COU 7: A Fully Parameterized Model of the Choice to Run

In this COU, you’ll analyze a version of the model of a potential candidate’s choice to run in which all three of the numerical values in the model – i.e. the potential candidate’s ideal point, the current candidate’s ideal point and the utility cost the potential candidate must bear to run – are parameterized.

Here’s the model:

A Fully Parameterized Model of the Choice to Run

A potential candidate chooses between two alternatives: to run or not run in an upcoming election
A current candidate is already running in the election, and is the only person other than the potential candidate who will run.
If the potential candidate does not run, the current candidate wins the election with certainty.
If the potential candidate runs, the potential candidate wins the election with certainty.
If the current candidate wins the election, he sets policy to the location of his ideal point, which is C.
If the potential candidate wins the election, she sets policy to the location of her ideal point, which is a number P.

Given the location x of policy that ultimately results from the election and the potential candidate’s choice to run or not run, the potential candidate’s utility level is

\begin{array}{ccc} \underbrace{ \begin{Bmatrix} x-P & \text{if $x < P$} \\ 0 & \text{if $x = P$} \\ P-x & \text{if $P < x$} \end{Bmatrix} } & - & \underbrace{ \begin{Bmatrix} R & \text{if the potential candidate runs} \\ 0 & \text{if the potential candidate does not run} \end{Bmatrix} } \\ \text{Policy Term} & & \text{Cost Term} \end{array} where R is a positive number that depicts the personal costs the potential candidate must bear to run.

Part 1

First, you’ll analyze the model for the case in which P \leq C – i.e. the potential candidate’s ideal point P lies to the left of the current candidate’s ideal point.

Prompt 1A

Assume P \leq C. Compute the expression for the potential candidate’s utility level if she runs. Show the steps you went through to compute it by first writing the utility function, with the two terms labeled as the “Policy Term” and the “Cost Term”. Then show the expressions for the values of each of these terms that result when the potential candidate runs. Then show the expression for the overall utility level.

Prompt 1B

Assume P \leq C. Compute the expression for the potential candidate’s utility level if she does not run. Show the steps you went through to compute it by first writing the utility function, with the two terms labeled as the “Policy Term” and the “Cost Term”. Then show the expressions for the values of each of these terms that result when the potential candidate does not run. Then show the expression for the overall utility level.

Prompt 1C

Using the expressions you derived in response to Prompts 1A and 1B, write the inequality that defines the conditions under which the potential candidate’s utility level from running is greater than or equal to her utility level from not running. Then, re-arrange the terms of this inequality so that the potential candidate’s ideal point P is by itself of one side of the inequality. After doing that, describe the steps you used to re-arrange the inequality. (e.g. “First I added 4 to both sides, then I multiple both sides by \frac{3}{16}.”)

Rubric for Part 1

Prompts 1A and 1B are each worth 1 point. On each, you get 1 point if you give the correct expression and show the steps through which you derived it as specified in the prompts, and 0 points otherwise.

On Prompt 1C you get 1 point if you write the initial inequality asked for, rearrange it as required (showing any intermediate steps or not), and describe a steps that correctly produces the final inequality from the initial inequality. You get 0 points otherwise.

Part 2

Now you’ll analyze the model for the case in which P > C – i.e. the potential candidate’s ideal point P lies to the right of the current candidate’s ideal point.

Prompt 2A

Assume P > C. Compute the expression for the potential candidate’s utility level if she runs. Show the steps you went through to compute it by first writing the utility function, with the two terms labeled as the “Policy Term” and the “Cost Term”. Then show the expressions for the values of each of these terms that result when the potential candidate runs. Then show the expression for the overall utility level.

Prompt 2B

Assume P > C. Compute the expression for the potential candidate’s utility level if she does not run. Show the steps you went through to compute it by first writing the utility function, with the two terms labeled as the “Policy Term” and the “Cost Term”. Then show the expressions for the values of each of these terms that result when the potential candidate does not run. Then show the expression for the overall utility level.

Prompt 2C

Using the expressions you derived in response to Prompts 2A and 2B, write the inequality that defines the conditions under which the potential candidate’s utility level from running is greater than or equal to her utility level from not running. Then, re-arrange the terms of this inequality so that the potential candidate’s ideal point P is by itself of one side of the inequality. After doing that, describe the steps you used to re-arrange the inequality. (e.g. “First I added 4 to both sides, then I multiple both sides by \frac{3}{16}.”)

Rubric for Part 2

Prompts 2A and 2B are each worth 1 point. On each, you get 1 point if you give the correct expression and show the steps through which you derived it as specified in the prompts, and 0 points otherwise.

On Prompt 2C you get 1 point if you write the initial inequality asked for, rearrange it as required (showing any intermediate steps or not), and describe a steps that correctly produces the final inequality from the initial inequality. You get 0 points otherwise.

Part 3

Prompt 3A

Using your answers to Prompts 1C and 2C, draw a diagram similar to the one that appears in the lesson just above the link to this COU. Unlike that diagram, this one should show the implications of the fully parameterized model in this COU for the potential candidate’s choice between running and not running. Like that diagram, it should show possible locations of P ranging from the left of the current candidate’s ideal point C to the right. It should also delimit and label the ranges of values of P where the potential candidate prefers to run instead of not run, and the ranges of values of P where the potential candidate prefers to not run instead of run. It should also use labels that show the exact expressions (in terms of the parameters C and R) for the exact limits of these ranges.

Prompt 3B

If you correctly answered Prompts 1C and 2C and correctly drew the diagram for Prompt 3A, then you know that the lower and upper limits range of values of P over which the potential candidate prefers to run instead of not run are each functions of the current candidate’s ideal point C and the personal cost of running R. Draw a graph showing how these limits change as C is held constant and R ranges from 0 to some arbitrarily chosen maximum value. Specifically, the graph should show various values of R on the horizontal axis from 0 on the left to your arbitrarily chosen maximum value on the right. Then, the graph should plot two functions of R – (a) the lower limit of the range of values of P at which the potential candidate prefers to run for office instead of not run and (b) the upper limit of the range of value of P at which he potential candidate prefers to run for office instead of not run. Note that at R = 0 these two limits are equal to one another. Make sure to label the value of these two limits along the vertical axis to show that you have correctly set the levels of the two functions at R = 0.

Rubric for Part 3

On Prompt 3A you get 1 point if you draw a diagram that correctly depicts the correct answer from Prompt 3, and includes all of the features required by the prompt. You get 0 points otherwise.

On Prompt 3B you get 1 point if you correctly draw the graph asked for by the prompt, including all the features required. You get 0 points otherwise.