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Apunte Inglés
Universidad Universidad Pompeu Fabra (UPF)
Grado International Business Economics - 1º curso
Asignatura Microeconomics I
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PART 1: CONSUMER CHOICE AND DEMAND TOPIC 1. THE ELEMENTS OF THE PROBLEM 1.1. THE REASON FOR CHOICE: PREFERENCES INTRODUCTION The consumer theory states that people choose the best things (preferences) they can afford (budget constraint). For this we assume that consumers know what they prefer, the they can rank consumption bundles. We call the objects of consumer choice consumption bundles; complete list of the goods and services involved in the choice problem.
The word complete deserves emphasis: when you analyse a consumer’s choice problem, make sure that you include all of the appropriate goods in the definition of the consumption bundle. If we are analysing consumer choice at the broadest level, we would want not only a complete list of the goods that a consumer might consume, but also a description of when, where, and under what circumstances they would become available. Notice we don’t value the same 10$ today and 10$ tomorrow, an umbrella in a sunny day or in a rainy day, two apples / 1 DVD / 2 weeks of vacation in Menorca… PREFERENCE RELATION Given two consumption bundles (X1, X2) and (Y1, Y2), the consumer can rank them as to their desirability: Strictly preferred: in the sense that she definitely wants the x-bundle rather than the y-bundle. (>,<) Indifference: means that the consumer would be just as satisfied, according to her own preference, consuming x-bundle as she would be consuming the y-bundle. (~) Weakly prefers: if the consumer prefers or is indifferent between the two bundles. (~<) RATIONAL PREFERENCES Complete: we assume that two bundles can be compared. It’s hardly objectionable, at least for the kinds of choices economists generally examine. It’s simply to say that the consumer is able to make a choice between any two given bundles: one might imagine extreme situations involving life or death choices.
Reflexive: we assume that any bundle is at least as good as itself.
Transitive: if the consumer thinks that X is at least as good as Y and that Y is at least as good as Z, then the consumer thinks that X is at least as good as Z.
INDIFFERENCE CURVES An indifference curve is the line through (X1, X2) consisting of all bundle of goods that leave the consumer indifferent to (X1,X2). With rational preferences, indifference curves representing distinct levels of preference cannot cross.
The general procedure for drawing indifference curves: we consider a bundle (X1,X2), for a given change in the amount off good 1 (increment X1), how does the amount of good 2 have to change (increment X2) to make the consumer just indifferent between (X1 + increment X1, X2 + increment X2) and (X1,X2) ? PERFECT SUBSTITUTES Two goods are perfect substitutes if the consumer is willing to substitute one goods for the other at a constant rate. Suppose, for example, that we are considering a choice between red pencils and blue pencils. The consumer only cares about the total number of pencils, not about their colours. Thus the indifference curves are straight lines with a slope of -1; if for example, we start at (10,10) in order to me give up one blue pen you have to give one black pen more (11,9). That’s why they have a constant slope.
PERFECT COMPLEMENTS Two goods are perfect complements if they are always consumed together in fixed proportions. For instance, the right shoes and left shoes. Having only one out of a pair of shoes doesn’t do the consumer a bit of good. Thus, the indifference curves are Lshaped. Suppose we pick the consumption bundle (10,10). Now add 1 more right shoe, so we have (11,10). By assumption this leaves the consumer indifferent.
BADS A bad is a commodity that the consumer doesn’t like. For instance the amount of light and noise in an apartment. The direction of increasing preference is down and to the right – that is, toward the direction of decreased noise and increased light. Thus, the slope is positive, in order to give up one unit of light I have to have one less unit of noise.
NEUTRALS A good is a neutral good if the consumer doesn’t care about it. For example, I don’t care about X2 (school district, since I don’t have kids). Therefore, if I change the amount of good X2, in order to me to be indifferent X1 has to be the same amount. Consequently, the general shape is a vertical line, if X2 is the neutral. The more X1 the better, but adding X2 doesn’t affect the consumer one way or the other.
