Group HW02

Directions

  1. All answers have to be handwritten using pen and physical paper. Scan your answers and then upload as a pdf file to Animospace. Complete solutions with a sufficient level of explanation are preferred over incomplete ones.
  2. Only one person from the group submits the answers. Make sure that the group agrees to who should submit the answers.
  3. The deadline is June 26, 2024 before 10:00 am.
  4. You are already part of a group and the collaboration is only allowed within your group. That is the extent of collaboration that is allowed.
  5. Each member of the group should submit an evaluation form for the group homework.
  6. Read and consult the late policy in the syllabus if you have a tendency to be late with respect to submissions.
  7. Every group member has a responsibility to oneself and to the other members of their group. Hold each other accountable and give credit where it is due.

Exercises

  1. A firm wants to estimate their cost function. Their cost function is a function of quantity produced \(q\geq 0\). It has fixed costs of forty thousand pesos in rent and overhead regardless of the level of \(q\). The firm believes that when \(q = 1000\), marginal cost equals 200 pesos and that when \(q = 5000\), marginal cost is 600 pesos. The firm believes that marginal cost can be approximated by a straight line.

    1. (1 point) Determine the marginal cost function given the information so far.
    2. (1 point) Using your knowledge of integration, determine what is the total cost function given your preceding answer.
    3. (2 points) At what level of \(q\) is average cost at a minimum? Show that you have indeed found a minimum.

  2. Assume that \(t\in \mathbb{R}\) instead of \(t\geq 0\) in Section 8.2 Example 1. Use the given information about \(c(t)\) and \(c^{\prime}(t)\) for this exercise.

    1. (1 point) Find stationary points of \(c(t)\).
    2. (2 points) Show that \[c^{\prime\prime}(t)=\frac{2t(t^2-12)}{(t^2+4)^3}.\] Use this information to find possible inflection points of \(c(t)\).
    3. (2 points) Show that \[c(t)=\frac{1/t^2}{1+4/t^2}\] and find \(\lim_{t\to\infty} c(t)\) and \(\lim_{t\to -\infty} c(t)\).
    4. (1 point) At what points do the function \(c(t)\) cross the horizontal axis? the vertical axis?
    5. (5 points) Create a table with column headers “Interval”, “Sign of \(c^\prime(t)\)”, “Sign of \(c^{\prime\prime}(t)\)”, “Remarks”.
    6. (1 point) Use all of the preceding information to create a decent sketch (by hand) of \(c(t)\).

  3. Consider the function in Section 8.1, Example 1(b).

    1. (1 point) What happens when you try finding stationary points?
    2. (3 points) Assume that \(x\geq 5\). Does the extreme value theorem guarantee the existence of both a maximum and a minimum? If it can guarantee the existence of both a maximum and a minimum, find them. If it cannot, suggest a modification which will enable you to find both a maximum and minimum and find them.

  4. The distance between a fixed point \((x_1,y_1)\) from another arbitrarily selected point \((x,y)\) is defined as \[\sqrt{(x-x_1)^2+(y-y_1)^2}.\] Consider the problem of finding the point on \(y=2x+5\) which is closest to the origin.

    1. (2 points) Define a squared distance function \(s(x)\) as a function of \(x\) alone which uses the definition of distance and incorporates the information in the problem.
    2. (3 points) Find the point \((x,y)\) which is closest to the origin. Make sure to check that the squared distance is at its smallest.

  5. The price \(P\) per unit obtained by a firm in producing and selling \(Q\) units is \(P=a+bQ^2\) with \(Q\geq 0\). The cost of producing and selling \(Q\) units is \(C=\alpha+\beta Q\).

    1. (2 points) What assumptions should we impose on the signs of \(a\), \(b\), \(\alpha\), \(\beta\) in order for us to interpret \(P=a+bQ^2\) as a demand function and \(C=\alpha+\beta Q\) as a total cost function?
    2. (2 points) Find the only stationary point of the profit function. Under what additional assumptions on \(a\), \(b\), \(\alpha\), \(\beta\) can you guarantee that \(Q\geq 0\)?
    3. (2 points) Under what additional assumptions on \(a\), \(b\), \(\alpha\), \(\beta\) can you guarantee that the stationary point is indeed a maximum?

  6. Let \(f(x)=\left(e^{2x}+4e^{-x}\right)^2\).

    1. (2 points) Show that \(f\) is convex over \(x \in \mathbb{R}\).
    2. (2 points) Does \(f\) have inflection points? Show why or why not.

  7. A firm produces \(Q=f(L)\) units of a commodity when \(L>0\) units of labor are employed. Assume that the price obtained per unit is 1 and price per unit of labor is \(w>0\). Let \(\pi(L)\) be the profit function. Assume that \(f^\prime*(L) >0\) and \(f^{\prime\prime}(L)>0\) for all \(L>0\).

    1. (1 point) What is the first-order condition for maximizing profits at \(L=L^*\)?
    2. (2 points) Use implicit differentiation on the first-order condition in order to find an expression for \(dL^*/dw\). Approximately how much does the optimal amount of labor change and in what direction when wages increase by a unit?
    3. (2 points) Derive \(d\pi(L^*) / dw\). What is the sign of the resulting expression? Provide an interpretation of your finding.