SCIE1500 Analytical Methods for Scientists

1. Answer the following questions about functions.
a) Write down the inverse for each of the following functions. Show your derivations
next two answer.
i) y = 2x − 3 : ...........
ii) y = 2 +
1
3 x : ...........
iii) y =
1
4 x
2
: ...........
iv) y = 4 −
1
x
: ...........
v) y = e 2x : ...........
b) Characterise each function below using an appropriate combination of descriptors
from this list: linear, quadratic, concave-up, concave-down, exponential, everywhere
increasing (function of x), everywhere decreasing (function of x)
i) y = 2 − 3x :
ii) y = 23 + 2x :
iii) y = −5 − 4x + 1.5x 2 :
iv) y = 5 + 0.4x − 3x 2 :
v) y = 0.5e 0.1x :
2. A researcher investigating fishing site choice among recreational fishers finds that their
choices are influenced by cost of travel to the fishing site and by how many fish the
person expects to catch at the site. She estimated the following utility function using
a logit probability model (Logit model is described in lecture L3.3):
Utility = 8 − 0.75 ∗ cost travel to site + 2.5 ∗ fish catch (1)
Suppose you have been given information on a group of fishers shown in the table below
and also in the attached Excel spreadsheet withe the name ”LogitFish Student.xlsx”.
Use the following Logit formula to calculate the probability of visit.
Prob(Site1) =
e Utility site1
e Utility site1 + e Utility site2
(2)
a) Complete the blank columns in the Excel spreadsheet by calculating for each site
and each fisher the utility value and the probability that the fisher goes fishing at
that site. When translating distances to travel costs, assume that the cost of travel
is $0.50 per kilometer. Take a screenshot of the completed spreadsheet and paste it
as your answer to this question.
b) Which fisher is most likely to go to site 1? Which fisher is most likely to go to site
2? Write the two FisherID’s as your answer to this question in the correct order.
c) What is the dollar value (or part-worth) of a single fish according to the estimated
utility function above? In other words, at what rate would a fisher trade off fish for
money according?
d) Suppose the sample of 10 fishers in our table is representative of the bigger population
of fishers who are likely to visit one or both of these sites. Which site would we
expect to attract a bigger portion of the population of fishers targeting the two
sites? Explain your answer.
3. In a study on factors contributing to a particular coronary disease, a researcher has
discovered that the probability of a person having the disease can be expressed as a
logistic function of the person’s sex (Female = 1 for a women and 0 for a man), how old
the person is (Age), whether s/he smokes (Smoker = 1 for a smoker and 0 otherwise),
and the number of cigarettes s/he smokes per day (CigarettesPerDay):
Probability of disease
?
=
e risk
e risk + 1
?
=
1
1 + e −risk
(3)
where: e is the natural number and risk can be expressed as a function of several
variables as follows:
risk = −4.5−0.60∗Female+0.1∗Age+1.2∗Smoker+0.1∗CigarettesPerDay (4)
The probability value is always between 0 and 1, because in the probability formulae
above the numerator and denominator are positive, and the denominator is at least as
big as the numerator, as e x is always non-negative.
a) Use the above formula to calculate the probability that a 70-year old women who is a
non-smoker contracts the disease. Repeat the same calculation for a 70-year old man
who is also non-smoker. Show the steps you followed to the calculation (maximum
3 lines per calculation, six lines in total).
b) The chart below shows how the predicted probability changes with age for a female
person who does not smoke (blue line) and also for a female person who smokes 12
cigarettes a day (red line). How would you summarise the relationship between the
probability of the disease, on the one hand, and the characteristics of the person, on
the other? Summarise your observations about the effect of smoking and age in 5
sentences.
20 30 40 50 60 70 80
0.0 0.2 0.4 0.6 0.8 1.0
Probability of disease for a female as function of age
age
probability
smoker (12 cig/day)
non−smoker
c) In the folder for this assignment, you will find a Jupyter Notebook with the file name
”probdisease.ipynb”. The file shows how you can set up a notebook to calculate the
probabilities but it is deliberately left incomplete so that you can fix it as an exercise
for this assignment. Read through the sections labelled ’Step 1’ to ’Step 3’. Those are
correctly done and you need to leave them intact. However, ’Step 4’ needs fixing as
indicated there so that it calculates correctly the probability values for a nonsmoker
female person. By looking at the earlier calculations done for a female smoker in
that same notebook, correct and complete the instructions in Step 4. In ’Step 5’,
fix/change the details of the chart (axis labels and titles) so that the chart has the
proper details.