SATIATION We sometimes want to consider a situation involving satiation, where there is some overall best bundle for the consumer, and the closer he is to that best bundle, the better off he is in terms of his own preferences. An example would be with ice-cream and cake. Suppose that the consumer has some most preferred bundle of goods (X1,X2) and the farther away he is from that bundle, the worse off he is. In this case we say that (X1,X2) is a satiation point or a bliss point. In this case the indifference curves have a negative slope when the consumer has “too little” or “too much” of both goods, and a positive slope when he has “too much” of one of the goods. When he has too much of one of the goods, it becomes a bad – reducing the consumption of the bad good moves him closer to his “bliss point”. If he has too much of both goods, they both are bads, so reducing the consumption of each moves him closer to the bliss point.
WELL-BEHAVED PREFERENCES A well-behaved preference involves monotonicity and convexity.
First, we will typically assume that more is better, that is, that we are talking about goods, not bads. Monotonicity means that a bundle that has more quantity is preferable. This implies that they have a negative slope.
For instance: A = 8 apples, 5 peers ; B = 5 apples, 5 peers ; C = 10 apples, 4 peers. We know that A>B, but we don’t know if A?C.
Secondly, we assume that the set of bundles weakly preferred to (X1,X2) is a convex set. Convexity implies that averages are better than extremes. Strictly convexity means that the weighted average bundle is strictly preferred to the two extreme bundles.
MARGINAL RATE OF SUBSITITUTION The slope of the indifference curve is the marginal rate of substitution = ΔX2 / ΔX1 = it’s the rate at which consumer is just willing to substitute on good for the other. For the monocity behavior MRS is usually negative when the consumer likes both goods, since reducing the consumption of one good implies getting more of another for. It will not be negative if there were bads or neutrals. For convex indifference curves the MRS decreases in absolute terms as X1 increases: diminishing marginal rate of substitution. This means that the amount of good 1 that the person is willing to give up for an additional amount of good 2 increases as the amount of good 1 increases. Stated in this way, convexity of indifference curves seems very natural: it says that the more you have of one good, the more willing you are to give some of it up in exchange for the other good.
EXAMPLES 1.2. REPRESENTATION OF PREFERENCES: UTILITY The utility is a way to describe the preferences of consumers. All that mattered about utility as far as choice behaviour was concerned was whether one bundle had a higher utility than another – how much higher didn’t really matter. A utility function assigns numbers to bundles such that more-preferred bundles are assigned higher numbers than less-preferred bundles.
ORDINAL UTILITY The ordinal utility is the term used to emphasize that the only important thing of utility assignment is how it orders the bundles of goods, the magnitudes of utility function is only important insofar as it ranks the different consumption bundles; the size of the utility difference between any two consumption bundles doesn’t matter. Therefore, there are different ways to assign utilities: Since only the ranking of the bundles matters, there can be no unique way to assign utilities to bundles of goods. What matters is the ordering of utilities not the absolute value. U2 and U3 are monotonic transformations, is a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers. Multiplying by -1 is not an example of monotonic transformation since it doesn’t preserve the order. Thus, a monotonic function always has a positive rate of change = positive slope = increasing function. We can say that a utility function is a way to label indifference curves. Since every bundle on an indifference curve must have the same utility, a utility function is a way of assigning numbers to the different indifference curves in a way that higher indifference curves get assigned larger numbers. Seen form this point of view a monotonic transformation is just a relabelling of indifference curves.
Example: Consumer 1, 2 and 6 have A > B > C.
Consumer 3 and 5 have A ~ B > C.
Consumer 4 has A ~ B ~ C CONSTRUCTIING A UTILITY FUNCTION To construct some other example we only have to equal the utility function to a number k and that will be our indifference curve. For instance: X1X2 = k. And X1^2*X2^2 is a monotonic function of the first one so the ordering is the same, the indifference curve is the same, the only thing changed is that k is now k^2.
PERFECT SUBSTITUTES The case of the red and blue pencil. All that mattered to the consumers was the total number of pencils.
