Lecture 6: Review

Housekeeping items:

  1. To reduce stress homework will now be completion only (now you get 10/10 for effort rather than ~9/10).
  2. <=1 assignments per week.
  3. cases are now just reading assignments that I will post a few days in advance, and you will report whether or not you read them on canvas.
  4. Excel or Python is essential to the course. If you are comfortable with one, try the other. Use this course to develop your abilities.
  5. I am passionate about this aspect of the course, but this is also the direction that the accounting profession is moving. The approach I am taking with this course is implementing the direction of the HKUST SBM Accounting Advisory Board.

"More about the difference between marginal cost and incremental cost, and how the questions in exam will be asked and how we answer it. If there is some examples to show us that will be better"

  • The incremental cost is the change in cost due to changing output by a specific increment, most often one unit. This is often calculated as the difference in costs before and after the change, and is the slope of the line through those two points on the cost curve.

  • The marginal cost is the rate of change of cost at a particular point. This is the slope of the line tangent to the cost curve at the point you are making the calculation.

  • These terms are often equivalent (e.g. when the cost curve is linear),

  • and the terms are often used interchangeably.
  • You should clarify in cases where the difference matters (i.e. if the two are meaningfully different).

The exam question on this topic will be a slightly modified version of P1.1. In all other cases I will specify.¶

Please cover shadow price and the difference between marginal cost and incremental cost

  • the shadow price is the benefit of relaxing the constraint, when we are maximizing profit this is the increase in profit from relaxing the constraint

Since this is a general question, let's just take a look at the slides and excel example.

"Will there be any sample questions or practice questions for us to do before the midterm or final exam?"

The exam questions will be simplified versions of the homework problems and case questions. So those should be your focus for review.

Cost in a Multiproduct Firm - is it possible to use excel and go through the assignments once?

Absolutely!

Would a cheat-sheet be allowed in the exam (both midterm and final)?

No, the exam is closed book. My focus is set-up and interpretation and I have found that allowing cheatsheets leads students to regurgitate formulas rather than answer the questions that I am actually asking.

Lecture 5 - you mentioned near the end of the class you do not recommend rounding up values if the solver / python results generate multiple decimal places as it may go over your constraints. In the Ava catering case, I tried to round down one variable and round up one and achieved higher profit without violating constraints. Is there a way to express the best integer pair in python or excel that can help fulfill all constraints thank you :)

This is exactly the correct way to think about this problem, and for the exam you will not need to worry about this.

I would do this by hand in excel, and in python I would write a function that tests the options against both constraints and picks the larger of the two.

With high production volumes this won't matter, but in the Ava example this does matter.

In [25]:
from gekko import GEKKO
m=GEKKO(remote=False)
# 1 choices
x=m.Var(name="banquets",lb=0)
x.value=1
y=m.Var(name="receptions",lb=0)
y.value=1
# 2 objective
m.Maximize(800*x+525*y)
def profit(x,y):
    return 800*x+525*y
# 3 constraints
m.Equation(12*x+4*y<35) # prep time constraint
m.Equation(9*x+6*y<40) # roofing constraint
# 4 solve
m.solve(disp=False)
In [26]:
pi_1=profit(x.value[0],y.value[0])
print("x:      ", int(x.value[0]))
print("y:      ", int(y.value[0]))
print("Profit: ",int(pi_1))

x:       1
y:       4
Profit:  3517
In [27]:
pi_1=profit(x.value[0],y.value[0])
print("x:      ", x.value[0])
print("y:      ", y.value[0])
print("Profit: ",pi_1)

x:       1.3888888889
y:       4.5833333333
Profit:  3517.3611111025
In [32]:
def check_const_pick_best(x,y):
    '''x,y are non-integer optimal choices'''
    x=int(x)
    y=int(y)
    # check if round x up is allowed
    if (12*(x+1)+4*y<35) and (9*(x+1)+6*y<40):
        profitx=800*(x+1)+525*y
    else:
        profitx=None
    # check if round y up is allowed
    if (12*(x)+4*(y+1)<35) and (9*(x)+6*(1+y)<40):
        profity=800*(x)+525*(y+1)
    else:
        profity=None
        
    if profitx is None and profity is None: 
        opt_x=x
        opt_y=y
    if profitx is not None and profity is None: 
        opt_x=x+1
        opt_y=y
    if profitx is None and profity is not None: 
        opt_x=x
        opt_y=y + 1
   
    if profitx is not None and profity is not None:
        if profitx<profity:
            opt_y=y+1
            opt_x=x
        if profitx>profity:
            opt_y=y
            opt_x=x+1
    print("Optimal x ",opt_x)
    print("Optimal y ",opt_y)
In [38]:
check_const_pick_best(x.value[0],y.value[0])
print(profit(1,4))
print(profit(1,4.5))
print(profit(1.4,4.58))
print(profit(1,5))

Optimal x  1
Optimal y  5
2900
3162.5
3524.5
3425

I want to ask what is the syllabus of the Midterm, from your website, it seems that we should be finished Ch.1 to Ch. 5 in the Zimmerman textbook. But so far we only covered about part of the Ch.1 and Ch. 2. Also, do you have recommendations about how to prepare for the Midterm? Like doing the exercise questions in the textbook, do you mind to provide the textbook answer to us?

As I mentioned above: The exam questions will be simplified versions of the homework problems and case questions. So those should be your focus for review.

