Blackboard listing for each student the interval of dates from which they should choose their article. 2 Mathematical model of education choice [8 marks] There is a worker with talent . There is a market of potential employers who are willing to pay a wage equalling the talent level. The worker knows the talent, but the employers do not. They believe talent to be normally distributed with mean and precision (which is dened as one divided by the variance) 1 = 2 . The worker has the option to make an eort to get educated. Eort is chosen from the real numbers. Eort is invisible to anyone except the worker, but higher eort leads (probabilistically) to a higher grade s. Specically, the grade is normally distributed with mean equal to the eort, and precision s . The employers can see the grade and use Bayes' rule to estimate the eort. Bayes' rule: If the previous belief about eort is normally distributed with mean e and precision e and the grade is normally distributed with mean e and precision s , then after observing grade s, the new estimate of eort is normally +s distributed with mean ee+s s and precision e + s . e Linear transformation of normal distribution: If a variable x is nor2 mally distributed with mean x and variance x , then the linearly transformed 2 variable ax + b is normally distributed with mean ax + b and variance a2 x . Suppose the employers believe that all workers choose their eort as a linear function of their talent: e = k1 + k2 . Here, k1 is a xed real number and k2 > 0. (1) Find the mean and variance of the normally distributed belief about the eort before the grade is seen. (2) Find the mean and variance of the normally distributed belief about the eort after the grade is seen. (3) Find the mean and variance of the normally distributed estimate of the talent after the grade is seen. The employers are risk-neutral and thus pay a wage equal to the mean of their belief about the talent. Wage is random, because it depends on the random grade (through the random mean of the belief). The worker's benet from the random wage is: the expected the wage times b, with b > 0. If the wage is linear in the grade, then to nd the expected wage, simply replace the grade in the wage by the mean of the grade. The worker maximizes the dierence between the expected wage and the cost of eort. The cost of eort e for a worker with talent is c(e ), with c positive, strictly increasing and strictly convex and > 0. (4) Write down the maximization problem of a worker with talent and solve it for e. Find k1 , k2 by equating the optimal e with k1 + k2 for each . Eort should turn out to be a linear function of the talent, which justies the belief of the employers that eort is linear in talent. 2 (5) How does eort change in the parameters , , , s , b? (6) Write the expected payo (benet minus cost) of the worker when eort is chosen optimally. Denote this payo by V . How does the expected payo change in the parameters , , , s , b? 3 Learning from data [6 marks] A worker would like to know how wage responds to grades. The assumption is that wage is linear in the grade (w = q1 + q2 s). (1) The worker has two observations: one person with wage w1 and grade s1 , another with dierent w2 and dierent s2 . Find q1 and q2 . (2) A third observation is added: w3 together with s3 . Assume w3 is dierent from w1 , w2 and s3 from s1 , s2 . The worker ts a line to the three observations by minimizing the sum of the squared vertical distances between the line and the data points. Write the minimization problem and solve for q1 and q2 . The vertical distance between a line y = ax + b and a point (x0 , y0 ) is |y0 ax0 b|. (3) The worker assumes that q2 0. Solve the previous minimization problem again, with this constraint added. 4 A consumer problem [8 marks] 1 The consumer maximizes the utility x + x subject to a budget 1 2 constraint p1 x1 + p2 x2 = I and nonnegativity constraints on x1 , x2 . Assume > 1, > 0, > 0, I > 0, p1 > 0, p2 > 0. (1) Solve the consumer's problem to nd x1 , x2 as a function of the prices and the parameters. Ignore second order conditions, they are too complicated. (2) Find how x1 and x2 change in the prices. (3) Take the limit as 0 of the optimal x1 , x2 you found in (1). (4) Multiply the utility by an appropriately chosen constant so that it takes 1 the form x + (1 )x with (0, 1). Express in terms of the 1 2 parameters. Take the natural log of this multiplied utility function, then take the limit as 0 using l'Hpital's rule. Solve the consumer's problem with o the limiting utility function you obtained to nd the optimal x1 and x2 . 3 Project description Type your project, including equations, and submit via Blackboard by 5 pm on May 26. The recommended program for doing this is LaTeX (see http: //www.latex-project.org/) together with a graphical user interface such as TeXstudio. There are many instruction manuals and help forums for LaTeX online. Simplify mathematical expressions to be easily readable. 1 Logical analysis of an article from print media [8 marks] 1. Find an article from English-language print media that presents a chain of reasoning (attempts a logical argument). Copy the paragraphs containing the reasoning, distinguishing them from the rest of your answer. 2. Identify the basic statements that are the building blocks of the logical argument. Assign letters to the statements. 3. Identify any implicit assumptions that are important in the reasoning. If the implicit assumptions are statements, assign letters to them. If the implicit assumptions are logical connections between statements that have been assigned letters, express them in the symbolism of logic. 4. Express the entire argument in the symbolism of logic. Then simplify it as much as possible, in simple steps. 