Microeconomic Theory -1- Introduction

Transcription

1 Microeconomic Theory -- Introduction. Introduction. Profit maximizing firm with monopoly power 6 3. General results on maximizing with two variables 8 4. Model of a private ownership economy 5. Consumer choice (constrained maximization) 6 6. Market demand and equilibrium in an exchange economy 55

2 Microeconomic Theory -- Introduction. Introduction Four questions What makes economic research so different from research in the other social sciences (and indeed in almost all other fields)?

3 Microeconomic Theory -3- Introduction. Introduction Four questions What makes economic research so different from research in the other social sciences (and indeed in almost all other fields)? What are the two great pillars of economic theory?

4 Microeconomic Theory -4- Introduction. Introduction Four questions What makes economic research so different from research in the other social sciences (and indeed in almost all other fields)? What are the two great pillars of economic theory? Who are you going to learn most from at UCLA?

5 Microeconomic Theory -5- Introduction. Introduction Four questions What makes economic research so different from research in the other social sciences (and indeed in almost all other fields)? What are the two great pillars of economic theory? Who are you going to learn most from at UCLA? What do economists do? Discuss in 3 person groups

6 Microeconomic Theory -6- Introduction. Profit-maximizing firm An example Cost function C( q) 0q q Demand price function p 65 4 q Group exercise: Solve for the profit maximizing output and price.

7 Microeconomic Theory -7- Introduction Two products MODEL Cost function C( q) 0q 5q q 3q q q Demand price functions p 85 q and p 90 q 4 4 Group exercise: How might you solve for the profit maximizing outputs? MODEL Cost function C( q) 0q 5q q 3q q q Demand price functions p 65 q and p 70 q 4 4 Group exercise: How might you solve for the profit maximizing outputs?

8 Microeconomic Theory -8- Introduction MODEL : Revenue R p q (85 q ) q 85q q, 4 4 R p q (90 q ) q 90q q 4 4 Profit R R C 85q q 90 q q (0q 5q q 3q q q ) q 75q q q 3q q

9 Microeconomic Theory -9- Introduction Think on the margin Marginal profit of increasing q 9 75 q 3q q Therefore the profit-maximizing choice is. q m ( q ) (75 3 q ) (5 q ) q 75q q q 3q q Marginal profit of increasing q Model : Profit-maximizing lines 75 3q q q 9. Therefore the profit-maximizing choice is q m ( q ) (75 3 q ) (5 q ). 9 3 The two profit-maximizing lines are depicted.

10 Microeconomic Theory -0- Introduction q m ( q ) (5 q ), q m ( q ) (5 q ) 3 3 If you solve for q satisfying both equations you will find that the unique solution is q ( q, q ) (0,0).

11 Microeconomic Theory -- Introduction MODEL Cost function C( q) 0q 5q q 3q q q Demand price functions p 65 q and p 70 q 4 4 Revenue R p q (65 q ) q 65q q, 4 4 R p q (70 q ) q 70q q 4 4 Profit R R C 65q q 70 q q (0q 5q q 3 q q q ) q 55q q q 3q q

12 Microeconomic Theory -- Introduction Think on the margin Marginal profit of increasing q 5 55 q 3q q. Therefore, for any q the profit-maximizing q is q m ( q ) (55 3 q ). 5 Marginal profit of increasing q 55 3q q q 5. Therefore, for any q the profit-maximizing q is Model : Profit-maximizing lines. q m ( q ) (55 3 q ) 5 The two profit-maximizing lines are depicted. If you solve for q satisfying both equations you will find that the unique solution is q ( q, q ) (0,0).

13 Microeconomic Theory -3- Introduction These look very similar to the profit-maximizing lines in Model. However now the profit-maximizing line for q is steeper (i.e. has a more negative slope). Model : Profit-maximizing lines Model : Profit-maximizing lines. As we shall see, this makes a critical difference.

