Microeconomic Theory -1- Introduction and maximization
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1 Microeconomic Theory -- Introduction and maximization Introduction Maximization. Profit maximizing firm with monopoly power 6. General results on maximizing with two variables 3. Non-negativity constraints 5 4. First laws of supply and input demand 7 5. Resource constrained maximization an economic approach 30 4 pages
2 Microeconomic Theory -- Introduction and maximization 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 and maximization 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 and maximization 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 and maximization 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 and maximization Maximization. Profit-maximizing firm Example : Cost function C( q) 5 q 3q Demand price function p( q) 0 q Group exercise: Solve for the profit maximizing output and price.
7 Microeconomic Theory -7- Introduction and maximization Example : 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 and maximization 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 and maximization 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 q 75 3q 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 and maximization 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 and maximization 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 Think on the margin
12 Microeconomic Theory -- Introduction and maximization 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 q 55 3q 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 and maximization 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 and maximization Model : Is q q q (, ) the profit-maximizing output vector? The profit-maximizing lines divide the positive quadrant into four zones. The arrows indicate the directions of in which ( q, q) increases. Consider the point 0 q. Output is higher in the diagonally shaded region Model : Profit-maximizing lines and lower in the dotted region. Thus the level set through 0 q must have a negative slope. A similar argument can be used in the other three quadrants.
15 Microeconomic Theory -5- Introduction and maximization 0 The level set ( q) ( q ) in the Profit is higher in the shaded region Z(, ) region Note that the level set is parallel to the q axis at the point of intersection with the maximizing line for q and is parallel to the horizontal axis at the point of intersection with the q maximizing line for q Model : Level set for profit
16 Microeconomic Theory -6- Introduction and maximization 0 The level set ( q) ( q ) 0 and superlevel set ( q) ( q ) are depicted opposite. Model : Level set for profit
17 Microeconomic Theory -7- Introduction and maximization 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).
18 Microeconomic Theory -8- Introduction and maximization The profit is depicted below (using a spread-sheet) 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.
19 Microeconomic Theory -9- Introduction and maximization 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 and so on If you pick any starting point you will find this process never leads to the intersection point q (0,0).
20 Microeconomic Theory -0- Introduction and maximization Local maximum q ( q,0) on the q axis By an essentially identical argument, there is a second local maximum q on the q axis.
21 Microeconomic Theory -- Introduction and maximization The profit function has the shape of a saddle. The output vector q where the slope in the direction of each axis is zero is called a saddle-point.
22 Microeconomic Theory -- Introduction and maximization. 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-)
23 Microeconomic Theory -3- Introduction and maximization 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, q ) 0 f and ( q, q) 0 q q (3-)
24 Microeconomic Theory -4- Introduction and maximization 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 the first and second order necessary conditions hold at q and the level sets are closed loops around q, then the function f( q ) has a local maximum at q Proposition: Sufficient conditions for a global maximum If the first and second order necessary conditions hold at q and the level sets are closed loops around q and the FOC hold only at q, then this is the global maximizer.
25 Microeconomic Theory -5- Introduction and maximization 3. Non-negativity constraints Many economic variables cannot be negative. Suppose this is true for all variables Let x ( x,..., x n ) solve Max{ f ( x)} x 0. We will consider the first variable. It is helpful to write the optimal value of all the other variables as x. Then x solves Case (i) x 0 Max{ f ( x, x )} x 0. This is depicted opposite. For x to be the maximizer, Case (i): Necessary condition for a maximum the graph of f ( x, x ) must be zero at x.
26 Microeconomic Theory -6- Introduction and maximization Case (ii) x 0 This is depicted opposite. For x to be the maximizer, the graph of f ( x, x ) cannot be strictly positive at x. Taking the two cases together, Case (ii): Necessary condition for a maximum f x ( x) 0, with equality if x 0 An identical argument holds for all of the variables. Necessary conditions f x ( x) 0, with equality if x 0 Case (ii): Necessary condition for a maximum
27 Microeconomic Theory -7- Introduction and maximization 4. Laws of supply and input demand The first law of firm supply As an output price p rises, the maximizing output q( p) increases (at least weakly). Case (i) p MC(0) Case (ii) p MC(0) Fig. : Profit-maximizing output As the output price rises, the profit-maximizing output increases (at least weakly).
