Economics Lecture Sebastiano Vitali

Transcription

1 Economics Lecture Sebastiano Vitali

2 Course Outline Consumer theory and its alications. Preferences and utility. Utility maimization and uncomensated demand.3 Eenditure minimization and comensated demand.4 Price changes and welfare.5 Labour suly, taes and benefits.6 Saving and borrowing

3 Firms, costs and rofit maimization. Firms and costs. Profit maimization and costs for a rice taking firm 3. Industrial organization 3. Perfect cometition and monooly 3. Oligooly and games

4 . Utility maimization and uncomensated demand

5 . Utility maimization and uncomensated demand. Budget line and budget set. Definition of uncomensated demand 3. Tangency and corner solutions 4. Finding uncomensated demand with Cobb-Douglas utility 5. The effects of changes in rices and income on uncomensated demand

6 6. Demand curves 7. Elasticity 8. Normal and inferior goods 9. Substitutes and comlements 0.Finding uncomensated demand with erfect comlements utility.finding uncomensated demand with erfect substitutes utility

7 Utility maimization and uncomensated demand. Budget line and budget set

8 . The budget set and budget line Assume that it is imossible to consume negative quantities. Notation quantities budget line rices income m budget set 0 Budget line + = m Budget set oints with 0, 0 + m.

9 What is the gradient of the budget line? budget line + = m Rearranging gives = - + m so = -( / ) + (m/ )

10 What is the gradient of the budget line? budget line + = m Rearranging gives = - + m so = -( / ) + (m/ ) gradient

11 Where does the budget line meet the aes? budget line + = m gradient /. = -( / ) + (m/ ) 0

12 Where does the budget line meet the aes? budget line + = m gradient /. = -( / ) + (m/ ) 0 m/

13 Where does the budget line meet the aes? m/ budget line + = m gradient /. = -( / ) + (m/ ) 0 m/

14 Utility maimization and uncomensated demand. Definition of uncomensated demand

15 Definition:. Definition of uncomensated demand functions The consumer s demand functions (,,m) and (,,m) maimize utility u(, ) subject to the budget constraint + m and non negativity constraints 0 0. Later we call this uncomensated demand. Some books use the term Marshallian demand.

16 To get uncomensated demand fi income and rices which fies the budget line. Get onto highest ossible indifference curve. 0

17 To get uncomensated demand fi income and rices which fies the budget line. Get onto highest ossible indifference curve. 0

18 To get uncomensated demand fi income and rices which fies the budget line. Get onto highest ossible indifference curve. 0

19 To get uncomensated demand fi income and rices which fies the budget line. Get onto highest ossible indifference curve. 0

20 To get uncomensated demand fi income and rices which fies the budget line. Get onto highest ossible indifference curve. 0 Comensated demand, Hicksian demand, is a demand function that holds utility fied and minimizes eenditures. Uncomensated demand, Marshallian demand, is a demand function that maimizes utility given rices and wealth.

21 Eamles of utility maimization (uncomensated demand)

22 Eamles of utility maimization (uncomensated demand). Cobb-Douglas utility. Perfect comlements 3. Perfect substitutes

23 Eamles of utility maimization (uncomensated demand) For each eamle we will look at Indifference curve diagram Effect of rices on demand, own rice and cross rice elasticities Effect of income on demand, normal and inferior goods, income elasticity Demand curve diagram

24 8 stes for finding uncomensated demand

25 8 stes for finding uncomensated demand with differentiable utility. Write down the roblem you are solving. What is the solution a function of? 3. Check for nonsatiation and conveity using calculus if the utility function has artial derivatives Elain their imlications. 4. Use the tangency and budget line conditions.

26 8 stes for finding uncomensated demand 5. Draw a diagram based on the tangency and budget line conditions. 6. Remind yourself what you are finding and what it deends on. 7. Write down the equations to be solved. 8. Solve the equations and write down the solution as a function. If at this oint 0 and 0 you have found the utility maimizing oint.