14 Microeconomic Theory -4- Introduction Is q the profit-maximizing output bundle (output vector)? Group Exercise: For model solve for maximized profit if only one commodity is produced. Compare this with the profit if q (0,0) is produced.

15 Microeconomic Theory -5- Introduction Model Suppose we alternate, first maximizing with respect to q, then q and so on. There are four zones. Z(, ) : The zone in which q is increasing and q is increasing Z(, ) : The zone in which q is increasing and q is decreasing Model : Profit-maximizing lines and so on If you pick any starting point you will find this process leads to the intersection point q (0,0).

16 Microeconomic Theory -6- Introduction The profit is depicted below (using a spread-sheet)

17 Microeconomic Theory -7- Introduction MODEL Suppose we alternative, first maximizing with respect to There are four zones. Z(, ) : q, then q and so on. The zone in which q is increasing and q is increasing Z(, ) : The zone in which q is increasing and q is decreasing and so on If you pick any starting point you will find this process never leads to the intersection point q (0,0).

18 Microeconomic Theory -8- Introduction The profit function has the shape of a saddle. The output vector q each axis is zero is called a saddle-point. where the slope in the direction of

19 Microeconomic Theory -9- Introduction 3. General results Consider the two variable problem Max{ f ( q, q )} q Necessary conditions Consider any q 0. If the slope in the cross section parallel to the q -axis is not zero, then by standard one variable analysis, the function is not maximized. The same holds for the cross section parallel to the q -axis. Thus for q to be a maximizer, the slope of both cross sections must be zero. First order necessary conditions for a maximum For q 0 to be a maximizer the following two conditions must hold f q ( q) 0 and f q ( q) 0 (3-)

20 Microeconomic Theory -0- Introduction Suppose that the first order necessary conditions hold at q. Also, if the slope of the cross section parallel to the q -axis is strictly increasing in q at q, then q is not a maximizer. Thus a necessary condition for a maximum is that the slope must be decreasing. Exactly the same argument holds for q. We therefore have a second set of necessary conditions for a maximum. Since they are restrictions on second derivatives they are called the second order conditions. Second order necessary conditions for a maximum If q 0 is a maximizer of f( q ), then q f q ( q) 0 and q f q ( q) 0 (3-)

21 Microeconomic Theory -- Introduction As we have seen, these conditions are necessary for a maximum but they do not, by themselves guarantee that q satisfying these conditions is the maximum. However, if the step by step approach does lead to q then this point is a least a local maximizer. Proposition: Sufficient conditions for a local maximum If in the neighborhood of q 0 satisfying the first order conditions (FOC) the step by step approach leads to q, then the function f( q ) has a local maximum at q Proposition: Sufficient conditions for a global maximum If the conditions above hold at q and the FOC hold only at q, then this is the global maximizer. If there is a unique q it is the global maximizer.

22 Microeconomic Theory -- Introduction 4. Model of a private ownership economy Commodities: The set of commodities is N {,..., n} Endowments: 0 the initial endowment of commodity (land, labor, coconuts ) Consumers: The set of consumers is H {,..., H}. Each consumer s preferences can be represented by a continuously differentiable strictly increasing h function U ( x ), h,..., H where x h ( x h,..., x h n ) Notation: If xi x for i,..., n and the inequality is strict for some i i we write x x. Then utility is increasing if x x implies that h h U ( x) U ( x).

23 Microeconomic Theory -3- Introduction Firms The set of firms is F {,..., F}. Transformers of inputs into outputs: f f f Let z ( z,..., z n ) be a vector of firm f s inputs. Each component is a quantity of one of the commodities. Let q f be a vector of the outputs. Production vector f f f y ( z, q ) is an ordered list of all the inputs and outputs. Each firm must choose among the production vectors that are feasible. We write this set of feasible plans as Private ownership: Shareholdings: f Y. hf k is the share-holding of consumer h in firm f.