Therefore the utility function could be X1 + X2 or any other monotonic transformation: (X1 + X2)^2. In general, preference for perfect substitutes can be represented by a utility function of the form aX1 + bX2, being a and b positive values. The slope of the typically indifference curve is given by -a/b.
PERFECT COMPLEMENTS The left shoe and the right shoe case. In these preferences, the consumer only cares about the number of pairs of shoes he has, so it’s natural to choose the number of pairs of shoes as the utility function. Thus, the utility function for perfect complements takes the form min (a*X1, b*X2), where a and b are positive numbers that indicate the proportions in which the goods are consumed.
QUASILINEAR PREFERENCES The general equation for an indifference curve takes the form X2 = u – f(X1), where u is a different constant for each indifference curve. This equation says that the height of each indifference curve is some function of X1 plus a constant k.
COBB-DOUGLAS PREFERENCES The general equation is X1^c * X2^d, where c and d are positive numbers. Cobb-Douglas indifference curves look just like the nice convex monotonic indifference curves that we referred to as “wellbehaves indifference curves”. Another example could be ln(A*X1^c*X2^d) = clnX1 + dlnx2. The indifference curves for this utility function will look just like the ones for the first Cobb-Douglas function, since the logarithm is a monotonic transformation.
MARGINAL UTILITY The marginal utility is the rate of change with respect to good 1: how does this consumer’s utility change as we give him or her a little more of good 1? We write it as MU1: Note that the amount of good 2 is held fixed in this calculation. It is important to realize that it depends on utility function transformation, which means that marginal utility itself has no behavioural content.
MARGINAL RATE OF SUBSTITUTION The marginal rate of substitution (MRS) can be interpretated as the rate at which a consumer is just willing to substitute a small amount of good 2 for good 1. Intuitively the willingness to exchange good 2 for one unit of good 1, given that the level of utility stays constant. Geometrically is equivalent to the slope of the indifference curve. The MRS is independent of the particular transformation of the utility function you choose to use.
EXERCISE 1.3. LIMITS OF CONSUMER CHOICE: BUDGET CONSTRAINT The budget constraint can be written as: The budget set is the set of affordable consumer bundles at prices p1 and p2, and a certain income. The budget line is the set of bundles that cost exactly, these are the bundles of goods that just exhaust the consumer’s income. The slope of the budget line measures opportunity cost of good 1 – how much of good 2 you must give up in order to afford more of good 1.
THE NUMERAIRE The budge line is defined by two prices and one income, but one of these variables is redundant. We could peg one of the prices, or the income, to some fixed value and adjust the other variable so as to describe exactly the same budget set. The numeraire price is the price set to 1 – the price relative to which we’re measuring the other price and the income.
EXAMPLE TOPIC 2. THE CHOICE 2.1. THE FORMAL PROBLEM OF MAXIMIZATION OF PREFERENCE Consumers choose the best things they can afford. We put preference and budget constraint together to determine what consumers will choose. The consumer problem is to choose the consumption bundle X that is best according to preferences subject to the constraint that the total cost of X is no greater than the consumer’s income.
OPTIMAL CHOICE WITH WELL BEHAVED PREFERENCES The optimal choice implies that the optimal consumption position is where the indifference curve is tangent to the budget line (usually, what is always true is that at the optimal point the indifference curve can’t cross the budge line). Moreover, the budget is exhausted. When preferences are well-behaved and solutions are interior, the tangency condition is fulfilled.
With well-behave preferences the slope of the indifference curve = the slope of the budget line – at interior optimum-; which means that the MRS has to be equal to -p1/p2. – What would happen if the MRS were different from the price ration. Suppose that the MRS is -1/2 and the price ratio is 1/1. Then this means that the consumer is just willing to give up 2 units of good 1 in order to get 1 unit of good 2, but the market is willing to exchange them on a one-to-one basis. Thus the consumer would certainly be willing to give up some of good 1 in order to purchase a little more of good 2. Whenever the MRS is different form the price ratio, the consumer cannot be at his or her optimal choice.
