# General expressions for average productivity of labor and capita

Please see the attached file for full problem description.

Solutions:1

a) If Twinkies cost $0.10 each and Slice costs $0.25 per cup, what is his budget constraint? Write down the budget equation and draw the budget line:

Budget constraint $0.10T + $0.25S = $1

to draw the budget line: for example consider this two points A( T=0; S=4) and B(T=10; S=0) AND c(T=5, s=2)

B) Where does the budget line intersect (touch) the indifference curve for U =10^1/2. To answer a question like this first, one should determine the TMS between T and S which is the point in which budget line is tangeant to the utility fonction. U't/U's=Price of T / price of S. (1/2*S^1/2*T^-1/2)/(1/2*T^1/2*S^-1/2) = $0.1/$0.25 After doing all the claculation we get S/T=0.1/0.25 which is S=0.4*T . we will now replace S in the budget equation to find T then S. T*=5 and S*=2. Thus, the intersection point is T=5 and S=2.

C) Show different combinations of T and S that satisfy the budget constraint, give the corresponding utility value, and verify that the combination of T and S obtained in part b, in fact, maximizes Paul's utility.

S T U

0 10.00 0

0.50 8.75 2.09

1.00 7.50 2.74

1.50 6.25 3.06

2.00 5.00 3.16

2.50 3.75 3.06

3.00 2.50 2.74

3.50 1.25 2.09

4.00 0 0

d. Suppose the school tries to discourage Twinkie consumption by raising the price to $0.40. But Paul's mother also increases his lunch allowance by $1. Can Paul achieve the same level of utility he received in part b with this extra allowance? What is the new combination of T and S that gives U= 10^1/2.

With this new situation: U't/U's = Price T / price S, thus S/T= 0.4/0.25 which gives S= 1.6 T replace this in the budget. 0.4 T + 0.25(1.6T)= 2 Then T= 2.5 and S=4

Solution 2. Show calculations for a, b, and include demand curves for c

a) she should purchase: Wf=1 and Wc=10

b) she will buy : Wf=2 and Wc=10

c) To draw the demand curve for Wf use the TMS

2Wc/wf= Pf/Pc thus Wf=Pc/pf*2Wc yes they are related goods. substitutes goods.

solution 3.

a)

i) Elasticity revenue: relative variationof quantities per relative variation of revenu.

ii) Elasticity price: relative variation of quantities per relative variation of price.

iii) If Emily's tastes change and she decides to spend only one-fourth of her income on clothing, how does her demand curve change? Draw a diagram to show the change in demand. What are her income elasticity and price elasticity now?

iii) By reducing the income allocated to clothing it will reduce the level of quantities bought.

b. Two drivers - Tom and Jerry - each drive up to a gas station. Before looking at the price, each places an order. Tom says, "I'd like 10 gallons of gas." Jerry says, "I'd like $10 worth of gas." What is each driver's price elasticity of demand? Draw diagrams of their respective demand curves.

b) Tom : elasticity to price will be very high. On the other hand jerry does not have any elasticity price because he knows the amount he will spend.

c) Qb=500-100Pb+2I-200Ps I=$1000, Pb=$2, Ps=$1 Qb= 2100 .eQ,P;eQI; eq,ps? to determine this is to do the variation relative of each variable P, I and Ps to see the changes in Qb. for examples : Pb increase from 2 to 2.5 then Qb = 2050. eQpb= (2050-2100)/2100)/(2.5-2)/2 = -0.092 . The elasticity demand Qb due to Pb: When the price Pb increase per 50% the demand will decrease by 9.2%. Please clarify what eQ,I and eQ,ps are equal to.

The production of barstools (q) is characterized by a production function of the form

a. Derive general expressions for average productivity of labor and capital for barstool production as functions of K and L. (Hint: By definition, = q/L and = q/K)

b. Graph and curve for K=100

c. Graph the q= 10 isoquant. Show all K-L combinations in a table.

For q = 10, we get K =

L K=100/L

1 100

2 50

3 33.33333

4 25

5 20

6 16.66667

7 14.28571

8 12.5

9 11.11111

10 10

11 9.090909

12 8.333333

13 7.692308

14 7.142857

15 6.666667

16 6.25

17 5.882353

18 5.555556

19 5.263158

20 5

21 4.761905

22 4.545455

23 4.347826

24 4.166667

25 4

26 3.846154

27 3.703704

28 3.571429

29 3.448276

30 3.333333

31 3.225806

32 3.125

33 3.030303

34 2.941176

35 2.857143

36 2.777778

37 2.702703

38 2.631579

39 2.564103

40 2.5

41 2.439024

42 2.380952

43 2.325581

44 2.272727

45 2.222222

46 2.173913

47 2.12766

48 2.083333

49 2.040816

50 2

51 1.960784

52 1.923077

53 1.886792

54 1.851852

55 1.818182

56 1.785714

57 1.754386

58 1.724138

59 1.694915

60 1.666667

61 1.639344

62 1.612903

63 1.587302

64 1.5625

65 1.538462

d. The point K=10, L=10 is one point on the q =10 isoquant. What value of K corresponds to L= 11 on that isoquant? What is the approximate value for the RTS at K = 10, L=10?

From the table above, corresponding to K = 10, L = 9.090909. What is the approximate value for the RTS at K = 10, L=10?

#### Solution Summary

Write down the budget equation and draw the budget line.