ECON 1530 Lecture Notes - Lecture 2: Elementary Algebra, Multiplication Sign, Natural Number

1.Write a function encipher(s, n) that takes as inputs an arbitrary string s and a non-negative integer n between 0 and 25, and that returns a new string in which the letters in s have been "rotated" by n characters forward in the alphabet, wrapping around as needed. For example:

>>> encipher('hello', 1) result: 'ifmmp'

>>> encipher('hello', 2) result: 'jgnnq'

>>> encipher('hello', 4) result: 'lipps'

Upper-case letters should be "rotated" to upper-case letters, even if you need to wrap around. For example:

>>> encipher('XYZ', 3)

result: 'ABC'

Lower-case letters should be "rotated" to lower-case letters:

>>> encipher('xyz', 3)

result: 'abc'

Non-alphabetic characters should be left unchanged:

>>> encipher('#caesar!', 2)

result: '#ecguct!'

Hints/reminders:

You can use the built-in functions ord and chr convert from single-character strings to integers and back:

>>> ord('a')

result: 97

>>> chr(97)

result: 'a'

You can use the following test to determine if a character is between 'a' and 'z' in the alphabet:

if 'a' <= c <= 'z':

A similar test will work for upper-case letters.

We recommend writing a helper function rot(c, n) that rotates a single character c forward by n spots in the alphabet. We have given you a template for this helper function in ps3pr3.py that checks to ensure that c is a single-character string. We wrote rot13(c) in lecture; rot(c, n) will be very close to rot13(c)! You can test your rot(c, n) as follows:

>>> rot('a', 1)

result: 'b'

>>> rot('y', 2)

result: 'a'

>>> rot('A', 3)

result: 'D'

>>> rot('Y', 3)

result: 'B'

>>> rot('!', 4)

result: '!'

Once you have rot(c, n), you can write a recursive encipher function.

Once you think you have everything working, here are three more examples to try:

>>> encipher('xyza', 1)

result: 'yzab'

>>> encipher('Z A', 2)

result: 'B C'

>>> encipher('Caesar cipher? I prefer Caesar salad.', 25)

result: 'Bzdrzq bhogdq? H oqdedq Bzdrzq rzkzc.'

2.Write a function decipher(s) that takes as input an arbitrary string s that has already been enciphered by having its characters "rotated" by some amount (possibly 0). decipher should return, to the best of its ability, the original English string, which will be some rotation (possibly 0) of the input string s. For example:

>>> decipher('Bzdrzq bhogdq? H oqdedq Bzdrzq rzkzc.')

result: 'Caesar cipher? I prefer Caesar salad.'

Here are two more examples:

>>> decipher('Hu lkbjhapvu pz doha ylthpuz hmaly dl mvynla lclyfaopun dl ohcl slhyulk.')

result: 'An education is what remains after we forget everything we have learned.'

>>> decipher('python')

result: 'eniwdc'

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def rot(c, n):
    """ your docstring goes here """
    # check to ensure that c is a single character
    assert(type(c) == str and len(c) == 1)

def letter_prob(c):
    """ if c is the space character (' ') or an alphabetic character,
        returns c's monogram probability (for English);
        returns 1.0 for any other character.
        adapted from:
       
    """
    # check to ensure that c is a single character   
    assert(type(c) == str and len(c) == 1)

    if c == ' ': return 0.1904
    if c == 'e' or c == 'E': return 0.1017
    if c == 't' or c == 'T': return 0.0737
    if c == 'a' or c == 'A': return 0.0661
    if c == 'o' or c == 'O': return 0.0610
    if c == 'i' or c == 'I': return 0.0562
    if c == 'n' or c == 'N': return 0.0557
    if c == 'h' or c == 'H': return 0.0542
    if c == 's' or c == 'S': return 0.0508
    if c == 'r' or c == 'R': return 0.0458
    if c == 'd' or c == 'D': return 0.0369
    if c == 'l' or c == 'L': return 0.0325
    if c == 'u' or c == 'U': return 0.0228
    if c == 'm' or c == 'M': return 0.0205
    if c == 'c' or c == 'C': return 0.0192
    if c == 'w' or c == 'W': return 0.0190
    if c == 'f' or c == 'F': return 0.0175
    if c == 'y' or c == 'Y': return 0.0165
    if c == 'g' or c == 'G': return 0.0161
    if c == 'p' or c == 'P': return 0.0131
    if c == 'b' or c == 'B': return 0.0115
    if c == 'v' or c == 'V': return 0.0088
    if c == 'k' or c == 'K': return 0.0066
    if c == 'x' or c == 'X': return 0.0014
    if c == 'j' or c == 'J': return 0.0008
    if c == 'q' or c == 'Q': return 0.0008
    if c == 'z' or c == 'Z': return 0.0005
    return 1.0

MULTIPLE CHOICE.

1) Which of the following is true of the typical relationship between marginal product (MP) and average product (AP)?

A) The AP curve intersects the MP curve at minimum MP.

B) If MP is less than AP, then AP is increasing.

