MicroeconomiaCorso E

John Hey

Chapter 7 – what do we know?

• Individual with given preferences and income m faces prices p1 and p2 for two goods:1 and 2.

• He or she is going to allocate his or her income buying quantities q1 and q2 of the two goods.

• What he or she buys/demands depends on her preferences.

• The relationship between q1 and q2 (the endogenous variables) and m, p1 and p2 (the exogenous variables) is called the demand function.

Chapter 7 – what do we know?

• Cobb-Douglas with parameter a

q1 = am/p1 and q2 = (1-a)m/p2

• Perfect Substitutes 1:a

if p1/p2 < a then q1 = m/p1 q2 = 0

if p1/p2 = a then....

if p1/p2 >a then q1 = 0 q2 = m/p2

• Perfect Complements 1 with a

q1=m/(p1 + ap2) and q2 =am/(p1 + ap2)• These formulas are in the promemoria.

The optimal point

• With indifference curves that are smoothly convex...

• ... the optimal point is the point of tangency between the budget line and the highest possible indifference curve...

• ...at which the relative price (the slope of the budget line) is equal to the marginal rate of substitution (the slope of the indifference curve).

Are we economists or not?

• Economists are ...• ... lazy ...• ... efficient.• In Chapter 7 income is in the form of money m. In

Chapter 6, income is in the form of endowments of the two goods: e1 and e2.

• What is the money value of this endowment? Call it m.

• We have m = p1e1 + p2e2.

• Let us just replace m with p1e1 + p2e2 everywhere!

From Chapter 7 we have

• Cobb-Douglas with parameter a q1 = a( m )/p1 and

q2 = (1-a)( m )/p2

• Perfect Substitutes 1:a if p1/p2 < a then q1 = ( m )/p1 and q2 = 0

if p1/p2 = a then....

if p1/p2 >a then q1 = 0 and q2 = ( m )/p2

• Perfect Complements 1 with a q1= ( m )/(p1 + ap2) and

q2 =a( m )/(p1 + ap2)

Hence for Chapter 6

• Cobb-Douglas with parameter a q1 = a(p1e1 + p2e2)/p1 and

q2 = (1-a)(p1e1 + p2e2)/p2

• Perfect Substitutes 1:a if p1/p2 < a then q1 = (p1e1 + p2e2)/p1 and q2 = 0

if p1/p2 = a then....

if p1/p2 >a then q1 = 0 and q2 = (p1e1 + p2e2)/p2

• Perfect Complements 1 with a q1= (p1e1 + p2e2)/(p1 + ap2) and

q2 =a(p1e1 + p2e2)/(p1 + ap2)

Chapter 6

• Finished!

Chapter 6

• We consider an individual who starts with an endowment of the two goods.

• We find his gross demands for the two goods.

• We analyse how these demands change when the prices and his income change. (These variables are exogenous for the individual).

• These are called comparative static exercises.

Chapter 6

• We start with an individual with Cobb-Douglas preferences with parameter a = 0.5.

• The Maple/html file contains other examples:• Cobb-Douglas with parameter a = 0.3;• Stone-Geary;• Perfect Substitutes;• Perfect Complements.• The shape of the demand curve depends

upon the preferences.

Chapters 6 and 7

• We use two spaces:• The first: to show the preferences of the

individual and the budget line:

• q1 on the horizontal axis and q2 on the vertical axis.

• The second: to show the effect of changes in an exogenous variable on the demand:

• q1 (and q2 ) on the horizontal axis and the exogenous variable on the vertical axis.

Chapter 6

• The indifference curves are given by the preferences.

• The budget constraint is given by the individual’s income and the prices of the two goods.

• We denote by (e1, e2) the endowment and by (q1, q2) the quantities chosen to consume. The budget line is given by the equation:

• p1q1 + p2 q2 = p1e1 + p2e2 • This is a line with slope• - p1/ p2

• which passes through the endowment point.

q2

q1

the budget line: p1 q1 +p2 q2 = p1 e1 +p2 e2

(p1 e1 +p2 e2 )/p1

(p1 e1 +p2 e2)/p2

has slope = -p1/p2

e2

e1

and passes through (e1,e2)

X

Cobb-Douglas with parameter a = 0.5

U(q1, q2) = q10.5q2

0.5

Chapter 6 – results

• Cobb-Douglas with parameter a

q1 = a(p1e1+p2e2)/p1 e q2 = (1-a)(p1e1+p2e2)/ /p2

• Perfect substitutes 1:a

if p1/p2 < a then q1 = (p1e1+p2e2)/p1 q2 = 0

if p1/p2 = a then....

if p1/p2 >a then q1 = 0 q2 = (p1e1+p2e2)/p2

• Perfetti complements 1 with a

q1= (p1e1+p2e2)/(p1 + ap2) e

q2 =a(p1e1+p2e2)/(p1 + ap2)

Chapter 6

• Goodbye!