5. Say whether the argument is logically valid and why. Print media includes online (and paper) newspapers and magazines, but not blogs, forums, books etc. The media can be from any part of the world, as long as it is in English. Sources must be properly referenced and the reference must include the url if an online source is used. Use the Harvard reference stylesearch online for the correct way to cite according to it. Example texts or arguments given by the instructor cannot be used in the project, e.g. a sample posted on the Blackboard course site analyses the argument (Q P ) (P Q), so analysing an argument of this form is not acceptable in your project. 1 Students should not choose the same article as another student. To achieve this, a table will be posted on Blackboard listing for each student the interval of dates from which they should choose their article. 2 Mathematical model of education choice [8 marks] There is a worker with talent . There is a market of potential employers who are willing to pay a wage equalling the talent level. The worker knows the talent, but the employers do not. They believe talent to be normally distributed with mean and precision (which is dened as one divided by the variance) 1 = 2 . The worker has the option to make an eort to get educated. Eort is chosen from the real numbers. Eort is invisible to anyone except the worker, but higher eort leads (probabilistically) to a higher grade s. Specically, the grade is normally distributed with mean equal to the eort, and precision s . The employers can see the grade and use Bayes' rule to estimate the eort. Bayes' rule: If the previous belief about eort is normally distributed with mean e and precision e and the grade is normally distributed with mean e and precision s , then after observing grade s, the new estimate of eort is normally +s distributed with mean ee+s s and precision e + s . e Linear transformation of normal distribution: If a variable x is nor2 mally distributed with mean x and variance x , then the linearly transformed 2 variable ax + b is normally distributed with mean ax + b and variance a2 x . Suppose the employers believe that all workers choose their eort as a linear function of their talent: e = k1 + k2 . Here, k1 is a xed real number and k2 > 0. (1) Find the mean and variance of the normally distributed belief about the eort before the grade is seen. (2) Find the mean and variance of the normally distributed belief about the eort after the grade is seen. (3) Find the mean and variance of the normally distributed estimate of the talent after the grade is seen. The employers are risk-neutral and thus pay a wage equal to the mean of their belief about the talent. Wage is random, because it depends on the random grade (through the random mean of the belief). The worker's benet from the random wage is: the expected the wage times b, with b > 0. If the wage is linear in the grade, then to nd the expected wage, simply replace the grade in the wage by the mean of the grade. The worker maximizes the dierence between the expected wage and the cost of eort. The cost of eort e for a worker with talent is c(e ), with c positive, strictly increasing and strictly convex and > 0. (4) Write down the maximization problem of a worker with talent and solve it for e. Find k1 , k2 by equating the optimal e with k1 + k2 for each . Eort should turn out to be a linear function of the talent, which justies the belief of the employers that eort is linear in talent. 2 (5) How does eort change in the parameters , , , s , b? (6) Write the expected payo (benet minus cost) of the worker when eort is chosen optimally. Denote this payo by V . How does the expected payo change in the parameters , , , s , b? 3 Learning from data [6 marks] A worker would like to know how wage responds to grades. The assumption is that wage is linear in the grade (w = q1 + q2 s). (1) The worker has two observations: one person with wage w1 and grade s1 , another with dierent w2 and dierent s2 . Find q1 and q2 . (2) A third observation is added: w3 together with s3 . Assume w3 is dierent from w1 , w2 and s3 from s1 , s2 . The worker ts a line to the three observations by minimizing the sum of the squared vertical distances between the line and the data points. Write the minimization problem and solve for q1 and q2 . The vertical distance between a line y = ax + b and a point (x0 , y0 ) is |y0 ax0 b|. (3) The worker assumes that q2 0. Solve the previous minimization problem again, with this constraint added. 4 A consumer problem [8 marks] 1 The consumer maximizes the utility x + x subject to a budget 1 2 constraint p1 x1 + p2 x2 = I and nonnegativity constraints on x1 , x2 . Assume > 1, > 0, > 0, I > 0, p1 > 0, p2 > 0. (1) Solve the consumer's problem to nd x1 , x2 as a function of the prices and the parameters. Ignore second order conditions, they are too complicated. (2) Find how x1 and x2 change in the prices. (3) Take the limit as 0 of the optimal x1 , x2 you found in (1). (4) Multiply the utility by an appropriately chosen constant so that it takes 1 the form x + (1 )x with (0, 1). Express in terms of the 1 2 parameters. Take the natural log of this multiplied utility function, then take the limit as 0 using l'Hpital's rule. Solve the consumer's problem with o the limiting utility function you obtained to nd the optimal x1 and x2 . 3 Project description Type your project, including equations, and submit via Blackboard by 5 pm on May 26. The recommended program for doing this is LaTeX (see http: //www.latex-project.org/) together with a graphical user interface such as TeXstudio. There are many instruction manuals and help forums for LaTeX online. Simplify mathematical expressions to be easily readable. 1 Logical analysis of an article from print media [8 marks] 1. Find an article from English-language print media that presents a chain of reasoning (attempts a logical argument). Copy the paragraphs containing the reasoning, distinguishing them from the rest of your answer. 