27 Why check for nonsatiation and conveity? If they are not satisfied there can be a tangency oint A where MRS = rice ratio that does not solve the roblem. referred set A Here nonsatiation fails. The tangency is at A but B maimizes utility. B. 0

28 Why check for nonsatiation and conveity? If they are not satisfied there can be a tangency oint A where MRS = rice ratio that does not solve the roblem. referred set A Here nonsatiation fails. The tangency is at A but B maimizes utility. B. 0

29 Why check for nonsatiation and conveity? A oint like A that is not a tangency cannot maimize utility. The oint B with > 0 and > 0 maimises utility. It must be a tangency oint. C referred set The oint C is a tangency oint but does not maimize utility. 0 A B Here A conveity fails

30 Why check for nonsatiation and conveity? A oint like A that is not a tangency cannot maimize utility. The oint B with > 0 and > 0 maimises utility. It must be a tangency oint. C The oint C is a tangency oint but does not maimize utility. 0 A B Here A conveity fails

31 Why check for nonsatiation and conveity? A oint like A that is not a tangency cannot maimize utility. The oint B with > 0 and > 0 maimises utility. It must be a tangency oint. C referred set The oint C is a tangency oint but does not maimize utility. 0 A B Here A conveity fails

32 Why check for nonsatiation and conveity? A oint like A that is not a tangency cannot maimize utility. The oint B with > 0 and > 0 maimises utility. It must be a tangency oint. C referred set The oint C is a tangency oint but does not maimize utility. 0 A B Here A conveity fails

33 Why check for nonsatiation and conveity? Here conveity fails. The tangency is A but B maimizes utility. B A referred set 0

34 Logic of first order conditions Very imortant. If the nonsatiation and conveity conditions are satisfied then any tangency oint at which referred set MRS = rice ratio 0, 0 budget set solves the utility maimizing roblem. 0

35 Finding a tangency solution The gradient of the indifference curve is MRS. The gradient of the budget line is /. If MRS = / the oint is tangent to some budget line with gradient /. B If in addition + = m the oint is on the budget line with income m. 0

36 We have already found that MRS u u so MRS = rice ratio requires that u u

37 Another way to look at the tangency condition Get Δ more units of increase in utility u Send more on gives more units of good / increase in utility u Send less on fall in utility u

38 Sending more on and less on good increases utility if Sending less on good and more on good increases utility if Utility maimization requires u u u u u u

39 Utility maimization requires u u or u u i.e. MRS = rice ratio

40 Finding uncomensated demand with Cobb-Douglas utility u(, ) = /5 3/5

41 4. Finding uncomensated demand with Cobb-Douglas utility Ste : What roblem are you solving? The roblem is maimizing utility u(, ) = /5 3/5 subject to non-negativity constraints 0 0 and the budget constraint + m. Ste : What is the solution a function of? Demand is a function of rices and income so is (,, m) (,,m)

42 Finding uncomensated demand with Cobb-Douglas utility Ste 3: Check for nonsatiation and conveity We have already done this, both are satisfied.

43 Easy to lose eam marks Failing to say that Because nonsatiation and conveity are satisfied any oint on the budget line at which MRS = rice ratio, 0 and 0 solves the utility maimizing roblem.

44 Finding uncomensated demand with Cobb-Douglas utility Because conveity and nonsatiation are satisfied any oint with + = m so it is on the budget line and MRS = solves the roblem here we have already found MRS = Ste 4: Use the tangency and budget line conditions u u / / / /

45 Finding uncomensated demand with Cobb-Douglas utility Ste 5: Draw a diagram based on the tangency and budget line conditions tangency condition MRS = = 3 + = m budget line 0

46 Finding uncomensated demand with Cobb-Douglas utility Ste 6: Remind yourself what you are finding and what it deends on. You are finding demand and which is a function of, and m. Ste 7: Write down the equations to be solved. The equations are + = m and = 3

47 Finding uncomensated demand with Cobb-Douglas utility Ste 8 solve the equations and write down the solution as a function. (You will do some algebra here.) Solving the equations simultaneously for and gives (uncomensated) demand which is a function of,, m. (,,m) = m (,,m) = 3 m 5 5 because the conditions 0, 0 are satisfied.