C) If MP is greater than AP, then AP is falling.

D) The MP curve intersects the AP curve at maximum AP.

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.

2) If, for a particular short-run production, we observe that marginal product is decreasing we can conclude that average product is decreasing as well.

MULTIPLE CHOICE.

3) An implicit cost is defined as:

A) the amount by which the money spent on an input to production exceeds its opportunity cost.

B) the difference between an input's explicit cost and its actual cost.

C) the opportunity cost of using a resource that is not explicitly paid out by the firm.

D) the amount by which economic profit exceeds accounting profit.

MULTIPLE CHOICE.

4) The amount of output a firm can produce with a given quantity of fixed and variable inputs is called:

A) total fixed product.

B) average variable product.

C) marginal product.

D) total product.

5) Which of the following statements is true of the relationship among the average cost functions?

A) AFC = ATC + AVC

B) ATC = AFC - AVC
C) AVC = AFC + ATC

D) AFC = ATC - AVC

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.

6) A firm's production function is the relationship between the factors of production and the resulting outputs of the production process.

MULTIPLE CHOICE.

Scenario 2:
Output (Q): 0 1 2 3 4 5 6

Total Cost (TC): $24 $33 $41 $48 $54 $61 $69

7) Refer to Scenario 2. Diminishing marginal returns starts to occur between units:

A) 2 and 3.

B) 3 and 4.

C) 4 and 5.

D) 5 and 6.

8) Which of the following would be classified as a short-run decision?

A) trucking firm's decision to move to a smaller facility.

B) A firm's decision to decrease the amount of electricity used in day-to-day operations by encouraging employees to adopt conservation strategies, e.g., shut off lights when leaving a room.

C) A university's decision to add a new residence hall.

D) A restaurant's decision to increase the number of patrons it can accommodate by adding on a new dining room.

9) Economic profit is equal to the difference between: 9)

1.

2.

3.

4.

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.

10) At the point where a firm incurs diminishing marginal returns, total product will begin to decline.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

11) In the context of a production function, the remote order takers in the fast food industry would be classified as:

A) a fixed input.

B) a marginal input.

C) an inframarginal input.

D) a variable input.

Scenario 3:
Total Product (Q): 0 1 2 3 4 5 6

Total Cost (TC): $50 $80 $105 $125 $155 $195 $250

12.

13.

increase; lower

decrease; lower

increase; higher

decrease; higher

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.

14.

15.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

16) A labor-intensive method of production is one that:
A) combines a small but sophisticated labor force with a large amount of capital.

B) requires employees to work harder than they would in other occupations.
C) relies on large quantities of labor and smaller quantities of capital equipment. D) relies exclusively on labor.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

17) Assume a firm produces 500 units of a good by using two inputs, capital and labor, whose per unit prices are $10 and $4. Assume also that the marginal physical product of the last unit of capital is 30 and the marginal physical product of the last unit of labor is 10. To minimize costs this firm should employ:

A) more labor and less capital. B) more capital and less labor.
C) the existing combination of resources. D) both more labor and more capital.

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false.

18) All else constant, a decrease in the per unit price of labor would create an incentive for a firm manager to substitute labor for capital in the firm's production process.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

19) All else constant, as the price of petroleum increases relative to the prices of other inputs to the production process, in their effort to minimize their total costs of production, we can expect to see firms employ:

A) the same amount of petroleum since there are no substitutes for petroleum. B) less petroleum and more of the other inputs to production.
C) less of each of the inputs of production.

D) more petroleum and less of the other inputs to production.

QUESTION 13

Label the next column (D) "6-month change." Here we will calculate how much prices have changed since the price six months ago. For example, if the price on January 1 were 224.1 and the price on July 1 is 324.1, the price change is 100. We can't calculate how much prices have changed over the past six months for the first half of 2008 because we don't know the price six months before. So, in column D, put an x in each cell for which we can't calculate the 6-month change. Starting in July, we can calculate the 6-month change. Using cell references, enter here the formula you should enter into cell D30 to calculate the 6-month change.

2 points

QUESTION 14

Copy and paste the formula from cell D30 (from the last question) into cells D31 to D55. What formula is now in cell D48?

2 points

QUESTION 15

What value is now in cell D50?

1 point

QUESTION 16

Label the next column (E) "% of Jan." In this column, we will divide every week's price by the price from January 7th. This will tell us as a percent how prices compare to their level at the start of the year. This time, we want an "absolute" reference so that it doesn't consider the position (i.e. "8 to the left and 2 down") but just locks in on B4 every time. To set up an absolute reference in a formula, we put the $ character into the reference, so instead of B4, we use $B$4. With the dollar signs in place, you can copy the formula and paste it, and in each case, it will go back to B4 regardless of where you are when you paste it. What formula should you type into cell E5 to get the "% of Jan" for January 14, 2008? Remember to use the absolute reference so it will be easy to copy later!

2 points

QUESTION 17

Copy and paste the formula from cell E5 into the rest of column E. (Well, at least from E6 down to E55.) What formula is now in cell E17?