MFE I - Homework 4
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MFE I - Homework 4
To upload on Gradescope by Tuesday, October 31st, midnight.
The below homework specifications will be enforced. If the specifications are not respected, points might be deducted, or the homework assignment may not be accepted for grading. Each exercise is worth 10 points.
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Exercise I: Elasticity and revenue I
Prerequisites: Elasticity
Let D(p) be the demand for a good at price p. The revenue function is the amount of
money gained by the producer when it fulfills the demand, namely
R(p) = pD(p).
1. Let E(p) be the elasticity of D(p). Compute R′ (p) in terms of D(p) and E(p) only.
2. Recall that, in general, the elasticity of the demand is negative, that is, E(p) < 0. We assume that this is the case from now on. Now, If the demand is elastic, what is the sign of R′ (p)? What happens to the revenue if we slightly increase the price? What if the demand is inelastic?
3. According to the previous question, if the revenue is maximal at p, can the demand be elastic at p? Inelastic? What can you conclude?
4. We now study a specific case. Assume that the demand for broccoli is given by D(p) = 5000 −500p2 , where D(p) is in tons and p is the price of a pound of broccoli. (a) Compute the elasticity of the demand as a function of price.
(b) Find p such that the demand at p is unit elastic. Gives the exact value and an approximation with two significant digits. What does this price mean in terms of revenue?
Exercise II: Elasticity and revenue II
You should complete Exercise I before starting this exercise.
Assume that the demand of chocolate of a local producer is given by q = 9 − √p in hundreds of kilograms, where p ∈ [0, 81] is the price of chocolate in dollars per kilogram.
1. Compute the elasticity of the demand as a function of p, and find where it is elastic, inelastic, and unit-elastic.
2. Compute the revenue R1 (p) in dollars as a function of p. Careful of units!
3. Compute R1(′)(25). Give its unit and interpret this quantity.
4. What is the sign of R1(′)(25)? Is it consistent with what you obtained in Question 1 and Question 2 of Exercise I?
5. Now, compute the revenue R2 (q) in dollars as a function of q. Careful of units!
6. Compute R2(′)(4). Give its unit and interpret this quantity.
7. Note that q = 4 corresponds to p = 25. Why do we obtain different interpretations in Questions 3 and 6?
Exercise III: Implicit differentiation
Prerequisites: Linear approximation, Implicit differentiation
Assume that a factory can produce a quantity
of goods, when x is the number of skilled workers, and y the number of unskilled workers.
1. Assume that the quantity produced is constant. Find y′ as a function of x and y.
2. Right now, there are 10 skilled and 10 unskilled workers. What is the quantity produced?
3. The factory wants to maintain the production at the level of Question 2. Find y′ at this point, and give its unit.
4. The management of the factory is considering replacing a skilled worker (paid s40/h) by unskilled workers (paid s15/h), while maintaining the production at the same level. Is this a good move? Explain precisely using the previous questions.
Exercise IV: Inverse functions
Prerequisites: Linear approximation, Inverse functions
1. Show that f(x) = x 3 + 2x is a bijection from (−∞, +∞) to (−∞, +∞), then ap-proximate the solution to f(x) = 3.1.
2. Use similar techniques to approximate the solution to
Explain as in the previous question.
3. Assume that f is a differentiable function such that f′ (x) = 1+f (x)2 for all x ∈ [a, b]. Show that f is a bijection from an interval to another interval (which you should determine), then compute the (f−1)′ (x). Your result should be written in terms of x (not f (x) or f−1 (x)).
2023-10-30