OPTIMAL CHOICE WITH PERFECT SUBSTITUTES We have three possible cases. If p2 > p1, then the slope of the budget line is flatter than the slope of the indifference curves. In this case, the optimal bundle is where the consumer spends all of his or her money on good 1. If p1 > p2, then the consumer purchases only good 2. Finally, if p1 = p2, there is a whole range of optimal choices—any amount of goods 1 and 2 that satisfies the budget constraint is optimal in this case.
OPTIMAL CHOICE WITH PERFECT COMPLEMENTS Note that the optimal choice must always lie on the diagonal where the consumer is purchasing equal amounts of both goods, no matter what the prices are.
X1* = X2 * = m/ p1+p2.
OPTIMAL CHOICE WITH CONCAVE PREFERENCES The optimal choice for these preferences is always going to be a boundary choice.
Note: depending on the slope of budget line/ exact format of IC, optimal choice could be: X1* = O and X2* = m/p2. But if concave preferences then the consumer specializes.
OPTIMAL CHOICE WITH CALCULUS Choose the consumption bundle X that is the best according to the utility function.
Subject to the constraint that the total cost of X is no greater than the consumer’s income. Which means that we have to choose X that maximizes utility and it’s subject to the budget constraint.
The optimal choice must satisfy the condition  And also the budget constraint  There are two methods to maximize utility analytically: 1. Substituting the constraint (BL) into the objective function (U).
2. Lagrange multiplier method.
2.2 DERIVATION OF THE DEMAND FUNCTION The consumer’s demand functions give the optimal amounts of each of the goods as a function of the prices and income faced by the consumer.
Important property of Cobb-Douglas: consumer spends fixed proportion of income in each good.
In the example of BOB (See more detail in the example of Cob Douglass, more easily explained).
COMPARATIVE STATISTICS Studying how a choice responds to changes in the economic environment is known as comparative statics.
The comparative statics questions in consumer theory therefore involves investigating how demand changes when prices and income changes.
With normal goods With inferior goods: for example, when my income increases I decrease my consumption of lowcost products and increase my consumption of brands.
INCOME CHANGES We have seen that an increase in income corresponds to shifting the budget line outward in a parallel manner. We can connect together the demanded bundles that we get as we shift the budget line outward to construct the income offer curve. This curve illustrates bundles of goods that are demanded at the different levels of income. If both goods are normal goods then the income expansion path (=income offer curve) will have a positive slope. We will have at the axis X1 and X2, and price will be fixed.
If we hold the prices of goods 1 and 2 fixed and look at how demand for one good changes as we change income, we generate a curve known as the Engel curve. The Engel curve is a graph of the demand for one of the goods as a function of income, with all price being held constant. Axis will be Xi and M (income).
REMARK: Engel curves are not necessarily straight lines. There are non-homothetic preferences.
ORDINARY GOODS AND GIFFEN GOODS Now let’s consider price changes. Suppose that we decrease the price of good 1 and hold the price of good 2 and money income fixed. Then what can happen to the quantity demanded of good 1? Intuition tells us that the quantity demanded of good 1 should increase when its price decreases.
Indeed, this is the ordinary case. When the price of good 1 decreases, the budget line becomes flatter.
Or said another way, the vertical intercept is fixed and the horizontal intercept moves to the right. The optimal choice of good 1 moves to the right as well: the quantity demanded of good 1 has increased.
It is logically possible to find well-behaved preferences for which a decrease in the price of good 1 leads to a reduction in the demand for good 1. Such a good is called a Giffen good.
Suppose that the two goods that you are consuming are gruel and milk and that you are currently consuming 7 bowls of gruel and 7 cups of milk a week. Now the price of gruel declines. If you consume the same 7 bowls of gruel a week, you will have money left over with which you can purchase more milk. In fact, with the extra money you have saved because of the lower price of gruel, you may decide to consume even more milk and reduce your consumption of gruel. The reduction in the price of gruel has freed up some extra money to be spent on other things—but one thing you might want to do with it is reduce your consumption of gruel! Thus, the price change is to some extent like an income change. Even though money income remains constant, a change in the price of a good will change purchasing power, and thereby change demand.