24 Microeconomic Theory -4- Introduction Feasible consumption vectors h h F F h h f f x z q h h f f, where (, ) f f f f y z q Y (production plans are feasible) total consumption total endowment - total inputs + total outputs Price taking equilibrium allocation An allocation to consumers, { x h } H, feasible production plans h { y f } F f and a price vector p 0 Such that (i) No consumer has a strictly preferred point in his/her budget set. (ii) There is no strictly more profitable feasible plan for any firm. (iii) All markets clear. h h F F h h f f x z q h h f f and 0 p if the inequality is strict.

25 Microeconomic Theory -5- Introduction We will look at simple economies to gain insights into how commodities are allocated via markets. For now consider one example. Alex likes only bananas. Bev likes only coconuts. A Alex has an endowment of 4 bananas and 6 coconuts (4,6). A Bev has an endowment of 0 bananas and 7 coconuts (0,7). Group exercise: What are market clearing prices in this economy?

26 Microeconomic Theory -6- Introduction 5. Consumers We begin by considering a consumer maximization problem. h Max{ U ( x) p x... p x I} x 0 n n Using the sumproduct notation a b ab... anbn, we can write this as follows: h Max{ U ( x) p I}. x 0 h We will assume that U( x ) is a strictly increasing function. This is an example of a maximization with a single linear resource constraint. As you will see by looking at sections A-C in the following slides, an almost identical argument holds for any maximization problem with resource constraints.

27 Microeconomic Theory -7- Introduction h Max{ U ( x) p I}. x 0 Let x be a solution. Note that, since U( x ) is strictly increasing, p I Example with commodities: Max{ U( x) x x p x p x I} x 0 Note that utility is zero if consumption of either commodity is zero. Therefore every component of the solution is strictly positive. (We write x 0 ). x

28 Microeconomic Theory -8- Introduction Geometry Max U x x x p x p x I x 0 { ( ) } In the commodity case we can represent preferences by depicting points for which utility has the same value. Such a set of points is called a level set. In the figure the level sets are the boundaries of the blue, red and green shaded regions. Note that utility is zero if consumption of either commodity is zero. Therefore every component of the solution x is strictly positive.(we write x 0 ).

29 Microeconomic Theory -9- Introduction Example with commodities: Max{ U( x) x x p x p x I} x 0 Level sets In mathematical notation the 4 level sets are { x U( x) 0},{ x U( x) }, { x U( x) },{ x U( x) 3}. Three of them are what economists often call indifference curves. Superlevel sets The set of points on or above a level set { x U( x) k} is called a superlevel set.

30 Microeconomic Theory -30- Introduction The set of points satisfying px px I I p Can be rewritten as x x p p p This is a line of slope p In the figure both components of the solution x ( x, x ) are strictly positive. (Mathematical shorthand x 0.) So the slope of the budget line is equal to the slope of the level set.

31 Microeconomic Theory -3- Introduction Necessary conditions for a maximum To a first approximation, if a consumer currently, choosing x commodity by, the change in utility is can increase consumption of U U ( x). This is depicted in the figure.. The slope of the tangent line at x is U ( x ). Fig. 4-: Utility as a function of

32 Microeconomic Theory -3- Introduction Necessary conditions for a maximum To a first approximation, if a consumer currently, choosing x commodity by, the change in utility is can increase consumption of U U ( x). This is depicted in the figure.. The slope of the tangent line at x is U ( x ). Fig. -3: Utility as a function of If the consumer has an additional E E dollars then E p x and so. p

33 Microeconomic Theory -33- Introduction Necessary conditions for a maximum To a first approximation, if a consumer currently, choosing x commodity by, the change in utility is can increase consumption of U U ( x). This is depicted in the figure.. The slope of the tangent line at x is U ( x ). Fig. 4-: Utility as a function of If the consumer has an additional The increase in utility is therefore E E dollars then E p x and so. p U U E U U ( x) ( x) ( x) E p p