The slides and posted examples will be the full set of solutions that you need for the exam.

The textbook readings are posted in the course schedule (lets look at them now).

Cost estimation

Constraints and linear programming

In my opinion, you have already illustrated the concepts very well. I am interested in how this will be tested.

The exam questions will be simplified versions of the homework problems and case questions.

For these two items the focus will be on the set up and interpretation steps.

There were a lot of questions about P5.5.

For question 5 in P5, is marginal cost of the first product is still kept at 30?

30 is the unit cost of x. When we consider the amounts of x and y independently this amount (30) is the marginal cost of x.

Because 6 is the cost of y from the unit change of x, is the total marginal cost 36 (including x and y)?

I think you are on the right track here, but I would put this slightly differently. When we frame the choice of y as dependent on the choice of x, then the resources we sacrifice to produce the marginal unit of x (the marginal cost of x) must include the effect of our choice of x on total profit through the impact of the choice of x on y.

Put differently, when we produce x we sacrifice 30x directly and 6x of profit (which is a resource) we would have earned had our choice of x not effected our choice of y.

For question 6 in P5, does it mean the cost of the product depends on the decision frame because two products can be no longer two independent choices?

Yes, this is exactly right. If we frame the choices as independent, we get one answer (30) if we frame them as related, we get a different answer (36).

"Lecture 4"

"MC of separate products in multi-product production firm ( the one covered in lecture 4)"

"I would like to ask for further explanation regarding the two implication of marginal cost you mentioned today. In the lesson, you have mentioned that the marginal cost's change is related to the capital change and labor change. Also, the number of labor and capital will affect the effeciency of production. However, I am not very clear about their relationship with the marginal cost's change and how it could affect the production efficiency. It would be appreciated if the above concept could be clarified for better learning process."

Recall that $q_1=\sqrt(KL_1)$ this means that the amount of $q$ you get from each additional unit of labor increases with K, but at a decreasing rate (this is because of the square root). So there are two factors driving the changes in marginal cost:

  1. The effect of the marginal unit on the consumption of capital. You can see this in the tables at the end of Lecture 4
  2. The scale of the firm at the margin.

Looking at the graphs and the marginal products of labor and capital here will help:

In [47]:
# quick set up for our plot
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

# create the framework for our plot
l1 = np.linspace(0, 300, 300)
k = np.linspace(0, 200, 300)
L1, K = np.meshgrid(l1, k)

# CALCULATE THE PRODUCTION FUNCTION 
Q1 = np.sqrt(L1*K)
In [48]:
# Create the figure and add a 3D axis
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# Plot the data
ax.plot_surface(L1, K, Q1)
# Set axis labels and show the plot
ax.set_xlabel('L1')
ax.set_ylabel('K')
ax.set_zlabel('Q1')
plt.show()
In [50]:
Q1_10 = np.sqrt(L1*10)
Q1_100 = np.sqrt(L1*100)
Q1_200 = np.sqrt(L1*200)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(L1, K, Q1_10)
ax.plot_surface(L1, K, Q1_100)
ax.plot_surface(L1, K, Q1_200)
ax.set_xlabel('L1')
ax.set_ylabel('K')
ax.set_zlabel('Q1')
plt.show()
In [52]:
L1=np.linspace(0,400,400)
Q1_10 = np.sqrt(L1*10)
Q1_100 = np.sqrt(L1*100)
Q1_200 = np.sqrt(L1*200)
plt.plot(L1,Q1_10,label='Q1 @ K=10')
plt.plot(L1,Q1_100,label='Q1 @ K=100')
plt.plot(L1,Q1_200,label='Q1 @ K=200')
plt.xlabel('Labor (L1)')
plt.legend()
plt.grid()
plt.axhline(0, color='grey')
Out[52]:
<matplotlib.lines.Line2D at 0x7ff3a9657a90>
In [64]:
def calcOutput(k,l):
    return np.sqrt(k*l)
In [65]:
k=10
l1=10 
base=calcOutput(k,l1)
print(base, "q1 at ",k," units of k and ",l1," units of l1")
print(calcOutput(k+1,l1)-base," change in q1 from incrementing k")
print(calcOutput(k,l1+1)-base," change in q1 from incrementing l1")

10.0 q1 at  10  units of k and  10  units of l1
0.4880884817015154  change in q1 from incrementing k
0.4880884817015154  change in q1 from incrementing l1
In [66]:
k=10
l1=100 
base=calcOutput(k,l1)
print(base, "q1 at ",k," units of k and ",l1," units of l1")
print(calcOutput(k+1,l1)-base," change in q1 from incrementing k")
print(calcOutput(k,l1+1)-base," change in q1 from incrementing l1")

31.622776601683793 q1 at  10  units of k and  100  units of l1
1.543471301870209  change in q1 from incrementing k
0.1577205624576159  change in q1 from incrementing l1
In [67]:
k=100
l1=10 
base=calcOutput(k,l1)
print(base, "q1 at ",k," units of k and ",l1," units of l1")
print(calcOutput(k+1,l1)-base," change in q1 from incrementing k")
print(calcOutput(k,l1+1)-base," change in q1 from incrementing l1")

31.622776601683793 q1 at  100  units of k and  10  units of l1
0.1577205624576159  change in q1 from incrementing k
1.543471301870209  change in q1 from incrementing l1