PRICE OFFER CURVE AND THE INVERSE DEMAND CURVE Suppose that we let the price of good 1 change while we hold p2 and income fixed. Geometrically this involves pivoting the budget line. We can think of connecting together the optimal points to construct the price offer curve. This curve represents the bundles that would be demanded at different prices for good 1.
We can depict this same information in a different way. Again, hold the price of good 2 and money income fixed, and for each different value of p1 plot the optimal level of consumption of good 1. The result is the demand curve. The demand curve is a plot of the demand function, x1(p1, p2, m), holding p2 and m fixed at some predetermined values. Ordinarily, when the price of a good increases, the demand for that good will decrease. Thus the price and quantity of a good will move in opposite directions, which means that the demand curve will typically have a negative slope. In terms of rates of change, we would normally have which simply says that demand curves usually have a negative slope. However, we have also seen that in the case of Giffen goods, the demand for a good may decrease when its price decreases. Thus it is possible, but not likely, to have a demand curve with a positive slope.
SUBSTITUTES AND COMPLEMENTS SUMMARY TOPIC 3. REVEALED PREFERENCES AND SLUTSKY EQUATION 3.1. REVEALED PREFERENCES 3.2. SLUTSKY EQUATION Economists are often concerned with how a consumer’s behaviour changes in response to changes in the economic environment; particularly we are interested in people’s reaction to changes in relative prices (income-price). We decompose the effect of a price change into simpler pieces to determine behaviour of whole.
When a price of good changes there are two sorts of effects: - Substitution effect: the change in demand due to the change in the rate at which you can change one good for another; increase demand for cheaper good, substituting away from other goods.
- Income effect: the change in demand due to the fact that total purchasing power of your income is altered; with the same income, consumer can now buy more (raise in purchasing power).
- The total price effect on demand = substitution effect + income effect.
The Slutsky decomposition means that we will decompose the total price effect on demand into a pure substitution effect and a pure income effect.
How can we separate substitution effect and income effect? First step: we will let the relative prices change and adjust money income so as to hold purchasing power constant. Second step: let purchasing power adjust while holding the relative prices constant.
The price of good 1 has declined. This means that the budget line rotates around the vertical intercept m/p2 and becomes flatter. Two steps: first pivot the budget line around the original demanded bundle and then shift the pivoted line out to the new demanded bundle. This “pivot-shift” operation is a way to decompose the change in demand into: the first step—the pivot—is a movement where the slope of the budget line changes while its purchasing power stays constant, while the second step is a movement where the slope stays constant and the purchasing power changes.
Step 1  Substitution effect: Let us calculate how much we have to adjust money income in order to keep the old bundle just affordable. Let m’ be the amount of money income that will just make the original consumption bundle affordable; this will be the amount of money income associated with the pivoted budget line.
Step 2  Income effect: we know that a parallel shift of the budget line is the movement that occurs when income changes while relative prices remain constant. We simply change the consumer’s income from m’ to m, keeping the prices constant at (p’1, p2).
Step 3  Total effect: Note: substitution effect must be negative (quantity moves in opposite direction of prices) due to revealed preference. Income effect can be negative or positive (depends on whether normal or inferior good).
3.3 THE LAW OF DEMAND The law of demand. If the demand for a good increases when income increases, then the demand for that good must decrease when its prices increases.
This follows directly from the Slutsky equation: if the demand increases when income increases, we have a normal good. And if we have a normal good, then the substitution effect and the income effect reinforce each other, and an increase in price will unambiguously reduce demand.
TOPIC 4. CONSUMER WELFARE 4.1. CONSUMER SURPLUS The consumer surplus measures the net benefit from consuming n units of the good: the difference between your willingness to pay and the actual price added up over all units purchased.
Your demand curve can be seen as your willingness to pay curve; for all units you consume, you actually pay the market price. The difference of how much you’re willing to pay for an extra unit of good and how much you’re actually paying is a measure of your benefit from purchasing one more unit.