28 Microeconomic Theory -8- Introduction and maximization The firm s supply curve For prices below MC (0), supply is zero. For higher prices the graph of marginal cost MC( q ) is the supply curve. Fig. : Firm s supply curve
29 Microeconomic Theory -9- Introduction and maximization The first law of input demand As an input price r rises, the maximizing input zr () decreases (at least weakly). The rate at which revenue rises as the input (and hence output) rises is called the Marginal Revenue Product (MRP). Fig. 3: Firm s input demand curve
30 Microeconomic Theory -30- Introduction and maximization 5. Resource constrained maximization - - an economic approach Problem: Max{ f ( x) b g( x) 0} x 0 NOTE: Always write a resource constraint as hx ( ) 0 Let x be the solution to this problem. Interpretation, if the firm chooses x it requires gx ( ) units of a resource that is fixed in supply (.e. Floor space of plant, highly skilled workers) We interpret q f ( x) as the output of the firm. The price of the output is so this is also the revenue of the firm. There is a single input z g( x). There are b units of this input available.
31 Microeconomic Theory -3- Introduction and maximization To solve this problem, we consider the relaxed problem in which the firm can purchase additional units at the price. Since this is a hypothetical opportunity, economists refer to the price as the shadow price of the resource rather than a market price. Suppose that the firm purchases g( x) L f ( x) ( g( x) b) f ( x) ( b g( x)) The relaxed problem is then b additional units. Its profit is then Max{ L f ( x) ( b g( x))} x 0 First Order Necessary Conditions: Necessary conditions for x( ) to solve Max{ L ( x, )} L f g ( x, ) ( x) ( x) 0, with equality if x 0,, x x x x 0
32 Microeconomic Theory -3- Introduction and maximization Let z g( x) be demand for the resource. In Section 4 it was argued that Demand, z() r declines as the input price rises. If the resource price is sufficiently high it is more profitable to sell all of the resource. Demand for the resource in the relaxed problem *
33 Microeconomic Theory -33- Introduction and maximization Let z g( x) be demand for the resource. In Section 4 it was argued that Demand, z() r declines as the input price rises. If the resource price is sufficiently high it is more profitable to sell all of the resource. Demand for the resource in the relaxed problem Case (i) z(0) b Supply exceeds demand at every price So the market clearing price 0. Demand for the resource in the relaxed problem
34 Microeconomic Theory -34- Introduction and maximization Case (ii): z(0) b At the price, demand for the resource is equal to b. Suppose we find such a price. SInce x is profit-maximizing, L f ( x) ( g( x) b) f ( x) ( b g( x)) ** Demand for the resource equals supply at price
35 Microeconomic Theory -35- Introduction and maximization Case (ii): z(0) b At the price, demand for the resource is equal to b. Suppose we find such a price. SInce x is profit-maximizing, L f ( x) ( g( x) b) f ( x) ( b g( x)) At the price, demand for the resource equals supply Demand for the resource equals supply at price It follows that L f ( x) f ( x) ( b g( x)) (*) *
36 Microeconomic Theory -36- Introduction and maximization Case (ii): z(0) b At the price, demand for the resource is equal to b. Suppose we find such a price. SInce x is profit-maximizing, L f ( x) ( g( x) b) f ( x) ( b g( x)) At the price, demand for the resource equals supply Demand for the resource equals supply at price It follows that L f ( x) f ( x) ( b g( x)) (*) Now consider the original problem, Max{ f ( x) b g( x) 0}. x 0 For any feasible x it follows that b g( x) 0. Appealing to (*), L f ( x) f ( x) ( b g( x)) f ( x) Thus x solves the original problem.
