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# INTERMEDIATE MICROECONOMICS ECON 102

UCR

GPA 3.9

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This 11 page Class Notes was uploaded by Sheila O'Hara IV on Thursday October 29, 2015. The Class Notes belongs to ECON 102 at University of California Riverside taught by Staff in Fall. Since its upload, it has received 75 views. For similar materials see /class/231762/econ-102-university-of-california-riverside in Economcs at University of California Riverside.

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Date Created: 10/29/15

Lecture Note 8 Cost curves Average costs Consider cost function 0 w1w2y that gives the minimum cost of producing output level y when factor prices are 101102 If we take the factor prices to be fixed7 we can write cost as a function ofy alone7 I Total costs of the firm is written as the sum of the variable costs7 cg y7 and the fixed costs7 F C31 cMy F I Average cost function measures the cost per unit of output I Average variable cost flmction measures the variable costs per unit of output I Average xed cost function measures the fixed costs per unit of output ACy07w0v3F M E 7 y y7AVCyAFCy AC AC AC A B C Figure 211 Construction of the average cost curve I Average xed cost decreases as y increases I Average variable cost is initially constant or even declining7 but eventually increases because of the fixed factors of production I Average cost curve is the sum of AFC and AVC and will generally have the Ueshape Marginal costs Marginal cost curve measures the change in costs for a given change in output Adv CyAt C21 Ay Ay Note that we can write the de nition of marginal costs in terms of the variable cost function ACU y i Cu 1 Ay Cu 1 Ay Ay MOW MOW because the xed costs don t change as y changes Note The variable costs are zero when zero units of output are produced Therefore for the rst unit of output produced MClt1gt cvlFcv0 F evil AVClt1gt ie the marginal cost for the rst unit equals the average variable cost for a single unit of output Consider an interval where average variable costs are decreasing Then it must be the case that the marginal costs are less than the average variable costs in this range This is because to decrease the average you need to add numbers that are less than the average In other words if the average is decreasing it must be that the cost of each additional unit produced is less than average up to that point Similarly if we are in a region where average variable costs are rising then it must be the case that the marginal costs are greater than the average variable costs it is the higher marginal costs that are pushing the average up Hence the MC curve must lie below the average variable cost curve to the left of its minimum point and above it to the right This implies that the marginal cost curve must intersect the average variable cost curve at its minimum point The same is true for the average cost curve if average costs are falling then MC must be less than the average costs and if average costs are rising the marginal costs must be larger than the average costs AC AVC MC AC MC AVC Figure 212 Cost curves To summarize o The AVC curve may initially slope down but need not However it Will eventually rise as long as there are xed factors that constrain production o The AC curve will initially fall due to declining xed costs but then rise due to the increasing average variable costs 0 The MC and AVC are the same at the rst unit of output 0 The MC curve passes through the minimum point of both the AVG and the AC curves Marginal costs and variable costs The area below the marginal cost curve up to y gives us the variable cost of producing y units of output if we add up the cost of producing each unit of output we will get the total costs of production except for xed costs MC MC Variable costs Figure 213 Marginal cost and variable costs Example Consider the cost function 2 y y2 1 Then 0 variable costs cu y y xed costs Cf y 1 average variable costs AVC y g y average fixed costs AFC y i 0 average costs AC y i y H79 0 marginal costs MC y 2y Where the last expression is obtained by differentiating c y with respect to y The following figure describes the cost functions above Long run costs In the long run a firm can choose the level of its fixed factors Important in the long run it will always be possible to produce zero units of output at a zero cost by definition of the long run A C AVC AC AVC I I I I I I I I I I I 1 Figure 214 Cost ccurves Suppose that plant size k is a xed factor of production in the short run Then the rm s short run cost function is cs y Note that here k i2 For any given level of output there will be some plant size that is the optimal size to produce that level of output Let s denote this plant size by k y which is just the rm s conditional factor demand for plant size as a function of output prices are omitted Then the long run cost function of the rm can be written as 09 Cs yaky ie it is its short run cost function where the amount of xed factor is xed at the optimal long run level In other words the longrun cost function of the rm is just the shortrun cost function evaluated at the optimal choice of the xed factors Obviously the rm must be able to do at least as