Microeconomia Corso E John Hey

Chapter 7 – what do we know? Individual with given preferences and income m faces prices p 1 and p 2 for two goods:1 and 2. He or she is going to allocate his or her income buying quantities q 1 and q 2 of the two goods. What he or she buys/demands depends on her preferences. The relationship between q 1 and q 2 (the endogenous variables) and m, p 1 and p 2 (the exogenous variables) is called the demand function.

Chapter 7 – what do we know? Cobb-Douglas with parameter a q 1 = am/p 1 and q 2 = (1-a)m/p 2 Perfect Substitutes 1:a if p 1 /p 2 < a then q 1 = m/p 1 q 2 = 0 if p 1 /p 2 = a then.... if p 1 /p 2 >a then q 1 = 0 q 2 = m/p 2 Perfect Complements 1 with a q 1 =m/(p 1 + ap 2 ) and q 2 =am/(p 1 + ap 2 ) 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: e 1 and e 2. What is the money value of this endowment? Call it m. We have m = p 1 e 1 + p 2 e 2. Let us just replace m with p 1 e 1 + p 2 e 2 everywhere!

From Chapter 7 we have Cobb-Douglas with parameter a q 1 = a( m )/p 1 and q 2 = (1-a)( m )/p 2 Perfect Substitutes 1:a if p 1 /p 2 < a then q 1 = ( m )/p 1 and q 2 = 0 if p 1 /p 2 = a then.... if p 1 /p 2 >a then q 1 = 0 and q 2 = ( m )/p 2 Perfect Complements 1 with a q 1 = ( m )/(p 1 + ap 2 ) and q 2 =a( m )/(p 1 + ap 2 )

Hence for Chapter 6 Cobb-Douglas with parameter a q 1 = a(p 1 e 1 + p 2 e 2 )/p 1 and q 2 = (1-a)(p 1 e 1 + p 2 e 2 )/p 2 Perfect Substitutes 1:a if p 1 /p 2 < a then q 1 = (p 1 e 1 + p 2 e 2 )/p 1 and q 2 = 0 if p 1 /p 2 = a then.... if p 1 /p 2 >a then q 1 = 0 and q 2 = (p 1 e 1 + p 2 e 2 )/p 2 Perfect Complements 1 with a q 1 = (p 1 e 1 + p 2 e 2 )/(p 1 + ap 2 ) and q 2 =a(p 1 e 1 + p 2 e 2 )/(p 1 + ap 2 )

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: q 1 on the horizontal axis and q 2 on the vertical axis. The second: to show the effect of changes in an exogenous variable on the demand: q 1 (and q 2 ) 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 (e 1, e 2 ) the endowment and by (q 1, q 2 ) the quantities chosen to consume. The budget line is given by the equation: p 1 q 1 + p 2 q 2 = p 1 e 1 + p 2 e 2 This is a line with slope - p 1 / p 2 which passes through the endowment point.

q2q2 q1q1 the budget line: p 1 q 1 +p 2 q 2 = p 1 e 1 +p 2 e 2 (p 1 e 1 +p 2 e 2 )/p 1 (p 1 e 1 +p 2 e 2 )/p 2 has slope = -p 1 /p 2 e2e2 e1e1 and passes through (e 1,e 2 ) X

Cobb-Douglas with parameter a = 0.5 U(q 1, q 2 ) = q q 2 0.5

Chapter 6 – results Cobb-Douglas with parameter a q 1 = a(p 1 e 1 +p 2 e 2 )/p 1 e q 2 = (1-a)(p 1 e 1 +p 2 e 2 )/ /p 2 Perfect substitutes 1:a if p 1 /p 2 < a then q 1 = (p 1 e 1 +p 2 e 2 )/p 1 q 2 = 0 if p 1 /p 2 = a then.... if p 1 /p 2 >a then q 1 = 0 q 2 = (p 1 e 1 +p 2 e 2 )/p 2 Perfetti complements 1 with a q 1 = (p 1 e 1 +p 2 e 2 )/(p 1 + ap 2 ) e q 2 =a(p 1 e 1 +p 2 e 2 )/(p 1 + ap 2 )

Chapter 6 Goodbye!