37 Microeconomic Theory -37- Introduction and maximization Summary: Necessary conditions for a maximum with a resource constraint Max{ f ( x) b g( x) 0} x 0 NOTE: Always write a resource constraint as hx ( ) 0 Consider the relaxed problem in which there is a market for the resource and the firm owns b of the resource. If the price of the resource is, then profit in the relaxed problem is units L f ( x) ( g( x) b) f ( x) ( b f ( x)). Since this market is a theoretical rather than an actual market we call the price a shadow price. *
38 Microeconomic Theory -38- Introduction and maximization Summary: Necessary conditions for a maximum with a resource constraint Max{ f ( x) b g( x) 0} x 0 NOTE: Always write a resource constraint as hx ( ) 0 Consider the relaxed problem in which there is a market for the resource and the firm owns b of the resource. If the price of the resource is, then profit in the relaxed problem is units L f ( x) ( g( x) b) f ( x) ( b f ( x)). Since this market is a theoretical rather than an actual market we call the price a shadow price. Suppose we find a shadow price 0 and x relaxed problem are satisfied and in addition, (i) b g( x) 0 0 (ii) 0 b g( x) 0. such that the Necessary First Order Conditions for the Then these conditions are the necessary conditions for the resource constrained problem.
39 Microeconomic Theory -39- Introduction and maximization Solving for the maximum Example : Output maximization with a budget constraint 3 Max{ q f ( x) x x x } where X { x 0 p x b} and p 0 x X 3 Preliminary analysis If q f ( x) takes on its maximum at x h( x) g( f ( x)) also takes on its maximum at x., then, for any strictly increasing function gq ( ), In this case the function g( q) ln q simplifies the problem since h( x) ln f ( x) ln x ln x ln x, where The derivatives of ln q are very simple since each term has only one variable. The new problem is x 0 3 Max{ h( x) ln x b p x 0}.
40 Microeconomic Theory -40- Introduction and maximization x 0 3 Max{ h( x) ln x b p x 0} We write down the profit in the relaxed problem in which there is a market price for the resource. Mathematicians call this the Lagrangian. If the firm sells b p x units of the resource, then the profit of the firm is 3 3 L ln x ( b p x ) Necessary conditions for profit maximization L x p 0 x, with equality if x 0,,,3. Note that as x 0 the first term on the right hand side increases without bound. Therefore the right hand side cannot be negative. Then L x p 0, x,,3. Therefore px,,,3
41 Microeconomic Theory -4- Introduction and maximization We have shown that px,,,3 (-) Summing over the commodities, b 3 3 p x, since 3 Appealing to (-) it follows that x b,,,3 p
42 Microeconomic Theory -4- Introduction and maximization Example : Utility maximization A consumer s preferences are represented by a strictly increasing utility function U( x ), where U( x) 0 if and only if x 0. The consumer s budget constraint is... n n p x p x p x I where the price vector p 0. The consumer chooses x to solve Max{ U( x) p x I}. x 0 Group Exercise: () Explain why x 0 and p x I (ii) Show that the FOC can be written as follows: U x p U xn.... p n (iii) Provide the intuition behind these conditions.
43 Microeconomic Theory -43- Introduction and maximization A graphical approach Suppose x solves Max{ U( x) p x I}. Define z ( x3,..., x n ). Then x 0 Hence ( x, x ) solves Max{ U( x, x, z ) px px p z I}. x 0 ( x, x ) solves Max{ U ( x, x, z ) px px I I p z}. x 0 We can illustrate this two variable problem in a figure showing the commodity budget constraint and level sets of the function U( x, x, z ). Choosing commodities and
44 Microeconomic Theory -44- Introduction and maximization The slope of the budget line is p p But what is the slope of the level set? Note that the level set implicitly defines a function x ( x). That is U( x, ( x ), z ) U( x, x,, z ) Choosing commodities and Differentiate with respect to x d (, ( ), ) U U x x z U ( x ) 0 dx x x Therefore the slope of the level set is U U ( x ) / x x
45 Microeconomic Theory -45- Introduction and maximization At the maximum the slopes are equal. Therefore p U U ( x, x, z ) / ( x, x, z ) p x x i.e. p U U ( x) / ( x) p x x Exactly the same argument holds for every Choosing commodities and pair of commodities. Therefore pi U U ( x) / ( x) for all i, p x x i Rearranging this equation, U ( x) xi p i U ( x) x. p
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