48 An alternative aroach: using Lagrangians You can also use Lagrangians to find demand. With two goods, the Lagrangian is not essential. You can base your analysis on grahs and simle algebra. With more than goods you have to use Lagrangians.

49 Homogeneity of uncomensated demand

50 5. The effects of changes in rices and income on uncomensated demand If all rices and income are multilied by a number k > 0 what haens?

51 If all rices and income are all multilied by what haens?. Demand for good increases.. Demand for good decreases. 3. Demand for good does not change. 4. Demand for good increases. 5. Demand for good decreases. 6. Demand for good does not change.

52 Mathematical definition of homogeneous functions A function f(z,z,z 3..z n ) is homogeneous of degree 0 if for all numbers k > 0 f(kz,kz,kz 3..kz n ) = k 0 f(z,z,z 3..z n ) = f(z,z,z 3..z n ). Multilying z,z,..z n by k > 0 does not change the value of f. A function f(z,z,z 3..z n ) is homogeneous of degree one if for all numbers k > 0 f(kz,kz,kz 3..kz n ) = k f(z,z,z 3..z n ) = kf(z,z,z 3..z n ) Multilying z, z z n multilies the value of f by k.

53 All rices and income are multilied by k > 0. m/ + = m gradient / budget set How does the budget line change? How does demand change? m/ The consumer s demand functions (,,m) and (,,m) are in rices and income. That is if k > 0 (k,k,km) = (k,k,km) =

54 All rices and income are multilied by k > 0. m/ + = m gradient / budget set How does the budget line change? no change How does demand change? m/ The consumer s demand functions (,,m) and (,,m) are in rices and income. That is if k > 0 (k,k,km) = (k,k,km) =

55 All rices and income are multilied by k > 0. m/ + = m gradient / budget set How does the budget line change? no change How does demand change? no change m/ The consumer s demand functions (,,m) and (,,m) are in rices and income. That is if k > 0 (k,k,km) = (k,k,km) =

56 All rices and income are multilied by k > 0. m/ + = m gradient / budget set How does the budget line change? no change How does demand change? no change m/ The consumer s demand functions (,,m) and (,,m) are homogeneous of degree 0 in rices and income. That is if k > 0 (k,k,km) = (k,k,km) =

57 All rices and income are multilied by k > 0. m/ + = m gradient / budget set How does the budget line change? no change How does demand change? no change m/ The consumer s demand functions (,,m) and (,,m) are homogeneous of degree 0 in rices and income. That is if k > 0 (k,k,km) = (,,m) (k,k,km) =

58 All rices and income are multilied by k > 0. m/ + = m gradient / budget set How does the budget line change? no change How does demand change? no change m/ The consumer s demand functions (,,m) and (,,m) are homogeneous of degree 0 in rices and income. That is if k > 0 (k,k,km) = (,,m) (k,k,km) = (,,m)

59 All uncomensated demand functions are homogeneous of degree 0 in rices and income This means that if all rices and income are all multilied by k > 0 demand does not change. With Cobb-Douglas utility u(, ) = /5 3/5 (,,m) = m (,,m) = 3 m 5 5 The values of these functions do not change when, and m are all multilied by k > 0.

60 Easy to lose eam marks Elain what haens to uncomensated demand when rices and income are all multilied by Saying nothing haens because uncomensated demand is homogeneous of degree 0 in rices This is a statement of what haens it is not an elanation.

61 Changes in demand and demand curves TP

62 The budget line moves from L to L. Is this due to. An increase in. A decrease in 3. An increase in 4. A decrease in 5. An increase in m 6. A decrease in m L L 0

63 increases, and m do not change. What haens to demand for goods and? With Cobb-Douglas utility u(, ) = /5 3/5 (,,m) = m (,,m) = 3 m 5 5

64 increases, and m do not change. What haens to demand for goods and?. Demand for good increases.. Demand for good decreases. 3. Demand for good does not change. 4. Demand for good increases. 5. Demand for good decreases. 6. Demand for good does not change.

