instruction: From your personal experience, identify two related actions (e.g., shopping and cooking, sleeping and studying, cardio and strength training, etc) you had to choose or are frequently choosing. Write down a constrained optimization problem with a quadratic objective function, at least one constraint, two choice variables, and with at least one parameter. Make sure that the problem captures essential aspects of your choice - What are you optimizing? Is "more always better"? Are there marginal or increasing costs associated with these two activities? How do your choices interact? Do they reinforce each other? What constraints do you face in making your choice? Solve the optimization problem you have set up. Find at least one comparative static and interpret your formal results in the context of the original setting.

Questions

1. A concise description of the real-world observation. This should include in plain English the decision(s) to be made, what you care(d) (e.g., cost, well-being, performance), and what else you were considering (e.g., convenience, physical and mental health, the opportunity cost of time).
2. Set-up of the model. This should include your objective function and at least one constraint. You should specify whether it is a min or a max problem and what the choice variables are. Your set-up must include at least one parameter.
3. Explanation that the set-up of your objective function captures your real-world decision problem. This should include an explanation of what the variable and parameter names mean, which real-world quantities they represent, and how essential features of the observation are captured in the model. Include in this discussion any interaction terms (or their absence) and explain why you choose the functional form (linear, log, quadratic, etc) for each term in your objective function and in your constraint.
4. Solving the model. This should include a guided solution of the optimization problem and include sufficiently many intermediate steps (setting up the Lagrangian, showing your FOCs, etc), so an interested peer can easily follow along. Make sure to discuss boundary points if that's appropriate (e.g., you cannot sleep a negative number of hours) and provide an argument or a verification that the solution you found is indeed a max (or min) for your optimization problem.
5. Statement of the formal solution of the model. Here, you should state the formal solution of the model, clearly identifying it as the optimal solution. If you found more than one solution, include all solutions.
6. A comparative static result with respect to your constraint. This part should briefly address how your realized outcome (e.g.,happiness, grade, well-being, cost incurred, etc) would change if your constrained was relaxed by one unit (e.g., if your budget increased by $1, if you had 1h more to spend, etc). Compare your answer with the value of your Lagrangian multiplier you found in part 4.
7. A comparative static result with respect to at least one parameter. This part should briefly address how your optimal choice and your realized outcome (e.g.,happiness, grade, well-being, cost incurred, etc) would change if the parameter you included in your optimization problem changed. Hint: You may want to consult section 14.7. The purpose of part 7 is for you to demonstrate mastery of the Envelope theorem for constrained optimization.
8. A discussion that interprets the formal result and the comparative static result in light of the original real-world observation. This section should translate the formal results (i.e. equations) back to plain-English text and interpret them in the context of your original decision-making setting. Compare the predictions of the formal result with your actual decision-making outcome. Note: The formal result may or may have captured features of your actual decision-making outcomes. To the extent that it did, describe how and in what form. To the extent that it did not, explain what was not captured. Discuss why this may be the case. Feel free to speculate how the model would need to be adjusted to more accurately capture your decision-making outcome.