Once you have corrected and finalised the notebook, execute Steps 4 and 5 to display
a proper chart. Then, take a screenshot of the contents of the notebook starting with
the contents of ’Step 4’ section and down to the chart you have generated at the end
of the notebook. Include that screenshot as your answer for this exercise.
4. The current notebook exercise aims to help you improve your interpretation skills using
a jupyter notebook implementation of the fishery management problem similar to that
explored in a previous lab (week 3). The notebook ”FishOptimalManagement.ipynb”
shows you how to set up and solve an optimal economic management problem for a
fishery. It differs from the one used in Lab 3 in two ways. First, parameter and price
settings are not the same. Second, there is a ’minimum population threshold’ (Smin)
element in the growth equation used here. The threshold makes the model more realistic
than the Schaefer fish growth model because it acknowledges that not all fish popula-
tions can recover if their stock levels are pushed to very low levels. Some fish stocks
might have minimum thresholds below which they collapse (growth becomes negative
and stock levels decline towards zero). This threshold effect can be accommodated by
modifying the growth equation to the following form:
G(S) = g(S − S min ) ? 1 −
S
K
?
(5)
where: G(S) is the growth in stock, g is the intrinsic (or potential) growth rate, S is
the fish stock, and K is the carrying capacity. For our present exercise, the S min is set
at 1500 tonnes while the carrying capacity is 12000 tonnes, and g is 0.2 or 20%.
If the stock level is less than S min , the above growth equation generates fish growth
(G(S) that is negative. This is consistent with the idea of a minimum threshold. How-
ever, if a negative G(S) calculated according to the equation is numerically bigger than
the stock size (S) that would mean that stock levels are going below zero (S+G(S) < 0),
which is impossible. Therefore, the growth equation needs to be qualified by the re-
quirement that when G(S) is negative and numerically bigger than S, then the G(S)
should be set equal to -S. This modification is implemented in the notebook.
The analysis is complete and correct. You do not need to make modification to how
the model is implemented . Your job is to understand what the instructions in the
notebook are accomplishing, and to write down your observations where it has been
requested that you do so. There are 12 such places in the notebook marked as ”Step
1”, ”Step 2”, etc. Specific instructions are provided on what and how many sentences
you should write for each Step.
First, write down the requested answers inside the notebook itself by modifying the
appropriate Markdown cells. Once you have done so, copy each of your answers for
Steps 1 to 12 and include them as answers to this question in the PDF file you submit.
5. Suppose you have been asked to derive the supply schedule (curve) for the fishery. A
supply curve shows the relationship between the price of a commodity (fish in this case)
and the amount of it marketed (which is assumed to be equal to what is harvested
here). The curve can be presented as a table (schedule) with a column of prices and
corresponding amounts of fish supplied, or as a plot of the same two variables.
a) Solve the fishery management model described above (Question 4) for optimal har-
vest levels using a modified set of parameters (e.g. K etc.) and at each of the
following fish prices (p values): 0.1, 0.25, 0.50, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0. For
each price level, identify the maximum economic yield (MEY) or optimal harvest
level. Write down the fish price, the level of effort, revenue, cost of effort and the
profit. For these exercises, assume the cost of fishing effort (c) is $7, the carrying
capacity (K) is 15000, the minimum population threshold (Smin) is 2500, the in-
trinsic growth model (g) is 0.10, and the catchability coefficient (e) 0.001.
Present as your answer for this question a table with one column showing the prices
listed above, a second column showing the amount of fish that would be supplied at
each of those prices, and a third column showing the optimal profits at each of those
prices.
Note: Generating the supply curve data requires you to run the model multiple times,
once for each price. The key instructions you need to solve the model from end to
end are shown at the end of the notebook provided for Question 4. It is up to you
whether you choose to run the model multiple times (each time changing the price
’by hand’) or you set up a loop to iterate through the prices you need to solve for.
b) Repeat the exercises above after changing the cost of fishing effort (c) to $9/unit.
Present a table of prices, and the new supply levels and optimal profits as your
answer for this part of the question.
c) Plot the two supply schedules above, with price on the horizontal axis and the two
supply curves on the vertical axis. Use proper legends to identify the supply curves
by their fishing cost effect values. Also use proper axes labels and chart title.
d) In a paragraph, summarise what you have learnt from the exercises above about the
relationship between fish price, cost of fishing effort, fish supply and profits.