65 With Cobb-Douglas utility u(, ) = /5 3/5 (,,m) = m (,,m) = 3 m 5 5 Demand for good is not affected by the rice of good. rice consumtion curve 0

66 6. Demand curves Demand (,,m) = m 5 To draw the demand curve with on the vertical ais rearrange this formula to get as a function of = m 5 0

67 Demand for good If increases from A to B there is a movement the demand curve, demand B A 0 B A

68 Demand for good If increases from A to B there is a movement on the demand curve, demand falls from A to B. B A 0 B A

69 Demand for good If changes there is in demand for good. 0

70 Demand for good If changes there is no change in demand for good. (imlied by Cobb-Douglas utility, not generally true.) 0

71 Demand for good If income m increases demand for good The demand curve 0

72 Demand for good If income m increases demand for good increases. The demand curve shifts outwards. 0

73 Elasticity

74 7. Elasticity Measuring the imact of changes in rices & income Own rice elasticity is % change in quantity % change in own rice Elasticity catures intuition better than numerical change in quantity numerical change in rice

75 A rice increase from to is large. A rice increase from to 0 00 is small. Elasticity does not deend on units ( $ or, kilos or ounds) because % changes do not deend on units.

76 Elasticity matters for every decision on rices, e.g. for a monooly or oligooly deciding on rices for a government deciding on taes.

77 either Own rice elasticity of demand Δ = = Δ (negative, Snyder & Nicholson, lectures) Δ Δ or Δ = = Δ Δ Δ Some authors makes elasticity ositive

78 Elasticity and demand curves rice of good A B 0 quantity of good Which demand curve is more elastic A or B?

79 Elasticity and demand curves rice of good A B 0 quantity of good Which demand curve is more elastic A or B?

80 5 5 income elasticity cross rice elasticity 5 5 own rice elasticity Find 5 ),, ( Uncomensated demand for goodis m m m m m m m m m (from Cobb Douglas utility)

81 5 5 income elasticity cross rice elasticity 5 5 own rice elasticity Find 5 ),, ( Uncomensated demand for goodis m m m m m m m m m (from Cobb Douglas utility)

82 5 5 income elasticity cross rice elasticity 5 5 own rice elasticity Find 5 ),, ( Uncomensated demand for goodis m m m m m m m m m (from Cobb Douglas utility)

83 5 5 income elasticity cross rice elasticity 5 5 own rice elasticity Find 5 ),, ( Uncomensated demand for goodis m m m m m m m m m (from Cobb Douglas utility)

84 5 5 income elasticity cross rice elasticity 5 5 own rice elasticity Find 5 ),, ( Uncomensated demand for goodis m m m m m m m m m (from Cobb Douglas utility)

85 5 5 income elasticity cross rice elasticity 5 5 own rice elasticity Find 5 ),, ( Uncomensated demand for goodis m m m m m m m m m

86 With Cobb-Douglas utility u(, ) = /5 3/5 (,,m) = m (,,m) = 3 m 5 5 MRS = rice ratio imlies = 3 income consumtion curve 0

87 Normal & inferior goods

88 8. Normal and inferior goods A good is normal if consumtion increases. when income A good is inferior if consumtion increases. m income elasticity m ositive negative if if is a is an m when income m

89 Normal and inferior goods A good is normal if consumtion increases when income increases. A good is inferior if consumtion increases. m income elasticity m ositive negative if if is a is an m when income m

90 Normal and inferior goods A good is normal if consumtion increases when income increases. A good is inferior if consumtion decreases when income increases. m m income elasticity m m ositive negative if if is a is an

91 Normal and inferior goods A good is normal if consumtion increases when income increases. A good is inferior if consumtion decreases when income increases. m m income elasticity m m ositive negative if if is a is an normal good

92 Normal and inferior goods A good is normal if consumtion increases when income increases. A good is inferior if consumtion decreases when income increases. m m income elasticity m m ositive negative if if is a is an normal good inferior good

93 Substitutes & comlements

94 9. Substitutes and comlements If demand for good increases when the rice of good increases goods and are substitutes. If demand for good decreases when the rice of good increases goods and are comlements.

