ECON30025/ECOM90020: COMPUTATIONAL ECONOMICS AND BUSINESS
SEMESTER 1, 2019
ASSIGNMENT B (INDIVIDUAL ASSESSMENT)
COVER SHEET
Due Monday May 20, 2019, 17:00
This assignment is worth 20% of your final grade.
Limit your answer to 8 pages (single-sided).
The assignment should be submitted as an online assignment on LMS. No late submissions will be accepted. Please ensure that the coversheet is attached. The assignment must be stapled with the coversheet. Write your name and student number at the top of each page of the answer sheet.
In our experience, neater assignments receive higher marks, conditional on content. As such, you are strongly encouraged (but not required) to type your assignments. Mathematical symbols and formulas can remain hand-written.
Please complete the declaration and list the student number and name of the student contributing to this assignment.
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We hereby certify that this assignment is our own work, based on our personal study and/or research and that we have acknowledged all material and sources used in the preparation of this assignment. We also certify that the assignment has not previously been submitted for assessment and that we have not copied in part or whole or otherwise plagiarised the work of other students or authors.
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ECON30025/ECOM90020 Computational Economics and Business Assignment B
You must attach the coversheet to your answers. Read the instructions on the coversheet. Try to keep your answers short and clear. This assignment has a total of 20 marks. There are 5 parts. Each Part carries 4 marks.
(CODE APPENDIX: In addition to the code snippets required as answers, please attached any additional code used as an Appendix to your assignment (e.g. for plots etc.). This code should be separated by Part, and clearly commented. This Appendix will not count toward your assignment page limit.)
Part I (4pts)
Consider a Leslie population model for the female population with the following parameters:
Prob(age 1-20 -> age 1-20) = 0.12 Prob(age 21-40->age 1-20) =0.70 Prob(age 41-60->age 1-20) =0.12 Prob(age 1-20 -> age 21-40) = 0.98 Prob(age 21-40 -> age 41-60) = 0.99 Prob(age 41-60 -> age 61+) = 0.8 Prob(age 61+ -> age 61+) = 0.2
The other transition probabilities are zero. The transition probability is interpreted as the conditional probability of being in the state 20 years later, conditional on being in the state now.
1.(1pt) Write down the model as a Markov model with four states.
2.(1pt) The initial state vector is [1 1 1 0]¡¯. Simulate the total female population for 400 years and report the result on a graph. Show your codes, and also report the value of the state vector in 1, 2, and 3 periods after the initial period of the model.
3.(1pt) Suppose medical technology increases the survival rate of people aged 61+ by 3 percentage points every 20 years. Simulate the total female population for 400 years and report the result on a graph. Show the new part of the code that generates this result.
4.(1pt) Suppose the government will trigger an immigration program whenever the
total female population drops strictly below 3. The program restricts the immigrants¡¯ age range to 21-40 and the total number of female immigrants is 3% of the total female population (both measured within a 20-year interval).
1
Simulate the total female population for 400 years and report the result on a graph. Show the new part of the code that generates this result.
Part II (4pts)
A factory employs x1 units of high-skilled labour and x2 units of low-skilled labour. One unit of high-skilled labour costs $8 and one unit of low-skilled labour costs $4. The factory needs to accomplish two tasks. First, it needs to produce at least 20 units of rubber. Second, it needs to produce at least 30 units of glass.
Each unit of rubber requires 1 unit of high-skilled labour, or 4 units of low-skilled labour, or a linear combination of both. Each unit of glass requires 1 unit of high-skilled labour, or 1 unit of low-skilled labour, or a linear combination of both. The factory also has to satisfy the regulation that at least 10 units of low-skilled labour are employed. Assume that products and labour are all continuous variables. Assume that rubber and glass production are not mutually exclusive tasks to labour, i.e., the labour can do both tasks at the same time if needed.
(a)(1pt) Write down the factory¡¯s cost minimization problem.
(b)(1pt) Draw all the constraints on a graph (x1 on the horizontal axis), label the constraints, and highlight the feasible area.
(c)(1pt) Without using software, compute the solution and the level of minimized cost. Show your working.
(d)(1pt) At what relative cost of labour will regulation become a binding constraint?
2
Part III (4pts)
We will use data envelopment analysis on the following data:
1.(1pt) Write down the linear programming problem for DMU2 under the assumption of variable returns to scale.
2.(1pt) Under variable returns to scale, suppose the optimal DMU weights of the linear programming problem for DMU2 are w1=0.33, w2=0, w3=0.67. Without using the EMS software, compute the efficiency score of DMU2. Show your working.
3.(1pt) Now assume constant returns to scale. Use the EMS software to compute both the efficiency score and optimal DMU weights for DMU2.
4.(1pt) Does the virtual DMU in question#3 produce more outputs than the virtual DMU in question#2? Briefly explain why.
Part IV (4pts)
1.(1pt) Consider an IEEE standard 8-bit floating point data type SEEEFFFF where S stores the sign (0 indicates positive), EEE stores the exponent with bias equal to 3, and FFFF stores the fractional part of the mantissa. What is the floating point number that represents ¡°01001100¡±? Show your working.
2.(1pt) What is the maximum floating point number that can be encoded by this data type?
3.(1pt) Use the linear congruential generator to simulate a standard normal distribution consisting of 1000 values. Plot a histogram with a bin width of 0.25 and an overlay of a normal density curve. (Hint: use I0=8, a=5, c=3, m=2501). Show your code, and also report the first 3 observations of the pseudo-data that you generate.
4.(1pt) Use the SAS built-in uniform distribution generator to simulate a standard normal distribution consisting of 1000 values. Show the new part of the code, and plot a histogram with a bin width of 0.25 and an overlay of a normal
DMU
Input
Output
X1
X2
Y1
Y2
Y3
1
5
14
9
4
16
2
8
16
3
4
5
3
7
12
4
9
13
3
density curve. (Hint: use seed number 32).
Part V (4pts)
We will modify the inventory model in the tutorial (tut_w10.pdf) as follows:
– Keep the order point fixed at 30.
– Evaluate the effect of setting the order size at 5, 10, 15, 20, 25, 30.
– Evaluate three outcomes: service level, inventory position (inv_p), and
number of orders.
Keep all other aspects of the model unchanged.
For your reference, the original code is posted below.
data inven;
do r_p = 25 to 35 ; do it = 1 to 100 ;
nday = 30 ; inv=50 ;
inv_p = 50 ; n_o=50 ;
do ii = 1 to nday ;
b_inv = inv ;
if ii = or_time then inv = inv + 50 ;
demand = rantbl(9897653,.01,.02,.04,.06,.09,.14,.18,.22,.16,.06,.02) – 1 ; if demand <= inv then sales = demand; else sales = inv ;unmet_d = abs(sales – demand) ;inv = inv – sales ;inv_p = inv_p – sales ;if inv_p <= r_p then do;or_time = ii + rantbl(9897653,.2,.6,.2) + 3 ;inv_p = inv_p + 50 ; end;output ; end;end;end; run;1.(2pt) Modify the original code to simulate the new model. Write down your new code and highlight where you have made changes.2.(1pt) Show three sets of side-by-side box plots by the order size: one for service level, one for inventory position, and one for the number of orders.3.(1pt) Briefly describe the results shown in the box plots. 4
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