34 Microeconomic Theory -34- Introduction We have seen that Therefore U U ( x) E p U U ( x ) E p In the limit as commodity E rises. approaches zero, this becomes the rate at which utility rises as expenditure on p U ( x ) is the marginal utility per dollar as expenditure on commodity rises

35 Microeconomic Theory -35- Introduction Suppose that the consumer spends dollar less on commodity. His change in utility is p U ( x ). He then spends the dollar on commodity i. The change in utility is U ( x ). The net change in utility is therefore p * U U ( x) ( x) p p i i i i

36 Microeconomic Theory -36- Introduction Suppose that the consumer spends dollar less on commodity. His change in utility is p U ( x ). He then spends the dollar on commodity i. The change in utility is U ( x ). The net change in utility is therefore p U U ( x) ( x) p p i i Case (i) x, x 0 i i i If the change in utility is strictly positive the current utility can be increased by consuming more of commodity i and less of commodity. If it is negative, utility can be increased by spending less commodity and more on commodity i that. Thus a necessary condition for x to be utility maximizing is U U ( x) ( x) p p i i

37 Microeconomic Theory -37- Introduction Case (ii) x x 0 i If the difference in marginal utilities is positive current Ux ( ) can be increased by spending a positive amount on commodity. Thus a necessary condition for x to be utility maximizing is that U U ( x) ( x) p p i i

38 Microeconomic Theory -38- Introduction Case (ii) x x 0 i If the difference in marginal utilities is positive current Ux ( ) can be increased by spending a positive amount on commodity. Thus a necessary condition for x to be utility maximizing is that U U ( x) ( x) p p i i Let be the common marginal utility per dollar for all those commodities that are consumed in strictly positive amounts. We can therefore summarize the necessary conditions as follows: Necessary conditions for a maximum If x 0 then U ( x ) p If x 0 then p U ( x ) Note: Since is the rate at which utility rises with income it is called the marginal utility of income

39 Microeconomic Theory -39- Introduction For the general problem, Max{ f ( x) h( x) b g( x) 0} x b g is the extra resources needed to increase commodity by. Therefore b g is the extra output you can get using an extra b of the resource. An almost identical argument then yields the following necessary conditions. f If x 0 then ( x) g f. If x 0 then ( x) g

40 Microeconomic Theory -40- Introduction Use the Lagrangian to write down the necessary conditions Problem Max{ f ( x) h( x) b g( x) 0} x The Lagrangian for the maximization problem is defined as follows: L ( x, ) f ( x) h( x) f ( x) ( b g( x)), where 0 **

41 Microeconomic Theory -4- Introduction Use the Lagrangian to write down the necessary conditions Problem Max{ f ( x) h( x) b g( x) 0} x The Lagrangian for the maximization problem is defined as follows: L ( x, ) f ( x) h( x) f ( x) ( b g( x)), where 0 Necessary conditions for x to be a solution to this maximization problem. L f h f g ( x, ) ( x) ( x) 0, with equality if x 0.,..., n () L ( x, ) h( x) b g( x) 0, with equality if 0. () *

42 Microeconomic Theory -4- Introduction Use the Lagrangian to write down the necessary conditions Problem Max{ f ( x) h( x) b g( x) 0} x The Lagrangian for the maximization problem is defined as follows: L ( x, ) f ( x) h( x) f ( x) ( b g( x)), where 0 Necessary conditions for x to be a solution to this maximization problem. L f h f g ( x, ) ( x) ( x) 0, with equality if x 0.,..., n () L ( x, ) h( x) b g( x) 0, with equality if 0. () From (), note that if x i and x 0, then f g f i g i. And if x 0 x, then i f g f i g i.