well by adjusting plant size as by having it xed Cy S 05 2119 for all levels of output Suppose that to produce y the optimal plant size is k k Then we can write 001 Csyak because at y the optimal choice of plant size is 19 Therefore at y the long run costs and the short run costs are the same If the short run cost is always greater than the long run cost and they are equal at one level of output then this means that the short run and the long run average costs have the same property AC 3 ACS y k and AC AC5 y k where 6 is the optimal size of the plant to produce y units of output Then the short run average cost curve always lies above the long run average cost curve and that they touch at one point y The same argument holds for levels of output other than y Then we can get a picture as follows The long run average cost curve is the lower envelope of the short run average cost curves What if there are only a few different levels of plant size to choose from Suppose that we have only four different choices 431162 kg and k4 To construct the long run average cost curve we choose the plant size that gives us the minimum cost of producing that output level AC SAC Cy k y Figure 216 Shortrun longrun average costs AC Short ru n average cost curves Long run average cost curve Figure 217 Shortrun and longrun average costs Lecture note 5 Technology When a rm makes choices it faces many constraints These constraints are imposed by its customers by its competitors and by nature First we will consider the latter source of constraints nature Inputs and outputs Inputs to production are called factors of production land labor capital and raw materials Capital goods or physical capital are those inputs into production that are themselves produced goods ie tractors buildings computers etc Financial capital is the term to describe the money used to start up or maintain a business Describing technological constraints Technological constraints only certain combinations of inputs are feasible ways to pro duce a given amount of output Therefore the rm must limit itself to technologically feasible production plans Production set is the set of all combinations of inputs and outputs that comprise a tech nologically feasible way to produce For example if we have only one input measured by x and one output measured by y Then a production set might have the following shape y OUTPUT y fX 2 production function Production set XINPUT Figure 181 A production set We say that some point 2631 is in the production set if it is technologically possible to produce 3 amount of output if you have C amount of input The production set shows the possible technological choices facing a rm Since the inputs to the rm are costly it makes sense to examine the maximum possible output for a given level of input Production function is the boundary of the production set7 which measures the maximum possible output that you can get from a given amount of input Production function concept applies to the cases with several inputs7 ie in the case of two inputs7 the production function f 351352 would measure the maximum amount of output y that we could get if we had 261 units of factor 1 and 262 units of factor 2 Isoquant is the set of all possible combinations of inputs 1 and 2 that are just suf cient to produce a given amount of output Note that isoquants are similar to indifference curves Important isoquants are labeled with the amount of output they can produce7 not with a utility level Thus the labeling of isoquants is xed by the technology and does not have the kind of arbitrary nature that the utility labeling has Examples of technology Fixed proportions Suppose that we are producing holes and that the only way to get a hole is to use one man and one shovel Extra shovels are not worth ahythihg7 and neither are extra men Thus the total number of holes that you can produce will be the minimum of the number of men and the number of shovels that you have7 ie f 17132 min 1317732 which has the following isoquants X2 Boquanw X Figure 182 Fixed proportions Note the similarity to the perfect complements case Perfect substitutes Suppose that now we are producing homework and the inputs are red pencils and blue pencils The amount of homework produced depends only on the total number of pencils therefore f172 1 2 which has the following isoquants CobbDouglas technology Consider the following form of the production function f 12 Az fzg This type of production function is a Cobb Douglas production function Notel the magnitude of the production function is important 0 The parameter A measures the scale of production ie how much output we would get if we used one unit of each input 0 The parameters a and b measure how the amount of output responds to changes in the inputs The Cobb Douglas isoquants have the same nice well behaved shape that the Cobb Douglas indifference curves have Properties of technology Similar to the case of consumers it is common to assume certain properties about technology 0 Monotonicity if you increase the amount of at least one of the inputs it should be possible to produce at least as much output as you were producing originally fab Z flt172 V gt 2 This is sometimes referred to as the property of free disposal if the rm can costlessly