Problem Set 4 - Solution PDF

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Problem Set 4 - Solution...

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Prof. Dr. Sebastian J. Goerg Prof. Dr. Michael Kurschilgen

Economics I Tutorial

Solutions for Problem Set 4 Question 1 (Budget constraint) (a) Given the income of 450, the household can afford 45 units of food or 30 units of clothes maximum.

(b) Budget restriction: 10qF + 15qC = 450

⇒ qC = 30 − 32qF ;

(c) 0): Budget line from a). 1): Given the new price for food (pF = 15), the housebold can now afford 30 units of food maximum. 2): Given the new price for clothes (pC = 10), the household can now afford 45 units of clothes maximum. 3): Given the higher income (m = 600), the household can now afford 40 units of food or 60 units of clothes maximum.

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Prof. Dr. Sebastian J. Goerg Prof. Dr. Michael Kurschilgen

Economics I Tutorial

Question 2 (Indifference curves and MRS ) (a) Indifference curve: 2q1 + 3q2 = 9 ⇒ q2 = 3 − 32 q1 ;

∂u

MRS1,2 = − ∂q∂u1 = − 23 ; ∂q2

The goods 1 and 2 are perfect substitutes. The household is always willing to give up 32 units of good 2 in order to get an additional unit of good 1. (b) Indifference curves: q1 · q2 = 1 ⇒ q2 = q11 ;

MRS1,2 = − qq12 ;

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Prof. Dr. Sebastian J. Goerg Prof. Dr. Michael Kurschilgen

Economics I Tutorial

(c) Indifference curve: min(q1 , 4q2 ) = 4; In order to achieve a utility level of 4, four units of good 1 and one unit of good 2 are necessary. Given this situation, an increase in only one of the goods, does ceteris paribus not lead to a higher utility level of the household. Therefore, these goods are perfect complementary goods.

For this utility function, the MRS is not defined.

Question 3 (Preferences and MRS ) (a) The two painkillers are perfect subtitute goods for consumers. They have a constant MRS (equivalent to a linear indifference curve), which means that the household is willing to change the goods in a constant ratio, irrespective of the consumed quantity of goods.

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Prof. Dr. Sebastian J. Goerg Prof. Dr. Michael Kurschilgen

Economics I Tutorial

(b) Gloves are just pairwise useful for the consumer. A single glove is useless. Right and left hand side gloves are therefore perfect complementary goods. They have right-angle indifference curves. The M RS is not defined for such utility functions.

(c) The rock musician just wears left hand gloves. For him, right hand gloves are useless. His utility soley depends on the number of left hand gloves in his possession. His indifference curve is parallel to the right hand gloves axis, with a MRSR,L = 0.

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Prof. Dr. Sebastian J. Goerg Prof. Dr. Michael Kurschilgen

Economics I Tutorial

(d) For non-smoker, cigarette smoke is bad. The higher the quantity of smoke, the more conviviality the consumer needs in order to achieve the same utility level. Therefore, her indifference curves have a positive slope and a positive M RSS,C .

Question 4 (Optimal consumption plan) Given: U (q1 , q2 ) = q1 · q2 ;

p1 = 1 and p2 = 4;

m = 160;

Optimality condition: MRS1,2 = −pp21 Budget restriction: p1 q1 + p2 q2 = m ∂u

(i) MRS1,2 = − ∂q∂u1 = − qq21 = − 41 ; ∂q2

⇒ q1 = 4q2 ;

(ii) q1 + 4q2 = 160

⇒ q1 = 160 − 4q2 ;

(i) in (ii): 4q2 = 160 − 4q2

⇒ q2 = 20; and q1 = 4 · 20 = 80;

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