When demand is continuous, area under the demand curve is an approximate measure of utility/consumer welfare (the difference is the income effect).
Exception: quasilinear utility area under demand curve is the exact measure (no income effect). In the special case of quasilinear utility there is “no income effect” since changes in income don’t affect demand, which means that the reservation prices are independent of the amount of money the consumer has to spend on other goods. Using the area under the demand curve to measure utility will only be exactly correct when the utility function is quasilinear.
We are interested in the change of consumer surplus that results from some policy change. The triangle T measures the value of the lost consumption of the xgood. And R measure the loss from having to pay more for the units he continues to consume.
Other ways of measuring utility changes or welfare changes different from consumer surplus are: compensating and equivalent variation. They measure utility and utility changes in monetary units. There is no need to know demand curve of consumers.
The compensating variation in income measures the change in income necessary to restore the consumer to his original indifference curve. It’s the change in income that will just compensate the consumer for the price change. (ex-post).
The equivalent variation in income measures how much money would have to be taken away from the consumer before the price change to leave him as well off as he would be after the price change. So, it measures the maximum amount of income that the consumer would be willing to pay to avoid the price change.
The compensating and equivalent variations are just two different ways to measure “how far apart” two indifference curves are. In each case we are measuring the distance between two indifference curves by seeing how far apart their tangent lines are. In general, this measure of distance will depend on the slope of the tangent lines, that is, on the prices that we choose to determine the budget line.
However, the compensating and equivalent variation are the same in one important case—the case of quasilinear utility. In this case the indifference curves are parallel, so the distance between any two indifference curves is the same no matter where it is measured.
4.2. PRODUCER SURPLUS The supply curve measures the amount that will be supplied at each price. The area above the supply curve measures the surplus enjoyed by the suppliers of a good.
The difference between the minimum amount she would be willing to sell the x∗ units for and the amount she actually sells the units for is the net producer’s surplus. Just as in the case of consumer’s surplus, we can ask how producer’s surplus changes when the price increases from p’ to p’’. In general, the change in producer’s surplus will be the difference between two triangular regions and will therefore generally have the roughly trapezoidal shape. The R measures the gain from selling the units previously sold anyway at p’ at the higher price p’’. The T region measures the gain from selling the extra units at the price p’’.
EXAMPLE 4.3. AGGREGATE DEMAND 4.4. ELASTICITY OF DEMAND The elasticity measures how responsive is the demand to changes in prices.
(The slope of the demand function presents some problems because it depends on the units in which you measure price and quantity).
Therefore, we use want to use a measure of price responsiveness which is independent of unit of measurement – comparable across different goods. The price elasticity of demand is defined to be the percent change in quantity divided by the percent change in price.
For instance, If a good has an elasticity of demand greater than 1 in absolute value we say that it has an elastic demand.
If the elasticity is less than 1 in absolute value we say that it has an inelastic demand. And if it has, exactly, 1 we say that is has unit elastic demand. An elastic demand curve is one for which the quantity demanded is very responsive to price: if you increase the price by 1 percent, the quantity demanded decreases by more than 1 percent. When are you hurt more by a price increase? When your demand is very inelastic.
In general, elasticity of demand for a good depends on availability of close substitutes, and goods with closer substitutes have more elastic demand. Time horizon also affect price elasticities.
EXAMPLE PART 2. THE FIRM AND THE SUPPLY FUNCTION TOPIC 5. TECHNOLOGY 5.1. TECHNOLOGICAL CONSTRAINT We will examine the constraints on a firm’s behaviour, when a firm makes choices it faces many constraints.
These constraints are imposed by its customers, by its competitors and by nature. Nature imposes the constraint that there are only certain feasible ways to produce outputs from inputs. The technology is a process by which inputs are converted into an output. Usually several technologies will produce the same product. But only a certain combination of inputs are feasible ways to produce a given amount of output.
Technology is a list of feasible production plans.