95 If and are substitutes. Demand for increases when increases.. Demand for decreases when increases.

96 If and are comlements. Demand for increases when increases.. Demand for decreases when increases.

97 Substitutes and Comlements If demand for good increases when the rice of good increases goods and are substitutes. If demand for good decreases when the rice of good increases goods and are comlements. cross rice elasticitity ositive if negative if and and are are

98 Substitutes and Comlements If demand for good increases when the rice of good increases goods and are substitutes. If demand for good decreases when the rice of good increases goods and are comlements. cross rice elasticitity ositive if negative if and and are are substitutes

99 Substitutes and Comlements If demand for good increases when the rice of good increases goods and are substitutes. If demand for good decreases when the rice of good increases goods and are comlements. cross rice elasticitity ositive if negative if and and are are substitutes comlements

100 Price of good A C Shifts in demand curves 0 Quantity of good Shift A to C. This is an increase in demand. Causes? Increase or decrease in rice of a comlement? Increase or decrease in rice of a substitute? Increase or decrease in income for a normal good. Increase or decrease in income for an inferior good.

101 Price of good A C Shifts in demand curves 0 Quantity of good Shift A to C. This is an increase in demand. Causes? Increase or decrease in rice of a comlement? Increase or decrease in rice of a substitute? Increase or decrease in income for a normal good. Increase or decrease in income for an inferior good.

102 Price of good A C Shifts in demand curves 0 Quantity of good Shift A to C. This is an increase in demand. Causes? Increase or decrease in rice of a comlement? Increase or decrease in rice of a substitute? Increase or decrease in income for a normal good. Increase or decrease in income for an inferior good.

103 Price of good A C Shifts in demand curves 0 Quantity of good Shift A to C. This is an increase in demand. Causes? Increase or decrease in rice of a comlement? Increase or decrease in rice of a substitute? Increase or decrease in income for a normal good. Increase or decrease in income for an inferior good.

104 Price of good A C Shifts in demand curves 0 Quantity of good Shift A to C. This is an increase in demand. Causes? Increase or decrease in rice of a comlement? Increase or decrease in rice of a substitute? Increase or decrease in income for a normal good. Increase or decrease in income for an inferior good.

105 Finding uncomensated demand with erfect comlements utility

106 0. Finding uncomensated demand with erfect comlements utility In general u(, ) = min(a,b ) here u(, ) = min(½, ) bike wheels, bicycle frames u 00 u(, ) = min( ½,00) if = wheels Getty Images

107 Perfect comlements utility: indifference curves u(, ) = min( ½, ) frames bicycle wheels, bicycle frames = ½ if < ½ increasing does not change utility if > ½ increasing increases utility. 0 wheels

108 Perfect comlements utility: indifference curves u(, ) = min( ½, ) frames = ½ bicycle wheels, bicycle frames if < ½ increasing does not change utility if > ½ increasing increases utility. 0 wheels

109 Perfect comlements utility: indifference curves u(, ) = min( ½, ) frames bicycle wheels, bicycle frames = ½ if < ½ increasing does not change utility if > ½ increasing increases utility. 0 wheels

110 Perfect comlements utility: indifference curves u(, ) = min( ½, ) frames bicycle wheels, bicycle frames = ½ if = ½ it is necessary to increase both and to increase utility 0 wheels

111 Nonsatiation in the indifference curve diagram with differentiable utility C. D 0 Nonsatiation means that any oint such as D inside or on the boundary of the shaded area is referred to C. Here starting from C increasing and/or increasing increases utility. Check for this by seeing if the artial derivatives of utility function are > 0.

