ECON 300 - Intermediate Microeconomics
Spring 2024
Problem Set 3
Due: May 2nd, 2024
This problem set is due on Thursday, May 2nd at 11:59 p.m. Late work will not be accepted without prior notification with a valid reason. The maximum score is 80 points. Be sure to review the Syllabus for details about Problem Sets and their grading. Feel free to contact me or via e-mail if you have specific questions about the assignment. Note that some exercises have several parts, and each part may conceal more than one task for you. Be sure to answer every question thoroughly for full credit!
You need to submit your responses on Canvas. Please read the following instructions (fol- lowing the instructions will help you get a better grade).
. I recommend you to work on a tablet or printout this problem set, so you can use the empty spaces I provided to answer the questions.
. Present all final answers and show any relevant calculations neatly, and please make sure your responses can always be clearly identified.
. I recommend you to highlight or circle your final answers and important steps in questions that need calculations.
. Please do NOT show your scratchwork.
. Please remember to attribute help received and collaboration.
1. (12 points; 3 points each) Below are production functions that turn capital (K) and labor (L) into output. For each of the production functions below, state and PROVE whether it is Constant/Increasing/Decreasing Returns to Scale. That is, you want to see how production changes when you increase all inputs (K,L,(M)) by a factor of α , where α > 1.
2. (10 points) For this question you will analyze the elasticity of substitution and the isoquant graphs for the following two production functions.
(a) (2 points) Graph the isoquant for F1 (K, L) that represents an output of 8. Be sure to show your work and label the axes clearly.
(b) (1 point) What is the marginal rate of technical substitution for F1 ?
(c) (2 points) What is the elasticity of substitution for F1 ?
(d) (2 points) Graph the isoquant for F2 (K, L) that represents an output of 8. Be sure to show your work and label the axes clearly.
(e) (1 point) What is the marginal rate of technical substitution for F2 ?
(f) (2 points) What is the elasticity of substitution for F2 ?
3. (18 points) A Brazilian baker produces brigadeiros (small chocolate candies) using ounces of chocolate c and cans of condensed milk m with a production function
F(c,m) = min[3c,27m]
This production function says that in order to make 27 brigadeiros, you need 1 can of condensed milk and 9 ounces of chocolate. The price of an ounce of chocolate is pc and the price of a can of condensed milk is pm.
(a) (4 points) Suppose the baker uses a special type of condensed milk from Brazil. The baker recently only bought¯(m) = 10 cans of condensed milk, so in the short run they are stuck with only 10 cans. Setup and solve the short run cost minimization problem for chocolate in the short run as a function of quantity cSR (q) and cost as a function of quantity and prices CSR (q, pc , pm ). (Note, there is an implicit limitation that q must be smaller or equal to 270 ).
(b) (6 points) Now look at the long run where the baker can flexibly choose choco- late and milk. Setup the long run cost minimization problem. Solve for the long run levels of chocolate c* (q, pc , pm ), milk m* (q, pc , pm ), and minimized cost C* (q, pc , pm ).
(c) (2 points) Compare and contrast the methods for solving the short and long run cost minimization problems. Why do the two problems require different methods for solving?
(d) (6 points) Now suppose we are back in a consumer theory problem with utility function u(x,y) = min(3x,27y). What are the compensated demand functions and the minimized expenditure function? Compare and contrast the problems and solutions for part d and part f. How are they similar? How is the economic interpretation different?
4. (10 points) Consider a flower shop that uses sunflowers (s) and daisies (d) to produce bouquets. The cost for a sunflower is ps and the cost for a daisy is pd. The quantity of bouquets is q. The flower shop sees these flowers as perfect substitutes such that bouquets are produced as:
F(s,d) = 6s + 2d
(a) (4 points) Suppose the delivery of sunflowers is delayed and are stuck with a certain number of sunflowers. In particular, they have ¯(s) = 50 sunflowers. Setup the short run cost minimization problem when ¯(s) = 50. Solve for the short run optimal amount of daisies dSR (q) and short run minimized cost CSR (ps ,pd , q).
(b) (5 points) Now suppose the flower shop is making orders for the long term. That is, they can freely choose sunflowers s and daises d. Setup the long run cost minimization problem. Let q = 600 and pd = 1. Solve for the long run optimal amount of sunflowers s* (ps ) and daisies d* (ps ).
(c) (1 point) If ps = 2, what is the long run minimized cost?
5. (30 points) Consider a firm with the production function
You will be solving the profit maximization for this firm with both the two step and 1 step methods and proving that the final answers are identical. This big problem is broken up into the following smaller parts:
(a) (11 points) Setup and solve the long run cost minimization problem for the long run optimal amount of capital K* (w,r, q) and labor L* (w,r, q), and the long run minimized cost C* (w,r, q).
(Hint: reduce the cost function for the next part can be helpful.
Note:
(b) (5 points) Setup and solve the profit maximization problem over quantity using the cost function you solved for in the previous part. Solve for the profit maximizing quantity q* (p,w, r), cost C* (p,w, r), and π* (p,w, r).
(c) (9 points) Setup and solve the profit maximization problem over inputs. Solve for the profit maximizing amounts of capital K* (p,w, r), labor L* (p,w, r), output q* (p,w, r) = F (K* , L* ), cost C* (p,w, r), and π* (p,w, r).
(d) (3 points) Prove that your answers in part (b) and part (c) give you the identical levels of profit maximizing output and profit. (Note: If you have already simplified the profit functions in parts (b) and (c), this requires little work.)
(e) (2 points) Does the firm exit in the long run? Prove how you know and show all of your work.