43 Microeconomic Theory -43- Introduction An alternative approach From the argument above, if both commodity i of their marginal utilities must be equal to the price ratio. and commodity are consumed, then the ratio To understand this consider a change in x i and x that leaves the consumer on the same level set. i.e. U( x, x ) U( x, x )

44 Microeconomic Theory -44- Introduction An alternative approach From the argument above, if both commodity i of their marginal utilities must be equal to the price ratio. and commodity are consumed, then the ratio To understand this consider a change in x i and x that leaves the consumer on the same level set. i.e. U( x, x ) U( x, x ) Above we showed that, to a first approximation, U U ( x). If we increase the quantity of commodity and reduce the quantity of commodity i, then the net change in utility is U U U ( x) ( x) i i

45 Microeconomic Theory -45- Introduction U U We have argued that U ( x) ( x) For this net change to be zero, i i i U i U

46 Microeconomic Theory -46- Introduction U U We have argued that U ( x) ( x) For this net change to be zero, i i i U i U In the figure, - is the slope of the level set at x. The ratio is the rate at which x must be substituted into the consumption bundle to compensate for a reduction in x Hence we call it the marginal rate of substitution of x for x. Definition: Marginal rate of substitution U i MRS( xi, x ) U

47 Microeconomic Theory -47- Introduction For x to be the maximizer the rate at which x can be substituted into the budget as reduced must leave total expenditure x is on the two commodities constant, i.e., pii p Then along the budget line 0 i p p i Graphically, the slope of the budget line must be equal to the slope of the indifference curve at x ie. U i pi MRS( xi, x ) U p

48 Microeconomic Theory -48- Introduction The example: Cobb-Douglas utility function: Max{ U( x) x x p x p x I} x 0 Necessary conditions for a maximum Method : Equalize the marginal utility per dollar To make differentiation simple, try to find an increasing function of the utility function that is simple.

49 Microeconomic Theory -49- Introduction The example: Cobb-Douglas utility function: Max{ U( x) x x p x p x I} x 0 Necessary conditions for a maximum Method : Equalize the marginal utility per dollar To make differentiation simple, try to find an increasing function of the utility function that is simple. Define the new utility function u( x) ln U( x) The new maximization problem is That is Max{ u( x) ln U( x) p x p x I} x 0 x 0 Note that Max{ ln x ln x p x p x I} u x.

50 Microeconomic Theory -50- Introduction Necessary conditions u u p p. For the example it follows that. p x p x Also px px I. Technical tip Ratio Rule: a b If a a a a a and b b 0 then b b b b b. **

51 Microeconomic Theory -5- Introduction Necessary conditions u u p p. For the example it follows that. p x p x Also px px I. Technical tip Ratio Rule: a b If a a a a a and b b 0 then b b b b b. Therefore * p x p x p x p x I

52 Microeconomic Theory -5- Introduction Necessary conditions u u p p. For the example it follows that. p x p x Also px px I. Technical tip Ratio Rule: a b If a a a a a and b b 0 then b b b b b. Therefore p x p x p x p x I We can then solve for x and x Cobb-Douglas demands x I p,

53 Microeconomic Theory -53- Introduction Method : Equate the MRS and price ratio U ( x) x x. Then U x x U and x x U MRS( x, x ) x x x x U x x x. Then to be the maximizer, x MRS( x, x) x p p As we have seen, it is helpful to rewrite this as follows: p x p x. Then proceed as before.

54 Microeconomic Theory -54- Introduction Data Analytics (Taking the model to the data) x ( p, I) Take the logarithm I p. x I p ln ln( ) ln ln The model is now linear. We can then use least squares estimation ln x a a (ln I ln p ) 0 or ln x a a ln I a ln p 0 Exercise: If U ( x) ( a x ) ( a x ), solve for the demand function x ( p, I )

55 Microeconomic Theory -55- Introduction 6. Market demand in a Cobb-Douglas economy with no production Consumer h H has some initial endowment (,..., n ). With a price vector p, the market value of this endowment is I h h. If the consumer sells his entire endowment he can then purchase p any consumption bundle h h h p x I p. h x satisfying **