2. Identify the basic statements that are the building blocks of the logical argument. Assign letters to the statements. 3. Identify any implicit assumptions that are important in the reasoning. If the implicit assumptions are statements, assign letters to them. If the implicit assumptions are logical connections between statements that have been assigned letters, express them in the symbolism of logic. 4. Express the entire argument in the symbolism of logic. Then simplify it as much as possible, in simple steps. 5. Say whether the argument is logically valid and why. Print media includes online (and paper) newspapers and magazines, but not blogs, forums, books etc. The media can be from any part of the world, as long as it is in English. Sources must be properly referenced and the reference must include the url if an online source is used. Use the Harvard reference stylesearch online for the correct way to cite according to it. Example texts or arguments given by the instructor cannot be used in the project, e.g. a sample posted on the Blackboard course site analyses the argument (Q P ) (P Q), so analysing an argument of this form is not acceptable in your project. 1 Students should not choose the same article as another student. To achieve this, a table will be posted on Blackboard listing for each student the interval of dates from which they should choose their article. 2 Mathematical model of education choice [8 marks] There is a worker with talent . There is a market of potential employers who are willing to pay a wage equalling the talent level. The worker knows the talent, but the employers do not. They believe talent to be normally distributed with mean and precision (which is dened as one divided by the variance) 1 = 2 . The worker has the option to make an eort to get educated. Eort is chosen from the real numbers. Eort is invisible to anyone except the worker, but higher eort leads (probabilistically) to a higher grade s. Specically, the grade is normally distributed with mean equal to the eort, and precision s . The employers can see the grade and use Bayes' rule to estimate the eort. Bayes' rule: If the previous belief about eort is normally distributed with mean e and precision e and the grade is normally distributed with mean e and precision s , then after observing grade s, the new estimate of eort is normally +s distributed with mean ee+s s and precision e + s . e Linear transformation of normal distribution: If a variable x is nor2 mally distributed with mean x and variance x , then the linearly transformed 2 variable ax + b is normally distributed with mean ax + b and variance a2 x . Suppose the employers believe that all workers choose their eort as a linear function of their talent: e = k1 + k2 . Here, k1 is a xed real number and k2 > 0. (1) Find the mean and variance of the normally distributed belief about the eort before the grade is seen. (2) Find the mean and variance of the normally distributed belief about the eort after the grade is seen. (3) Find the mean and variance of the normally distributed estimate of the talent after the grade is seen. The employers are risk-neutral and thus pay a wage equal to the mean of their belief about the talent. Wage is random, because it depends on the random grade (through the random mean of the belief). The worker's benet from the random wage is: the expected the wage times b, with b > 0. If the wage is linear in the grade, then to nd the expected wage, simply replace the grade in the wage by the mean of the grade. The worker maximizes the dierence between the expected wage and the cost of eort. The cost of eort e for a worker with talent is c(e ), with c positive, strictly increasing and strictly convex and > 0. (4) Write down the maximization problem of a worker with talent and solve it for e. Find k1 , k2 by equating the optimal e with k1 + k2 for each . Eort should turn out to be a linear function of the talent, which justies the belief of the employers that eort is linear in talent. 2 (5) How does eort change in the parameters , , , s , b? (6) Write the expected payo (benet minus cost) of the worker when eort is chosen optimally. Denote this payo by V . How does the expected payo change in the parameters , , , s , b? 3 Learning from data [6 marks] A worker would like to know how wage responds to grades. The assumption is that wage is linear in the grade (w = q1 + q2 s). (1) The worker has two observations: one person with wage w1 and grade s1 , another with dierent w2 and dierent s2 . Find q1 and q2 . (2) A third observation is added: w3 together with s3 . Assume w3 is dierent from w1 , w2 and s3 from s1 , s2 . The worker ts a line to the three observations by minimizing the sum of the squared vertical distances between the line and the data points. Write the minimization problem and solve for q1 and q2 . The vertical distance between a line y = ax + b and a point (x0 , y0 ) is |y0 ax0 b|. (3) The worker assumes that q2 0. Solve the previous minimization problem again, with this constraint added. 4 A consumer problem [8 marks] 1 The consumer maximizes the utility x + x subject to a budget 1 2 constraint p1 x1 + p2 x2 = I and nonnegativity constraints on x1 , x2 . Assume > 1, > 0, > 0, I > 0, p1 > 0, p2 > 0. (1) Solve the consumer's problem to nd x1 , x2 as a function of the prices and the parameters. Ignore second order conditions, they are too complicated. (2) Find how x1 and x2 change in the prices. (3) Take the limit as 0 of the optimal x1 , x2 you found in (1). (4) Multiply the utility by an appropriately chosen constant so that it takes 1 the form x + (1 )x with (0, 1). Express in terms of the 1 2 parameters. Take the natural log of this multiplied utility function, then take the limit as 0 using l'Hpital's rule. Solve the consumer's problem with o the limiting utility function you obtained to nd the optimal x1 and x2 . 3