112 Nonsatiation in the indifference curve diagram with erfect comlements utility frames Here starting from A increasing and increases utility. A Increasing only or only does not increase utility. (Think about frames & wheels.) 0 wheels The function u(, ) = min( ½, ) does not have artial derivatives when ½ = does not have MRS. Can t use calculus.

113 Perfect comlements: utility maimization budget line + = m u(, ) = min( ½, ) utility maimization frames = ½ imlies that (, ) lies at the kink of the indifference curves so 0 wheels = ½ and satisfies the budget constraint so + = m.

114 Perfect comlements: utility maimization = ½ + = m. Solving simultaneously for and gives = m = m ( + ) ( + )

115 Common mistake wheels, frames, wheels for each frame wheels frames Easy to think that utility should be u(, ) = min(, ) But this imlies that =, number of frames = (number of wheels) Utility is u(, ) = min(½, )

116 Demand curves and changes in rices and income with erfect comlements utility = m - demand curve diagram, rice on vertical ais quantity on horizontal ais 0 Increase in results in Increase in results in Increase in m results in

117 Demand curves and changes in rices and income with erfect comlements utility = m - demand curve diagram, rice on vertical ais quantity on horizontal ais 0 Increase in results in movement along demand curve. Increase in results in Increase in m results in

118 Demand curves and changes in rices and income with erfect comlements utility = m - demand curve diagram, rice on vertical ais quantity on horizontal ais 0 Increase in results in movement along demand curve. Increase in results in shift down in demand curve. Increase in m results in

119 Demand curves and changes in rices and income with erfect comlements utility = m - demand curve diagram, rice on vertical ais quantity on horizontal ais 0 Increase in results in movement along demand curve. Increase in results in shift down in demand curve. Increase in m results in shift u in demand curve.

120 Finding uncomensated demand with erfect substitutes utility: corner solutions again

121 Perfect substitutes utility In general u(, ) = a + b u(, ) = 3 +. indifference curves u = 3 + gradient 3/ 0

122 . Finding uncomensated demand with erfect substitutes utility Ste : What roblem are you solving? The roblem is maimizing utility u(, ) = 3 + subject to non-negativity constraints 0 0 and the budget constraint + m. Ste : What is the solution a function of? Demand is a function of rices and income so is (,, m) (,,m)

123 Finding uncomensated demand with erfect substitutes utility Ste 3: Check for nonsatiation and conveity u Getting 3 0, 3/ as a functionof u conveity is satisfied. 0 so nonsatiation is satisfied. 0 u and gives ( u 3 )/ so

124 Finding uncomensated demand with erfect substitutes utility Ste 4: Use the tangency and budget line conditions Because conveity and nonsatiation are satisfied any oint with + = m so is on the budget line and MRS = MRS = u u / 5 3 / 5 3 / 5 / 5 Problem What if / = 3/? It is better to use a diagram. 3

125 A B A B A B 0 C / < 3/ solution at 0 C / = 3/ solution at 0 C / > 3/ solution at

126 A B A B A B 0 C / < 3/ solution at C = m/, = 0 0 C 0 C

127 A B A B A B 0 C / < 3/ solution at C = m/, = 0 0 C / = 3/ solution at 0 C

128 A B A B A B 0 C / < 3/ solution at C = m/, = 0 0 C / = 3/ solution at any satisfying 0 0 and budget constraint 0 C

129 A B A B A B 0 C / < 3/ solution at C = m/, = 0 0 C / = 3/ solution at any satisfying 0 0 and budget constraint 0 C / > 3/ solution at

130 A B A B A B 0 C / < 3/ solution at C = m/, = 0 0 C / = 3/ solution at any satisfying 0 0 and budget constraint 0 C / > 3/ solution at A = 0 = m/.

131 What have we achieved? Model of consumer demand: given references satisfying listed assumtions. Show that references can be reresented by utility functions: mathematically convenient. Model shows how demand resonds to changes in own rice, rice of other good, income. Model has only two goods, but with more maths can easily be etended to many goods.