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 minimum point of the function \(f(x,y)=x^2+y^2-6x+8y+35\) over \(\mathbb{R}^2\) is \((x,y)=(3,-4)\). You are to show that this is the case in the following two ways:

    1. (3 points) No calculus is needed for this item. Use the procedure called completing the square to show that \(x^2+y^2-6x+8y+35\) could be written as \((x-3)^2+(y+4)^2+10\). Explain why \((x,y)=(3,-4)\) will minimize \(f(x,y)\) given its new form.
    2. (3 points) Note 2 in page 467 of EMEA states the definition of convex and concave functions. Note that \(\mathbb{R}^2\) is a convex set. Show that \(f(x,y)\) is convex.

  2. Let \(f(x,y) = 3x^4+3x^2y-y^3\).

    1. (3 points) Find the stationary points of \(f\).
    2. (3 points) Classify these as local maximum, local minimum, saddle point, or “cannot tell”.
    3. (2 points) Consider the trace of the function at \(y=0\). Given this information, determine if it is possible to find global maxima.

  3. Consider Section 13.2 Exercise 7.

    1. (2 points) Convert the minimization problem involving three variables into a minimization problem involving two variables only. There are some choices to be made here, so try to choose the easiest approach if you could. You may have to introduce some new notation here to help you communicate your solution.
    2. (8 points) Find the smallest value of \(x^2+y^2+z^2\) and make sure to show that you have indeed achieved the minimum.
    3. (5 points) Extend the setting in the exercise to the case where instead of \(4x+2y-z=5\), we have \(ax+by+cz=d\) with \(a,b,c,d\) are all not equal to zero. Find the smallest value of \(x^2+y^2+z^2\) and make sure to show that you have indeed achieved the minimum. Do we need to impose additional restrictions on the values of \(a,b,c,d\) to guarantee a minimum?

  4. Suppose that a firm has a production function of the form \(Q=L^a K^b\). It faces output price \(p>0\) and input prices for labor \(L\) and capital \(K\) denoted by \(w>0\) and \(r>0\), respectively.

    1. (3 points) Solve the first-order conditions for a profit maximizing input bundle.
    2. (4 points) Use the second-order conditions to determine the values of the parameters \(a\) and \(b\) for which this solution has a local maximum.
    3. (1 point) Connect your previous finding to your basic microeconomics. What kind of production functions are required for profit maximization

  5. (7 points) Pick up where I finished from the video related to solving Section 13.5 Exercise 3. Complete the remaining parts of the solution and solve the maximization problem. Organize your solution.

  6. Suppose the production function is given by \(Q=K^a+L^b\), for parameters \(a, b, c \in (0,1)\). It faces output price \(p>0\) and input prices for labor \(L\) and capital \(K\) denoted by \(w>0\) and \(r>0\), respectively.

    1. (2 points) Assuming that a maximum exists, find the values fo \(K\) and \(L\) that maximize the firm’s profits.
    2. (2 points) Let \(\pi^*\) denote the optimal profit as a function of the three prices. Compute the partial derivative \(\partial \pi^*/\partial r\).
    3. (2 points) Verify the envelope theorem in this case.