dispose of any inputs haVing extra inputs around can7t hurt it o Convexity if you have two ways to produce y units of output 12 and 2122 then their weighted average will produce at least y units of output Suppose you have a way to produce 1 unit of output using 11 units of factor 1 and 12 units of factor 2 and that you have another way to produce 1 unit of output using b1 units of factor 1 and b2 units of factor 2 We call these two ways to produce output production techniques Suppose that you are free to scale the output up by arbitrary amount so that 10011 100 and 100191 100192 will produce 100 units of output each If you have 25a1 75b1 units of factor 1 and 25612 75b2 units of factor 2 you can produce 25 units of the output using the 77a77 technique and 75 units of the output using the 77b77 technique Numerical example Suppose two technologies are a 2 units of factor 1 labor and 1 unit of factor 2 capital7 and b 1 unit of factor 1 labor and 3 units of factor 2 capital So that 21 and 13 are two different ways to produce a unit of output Consider production of 100 units of output each technology 200100 and 100300 will produce 100 units of output Suppose that you have 25 2 75 1 125 units of labor and 25 1 75 3 250 units of capital Then you can produce 25 units of the output using technique a 50 25 and 75 units using technology b 75 225 Note that to do this you need 125 units of labor and 250 units of capital X2 10032 25a1 75b1 25a2 75132 100b2 Isoquant 100a1 mob1 X1 Figure 184 Convexity The marginal product Suppose we are operating at some point x1 x2 and that we consider using a little bit more of factor 1 while keeping factor 2 xed at the level 902 How much more output will we get per additional unit of factor 1 Marginal product of factor 1 f1Ax172 f12 A151 A1171 Marginal product of factor 2 i 160617962 Ax2 f12 A272 A372 We will denote marginal product of factors 1 and 2 by MP1x12 and MP2 12 re spectively Notel marginal product is a rate the extra amount of output per unit of extra input The technical rate of substitution Suppose that we are operating at some point 1 2 and that we consider giving up a little bit of factor 1 and using just enough more of factor 2 to produce the same amount of output y How much extra of factor 2 Am do we need if we are going to give up a little bit of factor 1 A21 This is just the slope of the isoquantl Technical rate of substitution TRS measures the tradeoff between two inputs in production It measures the rate at which the rm will have to substitute one input for another in order to keep output constant Consider a change in our use of factors 1 and 2 that keeps output xed Ay MP1lt12A1 MP2 12 A2 0 MP1lt12A1 MP2 12 A2 0 2 A MP1lt12 MP2 12 E 0 1 Ax MP2 12 A7 7MP112 1 A2 MP1lt12 TRSz1z2 A71 m Diminishing marginal product If we have a strictly monotonic technology we know that the total output will go up as we increase the amount of factor 1 But it is natural to expect that it will go up at a decreasing rate Example Suppose that one man on one acre of land might produce 100 bushels of corn If we add another man and keep the same amount of land we might get 200 bushels of corn so in this case the marginal product of an extra worker is 100 If we keep adding workers to this acre of land eventually the extra amount of corn produced by an extra worker will be less than 100 bushels eg after 4 or 5 people are added the additional output per worker will drop to 90 80 70 or even fewer bushel of corn If we get hundreds of workers crowded together on this one acre of land an extra worker may even cause output to go down Therefore we would typically expect that the marginal product of a factor will diminish as we get more and more of that factor we call this the law of diminishing marginal product Importantl in the example above all other inputs are xed and the change concerns only the labor input Diminishing technical rate of substitution As we increase the amount of factor 17 and adjust factor 2 so as to stay on the same isoquant7 the technical rate of substitution declines the slope of an isoquant must decrease in absolute value as we move along the isoquant in the direction of increasing 31 Important diminishing technical rate of substitution and diminishing marginal product are related but not exactly the same things a Diminishing marginal product is an assumption about how the marginal product changes as we increase the amount of one factor7 holding the other factors xed a Diminishing TRS is about how the ratio of the marginal products the slope fo the isoquant changes as we increase the amount of one factor and reduce the amount of the other factor so as to stay on the same isoquant The long run and short run We may want to distinguish between the production plans that are immediately feasible and those that are eventually feasible Short run there are some factors of production that are xed at predetermined levels7 eg land Then production function has the form y f 51317532 Y XV E2 Xi Figure 185 Production function Note that the short run production function is flatter for larger values of 21 This is because of the law of diminishing marginal product Long run all the factors of production can be varied y f 513175132

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