Inputs to production are called factors of production. They are often classified as: land, labor, capital goods1 or physical capital (inputs to production that are themselves produced goods: machinery, computers, buildings, vehicles…) and raw materials. We will usually want to think of inputs and outputs as being measured in flow units: a certain amount of labor per week and a certain amount of output a week The production set is the set of all combinations of inputs and outputs that comprise a technological feasible way to produce. Suppose, for example, that we have only one input, measured by x, and one output, measured by y. To say that some point (x, y) is in the production set is just to say that it is technologically possible to produce y amount of output if you have x amount of input. The production set shows the possible technological choices facing a firm.
The production function is the upper boundary of the production set and it measure the maximum possible output you can get from a given amount of input.
1 Different from financial capital: Money.
Of course, the concept of a production function applies equally well if there are several inputs. If, for example, we consider the case of two inputs, the production function f(x1, x2) would measure the maximum amount of output y that we could get if we had x1 units of factor 1 and x2 units of factor 2. In the two-input case there is a convenient way to depict production relations known as the isoquant. An isoquant is the set of all possible combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output. Isoquants are similar to indifference curves. But there is one important difference between indifference curves and isoquants. Isoquants are labeled with the amount of output they can produce, not with a utility level. Thus the labeling of isoquants is fixed by the technology and doesn’t have the kind of arbitrary nature that the utility labeling has.
5.2. DIFFERENT TYPES OF TECHNOLOGY FIXED PROPORTION 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 aren’t worth anything, 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 have. We write the production function as f(x1, x2) = min{x1, x2}.
PERFECT SUBSITUTES Suppose now that 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, so we write the production function as f(x1, x2) = x1+x2.
COBB-DOUGLAS FUNCTION If the production function has the form f(x1, x2) = A(x1^a * x2^b), then we say that it is a Cobb-Douglas production function. This is just like the functional form for Cobb-Douglas preferences that we studied earlier. The numerical magnitude of the utility function was not important, so we set A = 1 and usually set a + b = 1. But the magnitude of the production function does matter so we have to allow these parameters to take arbitrary values. The parameter A measures, roughly speaking, the scale of production: how much output we would get if we used one unit of each input. The parameters a and b measure how the amount of output responds to changes in the inputs.
PROPERTIES OF TECHNOLOGY Technologies are positive monotonic: 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. This is sometimes referred to as the property of free disposal: if the firm can costlessly dispose of any inputs, having extra inputs around can’t hurt it.
We will often assume that the technology is convex. This means that if you have two ways to produce y unit of output, then their weighted average will produce at least y units of output.
5.3. MARGINAL PRODUCT AND MARGINAL RATE OF TECHNICAL SUBSTITUTION The marginal product is how much extra output you get from increasing the input factor i, holding factor j fixed.
The marginal product is decreasing. More and more of a single product produces more output, but at a diminishing rate  law of diminishing returns.
The technical rate of substitution is the slope of the isoquant, if you increase the input of good 1 a little bit how much input of good 2 has to vary to get the same level of production as before? How much extra of factor 2 do we need if we are going to give up a little bit of factor 1? So, it measures the rate at which the firm will have to substitute on input for another in order to keep output constant.
The technical rate of substitution is diminishing, means that the slope of the isoquant must decrease in absolute value as we move along the isoquant in the direction of increasing X1. Which means that the isoquant will have the same sort of convex shape that well-behaved indifference curves have.
Diminishing marginal product is an assumption about how the marginal product changes as we increase the amount of one factor, holding the other fixed. Diminishing TRS is about how the ratio of the marginal product - the slope of the isoquant - changes as we increase the amount of one factor and reduce the amount of the other factor and reduce the amount of the other factor so as to stay on the same isoquant.
5.4. RETURNS TO SCALE What happens to the production level when we increase the amount of all inputs proportionately = jump to a higher isoquant? Constant returns to scale  multiply the amount of both inputs in the production function by t > 1 then the amount produced will be t times higher than before Increasing returns to scale  we scale up both inputs by some factor t and we get more than t times as much output Decreasing returns to scale  we scale up both input by some factor t and we get less than t times as much output. The usual way in which diminishing returns to scale arises is because some input is being held fixed, it’s a really short-term phenomenon.
