Economic Principles - Assignment A

Economic Principles Question 1 i. The marginal utility of a good provides the rate of increase or decrease in the utility from that good as the total consumption of it rises. It is not a measure of the total utility of a good; the marginal utility is based in the idea that, for goods that can be consumed repeatedly (like a bottle of water, whose content is drinkable in a series of gulps), the utility derived from the nth gulp in inferior to the utility derived from the very first gulp. The marginal utility provides a rate of this decrease in utility as the consumption increases. ii. Whenever consumers are faced with the task of choosing among different baskets (which are essentially a collection of goods and services), their choices are dictated by rational criteria. Let's assume that all such rational behaviour conforms to three assumptions: 1. preferences are complete, which means the consumer is able of ranking the baskets; 2. preferences are transitive, which means they are internally consistent with each other; 3. more is always preferred to less. In order to decide which basket to choose, consumer must rank the baskets. The ranking criteria is the utility derived from each basket; The consumer will rationally choose the basket from which he or she gets the most utility. Consumers can usually rank the utility of the basket in a qualitative way – saying they prefer one basket to the other, but not by how much. This measure is the qualitative utility derived from that basket – consumers can say which brings them the most utility, but not by how much. This means there is no information about the strength of the preference. When consumers can actually rank the intensity of the utility, we get information about the rate of preferences. This means that we are is a position to say that one basket is preferred to another by, for instance, ten percent more.

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Economic Principles iii.

Perfect Substitutes U=x+6y 1,8 1,6 1,4

Pencil, y

1,2

U=6 U=8

1 0,8 0,6

U=10

0,4 0,2 0 0

2

4

6

8

10

12

Pen, x

Perfect Complements U=3min(x,y) 4,5

Coca-Colas , y

4 3,5 3 2,5

U=3 U=6

2 1,5

U=9

1 0,5 0 0,5

1

1,5

2

2,5

3

3,5

4

4,5

BigMacs , x

This example considers that the economic agent wants to have a Coca-Cola for every BigMac he consumes. 2

Economic Principles

Satiated Preferences X2

X1

iv. Consider that a consumer obtains a fixed amount of utility from a basket composed of two goods, x and y, and he or she is presented with the opportunity of substituting a certain amount of one good for a certain amount the other. The rate at which the consumer is willing to substitute one product for the other, while maintaining the utility constant, is the Marginal Rate of Substitution.

v. The following chart depicts a generic utility function for two goods, x and y, such that Uxy = xy:

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Economic Principles 18

A 16

14

12

10

B

Y 8

U=4

6

C 4

D E

2

0 0

2

4

6

8

10

12

14

16

18

X

The Marginal Rate of Substitution of x for y on any specific point on the indifference curve is given by the slope of a line tangent to that point. Consider the lines tangent to points A to E in the example above: their slopes equal the absolute value of the MRS xy at each point:

Point A B C D E

Good X

Slope/MRSxy

Good Y 1 2 4 8 16

16 8 4 2 1

16 4 1 0,25 0,06

This shows that along the same utility curve (along which the consumer always gets

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Economic Principles the same utility), when a consumer has a lot of good x and little of good y, he is prepared of giving up lots of x for a little more y; but as he gets more y he is less willing to give up any of x for a greater amount of y – hence the diminishing slope of the utility curve and the diminishing Marginal Rate of Substitution of x for y. mar. This circumstance only holds for utility functions that, like the one depicted here, are bowed towards the origin.

Question 2 i. a = 0.5, b = 0.5 and u = 1,2,3,4,5.

Indifference Curves U(x1,x2)=x1^a*x2^a; a=b=0,5 30

25

20

x2

U=1 U=2 U=3 U=4

15

U=5 10

5

0 0

1

2

3

4

5

6

x1

5

Economic Principles

ii. a = 0.2, b = 0.8 and u = 1,2,3,4,5.

Indifference Curves U(x1,x2)=x1^0,2*x2^0,8 25

20

15

U=1 U=2

x2

U=3 U=4 U=5

10

5

0 0

1

2

3

4

5

6

x1

6

Economic Principles

Question 3 This is a Cobb-Douglas function, so it has 2 useful properties: 1. MRSx1,x2 and MRSx2,x1 are both positive 2. Diminishing rate of substitution with the utility functions bowing towards the origin. The optimal basket will be on the budget line; so: Px1x1+Px2x2=I Since u(x1,x2)=x1a+x2b, we get lnu(x1,x2)=alnx1+blnx2 MRSx1,x2=u(x1,x2)=x1a+x2b, dx1 = a/x1 and MRSx2,x1=u(x1,x2)=x1a+x2b dx2 = b/x2 Since, at the maximum point in the logarithmic function MRSx1,x2=MRSx2,x1 a/x1=b/x2 x2=(bx1)/a and x1=(ax2)/b And we can substitute it in the first equation: Px1x1+Px2x2=I Px1x1+(Px2bx1)/a=I x1=(aI)/(aPx1+bPx2) – Demand Curve for x1, depending on price Px1x1+Px2x2=I (Px1ax2)/b+Px2x2=I x2=(bI)/(aPx1+bPx2) – Demand Curve for x2, depending on price

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Economic Principles Question 4 Suppose two goods, x and y, with x being the good whose demand curve we are trying to determin, and y being a composite good, with a price of 1. These two goods account for all the goods the individual buys, and the individual will maximize his utility, spending all of his income in buying those goods. The resulting bundle of goods bought will therefore depend on the income of the individual and on the relative prices of the goods – and, since y is a composite with a price of 1, it will depend on the price of x and on the individual's income. We will assume the individual's income is 20. Our objective is to derive the individual's budget curve for various prices of x. Imagining the price of y and the individual's income both remain constant, chart 1 shows three budget curves for three different possible prices of x: Px=10, Px=5 and Px=2. The chart displays the budget lines at each of these prices of x, and it also displays the individual's indifference curves for each of the budget lines – U1, U2 and U3.

Budget Lines 25

20

Units of y

15 BL1 BL2

U3

BL3

10

U1

U2

5

0 0

2

4

6

8

10

12

Units of x

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Economic Principles The point at which each of the indifference curves touches (is tangent) to the respective budget line gives us the number of units of x and y that the subject will acquire for that price, maximizing his utility in the process. These numbers of units of x and their respective prices may than be plotted into a demand curve for that individual:

Demand Curve for Good x 12

10

Price

8

6

4

2

0 0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

5,5

Quantity

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Economic Principles Question 5 The uncompensated (Marshallian) demand curve gives us information about the amount of a good demanded for a certain level of price, keeping the subject's income constant. This way, it's the utility of the subject that may vary along the line. Thanks to the uncompensated demand curve, it is possible to observe the income effect and the substitution effect. On the other hand, the compensated (Hicksian) demand curve gives us information about the amount of good demanded for a level of price, keeping the subject's utility constant. This means that, to keep the subject in the same utility curve as the price of the good changes, it's the income that must change, meaning the consumer is “compensated”. The compensated demand curve cannot depict the income effect, but it gives information about the substitution effects, should it occur.

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