INDIFFERENCE CURVE

QUESTION

UNIVERSITY OF THE VIRGIN ISLANDS, ST. CROIX CAMPUS
MICROECONOMICS March 292012
1. Define THREE of the following terms: [5 marks each]
• Total utility
• Optimal consumption curve
• indifference curve
• Elasticity
• Long-run cost
—
~
2. The Universityof the Virgin Islands has $100,000 to allocate for_grmrts~
students”‘iIf Economics and students in Business. Tbe:-~axinllim ‘amount a
student in Economics can get is $1,200. The maximum amount a student
in Business can get is $1,000. The University’s utility function is a
function of the two types of students.
a. Write the utility function of the University [2 marks]
b. Write the budget constraint of the University [2 marks]
c. Maximize the University’s utility function subject.to its budget
constraint [15 points].
d. Draw a graph to show your answer [3 points]
e. Prove your answer. [_3points_l _
3. Given a demand for Electricity is P = $120 ~
C = 1000 + .05Q2
Maximize the profit for Electricity in this market structure. What market
structure is it? Derive all of the values for the market structure. That is,
find the values for Total Revenue, Total Cost, Marginal Revenue, Marginal
Cost, and Profit. Draw curve to represent the market structure.
4. Given the data in the Table below
a. Complete the table [15 marks]
-L
1
2 3
4 5 6
Shirts per Total
Total Total Marginal
Marginal
Day
Revenue
Cost Profit
Cost
Revenue
0
$0 12
1
30
40
2 56 56
3
78
,66
4 96
74
5
110
80
6
120
87
7
126
96
8
128
112
9
126
144
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–

SOLUTION

INSolution 1

Indifference Curve:

Indifference curves are used for building models that illustrate relation between consumption of goods having different utilities. The definition of indifference curve may well be, Indifference Curve shows all combination of products that will yield same level of satisfaction. This may also be called utility. This kind of analysis by indifference curves is based ordinal theory of utility rather than cardinal utility theory.

The above figure illustrates a ‘map’ of indifference curve that a hypothetical person may possess for product A and Product B. The curved line indicates one indifference curve for the person and represents combination that provide person with given level of desirability.

Indifference curves however in practical matter cannot be used to make decisions. However the utility function can be used to derive utility function.

Total Utility

The concept of Total Utility has to be derived from marginal utility. Total utility is the aggregate fulfillment from a particular goods or services by consuming it. The marginal utility for every goods or services will be different and thus the total utility will also be different. Mathematically Total Utility can be said that it is the aggregate of all the marginal utilities of a particular entity. It can be clearly understood that every consumer will stride forward to attain maximum possible total utility.

One of the ways to attain the maximum total utility is to have a combination of products.

As per the law of diminishing marginal utility the satisfaction will drop as the consumption will increase. Thus the total utility will increase at lesser rate as the consumption of a particular goods or services increases.

Elasticity

Elasticity can be defined as degree of change in demand or supply of a product. Elasticity in other words can be said to be the responsiveness of demand or supply to change in price. The variation in elasticity can be seen from product as the elasticity of a particular product may be different from that of the other.  For example necessities show insensitiveness to change in price as the demand of such products has to be fulfilled. Consumer will also be buying to fulfill the necessities whether the price increase or not. However it can be said that the demand may increase more if price reduces. Thus the elasticity will be high upto certain level.

Below is the formula for elasticity
Elasticity = (% change in quantity / % change in price)
Solution 2

1. The ratio of grant to economics student to that of grant to students of business is 6:5. Thus in such case to divide the grant money equally the ratio of  number of student of business getting the grant to students of economics getting grant is set as 6:5.

Thus the utility function can be given as

u(E1; B2) = 5E1 + 6B2

Since no information is given regarding the type of utility function the function has been set in a way that the utility function is dependent on two variables and are complimentary.

1. The budget constraint as per the given information can be set as

1200E1 + 1000B2<=100000

Where 1200 is the grant to the Economics student and 1000 is the grant to business students. E1 is the number of students from Economics and B2 is the number of students from business

1. If we look at it closely it can be said that whole of the grant can be given to economics students only and other other way around the whole of the grant can be given to only business students. Thus it can be clearly seen that the utility function will have the maximum value in case the complete grant is given to business students only.

Also mathematically, If the grant is spent on business students only the utility function  will  be u(0; B2) = 6 X 100,000/ 10,000= 60

If the grant is spent on economics students only the utility function will  be u(0; B2) = 5 X 100,000/ 12,000= 41.66

Thus the grant spent only on Business students will have higher value for utility function.

1. The graph has been drawn below

E1

(0,41.66)

(60,0)

B1

1. Since the absolute value of the utility function line is always lower than the grant ratio the university will not give the grant to economics student as per the utility function taken.

The grant to economics student will be given when 60<5 X100000/Grant (Economics Students)

Thus  grant will be given when grant for each student is less than 8333.34

Solution 3

As given in the problem

Price= $120

C=1000+0.05Q²

Thus the profit equation will be given as

120Q-(1000+0.05Q²)

Maximization of Profit

To get the point of maximum profit the first derivative of profit equation will be equated to zero.

Thus the point of maximum profit will be

120-0.1Q=0 or Q=1200

Thus profit will be maximum for Q=120

To derive all of the values for the market structure, that is, Total Revenue, Total Cost, Marginal Revenue, Marginal Cost, and Profit the value of quantity is taken as that of point of profit maximization. These calculations have been shown below:

Total Revenue: P X Q = 120 X 1200 = $144,000

Total Cost: 1000 + 0.05 Q²  = 1000 + (0.05 X 1200 X 1200)= $ 73000

Marginal Revenue: The equation for Total Revenue will be P X Q. Thus to get the marginal revenue first derivative is taken which will be P. Thus the Marginal Revenue will be $ 120

Marginal Costs: To get the marginal Cost first derivative is taken which will be 0.1Q. Thus the Marginal Revenue will be 0.1 X 1200 = $120

Profit at Q=1200 will be Total Revenue – Total Cost

Which will be $144,000 – $ 73,000 = $ 71,000

Shown below is the table showing the per unit cost with the increase in number of units produced.

Based on the above information it can be said that the cost is increasing as the number of units being produced is increasing, but the price has been kept fixed. Thus it shows that this is the perfectly competitive market as the total profit of the company is based on the number of units sold although the profit margin reduces.

Solution 4

Based on the given table Total Profit, Marginal Revenue, Marginal Costs and Marginal Contribution have been computed as shown below:

Total Profit= Total revenue- Total Costs

Marginal Revenue= Increase in Revenue= (Total Revenue)_{N+1 Shift} – (Total revenue)_{N Shift}

Marginal Costs= Increase in Costs= (Total Costs)_{N+1 Shift} – (Total Costs)_{N Shift}

Based on these formulas the table has been completed as shown below:

Optimum level of profit

To obtain the optimum level of profit marginal profit has been calculated. It has been calculated as follows:

Marginal Profit = Marginal Revenue- Marginal Costs

Marginal Profit for all possible number of shifts has been shown below

As shown above the marginal contribution is $3 for 6 shifts per day and $-3 for 7 shifts per day.

Thus the optimal level of profit is when the number of shifts in a day is 6 as the marginal contribution is becoming negative if the number of shifts are increased beyond this.

JG14

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