Technology can exhibit different kinds of returns to scale at different levels of production.
5.5. THE LONG AND THE SHORT RUN In the short run there will be some factors of production that are fixed; a firm is restricted in some way in its choice of at least one input level.
In the long run all factors can be varied; a firm is unrestricted in its choice of all input levels.
TOPIC 6. BENEFIT MAXIMIZATION AND COST MINIMIZATION In this chapter we will assume there is a competitive market where the individual producers take the prices as outside their control. Input and output process are fixed and firm has to choose quantity.
6.1. THE FORMAL PROBLEM OF PROFIT MAXIMIZATION Notice that costs should include all factors of production, valued at market price, including compensation for the owner of the firm – opportunity costs.
Economic costs are often referred to as opportunity cost, the name come from the idea that if you’re using your labour, your land in one application you forgo the opportunity of employing it elsewhere. Therefore, those lost wages are part of the cost of production. The economic definitions of profit require that we value all inputs and output at their opportunity cost. Profits as determined by accountant do not necessarily accurately measure economic profits, as they typically use historical costs – what a factor was purchased for originally – rather than economic costs – what a factor would cost if purchased now.
Output and input levels are typically flows (labor hour per week and so many machines hours per week will produce so much output per week). Consequently, profit is typically a flow also.
SHORT RUN PROFIT MAXIMIZATION The condition for the optimal choice of factor 1 is not difficult to determine. If x* is the profit-maximizing choice of factor 1 then the output price times the marginal product of factor 1 should equal the price of factor 1. In other words, the value of the marginal product of a factor should equal its price.
(- if value of MP greater than its costs, then profits can be increased by increasing output a little bit.) (- if vale of MP smaller than its costs, then profits can be increased by decreasing output a little bit.) Isoprofits are all the combinations of inputs and output that gives the same level of profits.
 The profit-maximization problem is then to find the point on the production function that has the highest associated isoprofit line. As usual it is characterised by a tangency condition: the slope of the production function should equal the slope of the isoprofit line. So, since the slope of the production function is the marginal product we have that: COMPARATIVE STATICS We can use this graph to analyse how a firm’s choice of inputs and outputs varies as the prices of inputs and outputs vary. This gives us one way to analyse the comparative statics of firm behaviour.
LONG RUN PROFIT MAXIMIZATION If the firm has made the optimal choices of factors 1 and 2, the value of the MP of each factor should equal its price. At the optimal choice, the firm’s profits cannot increase by changing the level of either input.
The only reasonable long-run level of profits for a competitive firm that has constant returns to scale at all levels of output is a zero level of profits.
6.2. THE FORMAL PROBLEM OF COST MINIMIZATION Our strategy will be to maximize profits by minimizing costs, otherwise there could be a cheaper way of producing, meaning that profits were not maximized. There are two pieces: - Minimizing costs of producing y - Finding y that maximizes profits MINIMIZING COSTS OF PRODUCING Y The solution to cost-minimization problem, the minimum costs necessary to achieve the desired level of output, will depend on w1, w2, and y, so we write it as c(w1, w2, y). This function is known as the cost function.
The isoquants gives us the technological constraint. And the isocost lines is a curve that contains all of the input bundles that cost the same amount.
Thus our cost-minimization problem can be rephrased as: find the point on the isoquant that has the lowest possible isocost line associated with it. The slope of the isoquant must be equal to the slope of the isocost curve, in other words, the technical rate of substitution must equal the factor price ratio.
The average cost function is simply the cost per unit to produce y units of output: If the technology exhibits constant returns to scale, the cost per unit of output will be constant no matter what level of output the firm wants to produce.
If the technology exhibits increasing returns to scale, then the costs will increase less than linearly with respect to output, so the average costs will be declining in output: as output increases, the average costs of production will tend to fall.
If the technology exhibits decreasing returns to scale, then average costs will rise as output increases.
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