56 Microeconomic Theory -56- Introduction 6. Market demand in a Cobb-Douglas economy with no production Consumer h H has some initial endowment (,..., n ). With a price vector p, the market value of this endowment is I h h. If the consumer sells his entire endowment he can then purchase p any consumption bundle h h h p x I p. h x satisfying Consider a -commodity economy in which each consumer has the same Cobb-Douglas preferences. U ( x ) ( x ) ( x ). h h h *

57 Microeconomic Theory -57- Introduction 5. Market demand in a Cobb-Douglas economy with no production Consumer h H has some initial endowment (,..., n ). With a price vector p, the market value of this endowment is I h h. If the consumer sells his entire endowment he can then purchase p any consumption bundle h h h p x I p. h x satisfying Consider a -commodity economy in which each consumer has the same Cobb-Douglas preferences. U ( x ) ( x ) ( x ). h h h From section 4, demand for commodity is x h h h I ( p, I ) p where

58 Microeconomic Theory -58- Introduction Suppose that all consumers have the same preferences Then a consumer with a budget of has a consumption of x ( p,) h s demand is simply scaled up by her income. p. Therefore consumer x h (, h ) h (,) p I I x p x ( p,) where p h (, h ) h x (,) p I I x p Summing over consumers, Cobb-Douglas market demand for two consumers is x ( p) x ( p, I ) x ( p, I ) x ( p,) I x ( p,) I x ( p,)( I I ) h h For H consumers, market demand is H h h H H h x ( p) x ( p, I ) x ( p,) I... x ( p,) I x ( p,)( I... I )

59 Microeconomic Theory -59- Introduction Consider a very simple exchange economy No production. On one end of the island there are banana plantations. At the other end there are coconut plantations. Given a price vector p, if consumer h I p p p. h h h h h h h has an endowment of (, ), this has market value She can trade this for any other bundle of bananas and coconuts costing less. So her maximiaation problem is h h h h h Max{ U ( x p x p I }. h x 0 The vale of all the endowment is h where. h I... I p H h Therefore if all consumers have the same Cobb-Douglas preferences. x ( p) p p

60 Microeconomic Theory -60- Introduction h h h Equilibrium in an exchange economy with Cobb-Douglas utility U ( x ) ( x ) The total supply of commodity is x ( p) p ( p p ) 0 p p Solving p ( p p ) p p. The market commodity clears if

61 Microeconomic Theory -6- Introduction Additional example: (A bit technical for a closed book exam) U( x) 9 x x, U ( x ) x U x 9, ( ) x Solve the budget problem: Max{ U( x) x 4 x ****.

62 Microeconomic Theory -6- Introduction Additional example: (A bit technical for a closed book exam) U( x) 9 x x, U ( x ) x U x 9, ( ) x Solve the budget problem: Max{ U( x) x 4 x Step : x 0 (why?) U U 9. *** p p x x 4.

63 Microeconomic Theory -63- Introduction Additional example: (A bit technical for a closed book exam) U( x) 9 x x, U ( x ) x U x 9, ( ) x Solve the budget problem: Max{ U( x) x 4 x Step : If x 0 U U 9. p p x x 4 Step : Make the denominators linear in each commodity by taking the square root. 3 x x **.

64 Microeconomic Theory -64- Introduction Additional example: (A bit technical for a closed book exam) U( x) 9 x x, U ( x ) x U x 9, ( ) x Solve the budget problem: Max{ U( x) x 4 x Step : If x 0 U U 9. p p x x 4. Step : Make the denominators linear in x by taking the square root. 3 x x Step 3: Convert the denominators into expenditure and appeal to the Ratio Rule * x 4x x 4x

65 Microeconomic Theory -65- Introduction Additional example: (A bit technical for a closed book exam) U( x) 9 x x, U ( x ) x U x 9, ( ) x Solve the budget problem: Max{ U( x) x 4 x Step : If x 0 U U 9. p p x x 4. Step : Make the denominators linear in x by taking the square root. 3 x x Step 3: Convert the denominators into expenditure and appeal to the Ratio Rule x 4x x 4x Step 4: Solve for the demands x 3 x 4.5