Intermediate Microeconomics 2024 PDF

Unit 1: Theory of Consumer Behaviour Section 1: Consumer Behaviour 1.0. Introduction The traditional theory of demand starts with examination of consumer behaviour, because the market demand is assumed to be summation of the demands of individual consumers. The theory of consumer behaviour is an e...

Unit 1: Theory of Consumer Behaviour Section 1: Consumer Behaviour 1.0. Introduction The traditional theory of demand starts with examination of consumer behaviour, because the market demand is assumed to be summation of the demands of individual consumers. The theory of consumer behaviour is an explanation of how consumers allocate incomes to the purchase of different goods to maximise their satisfaction. The theory explains how consumers allocate their incomes across goods and how these allocation decisions determine the demands for various goods. Simply put, consumer behaviour illustrates how a consumer decides on a basket of goods and services he consumes. Understanding consumer purchasing decisions helps to understand how changes in income and prices affect the demand for goods and why the demand for some goods is more sensitive than others to changes in prices and income. 1.1. The Consumer A consumer is the one who uses a good or service. The consumer is seen as a decision- making unit. A consumer may be an individual or a household. If the decision involves the buying of a good and service for common use, such as furniture and home, then the household is the decision-making unit but if it is purchase of a good for individual use, such as a book then the individual is the decision-making unit. The consumer is assumed to be rational, the consumer is a utility maximiser, has limited income which is a constraint to his consumption, and has information necessary for his consumption. 1.2. Consumption Bundle Consumption bundle or consumption basket is a specific combination of goods that the consumer would like to consume. Consumers select bundles that make them as well off as possible. There are many possible goods consumers can afford to consume. We assume there is a finite, fixed, but arbitrary number of n different goods ( x1 , x2 , , xn ). Each of those goods can be represented in a given basket by a nonnegative quantity of all the possible goods. Any given basket could have zero of one or more of those goods. To depict the consumer’s choice behaviour graphically, we assume the consumption bundle consists of two goods. If x1 denotes the amount of good 1 and x2 denotes the amount of good 2, then the complete consumption bundle is stated as ( x1 , x2 ). This consumption bundle can conveniently be stated as X. We assume that X is a closed and convex set. The convexity of a consumption set means that every good is divisible and can be consumed by fraction units. Pareto Preferred Consumption Bundle Consider two consumption baskets, A and B. We say A is Pareto preferred if every consumer is at least as well off under A as they are under B, and at least one consumer is better off under A. The Pareto criterion states that A is an improvement on B if at least one consumer would be strictly better off with A than B (in other words, would strictly prefer A to B) and no consumer would be worse off. We say that A Pareto dominates B. Suppose these consumption bundles for two consumers, Kofi and Ami: 1 | e lawluvi – intermediate microeconomics  Consumer Bundle X Bundle Y Bundle Z Kofi 5 5 4 Ami 5 6 8 Bundle Y Pareto dominates bundle X (Ami is better off; Kofi is not worse off). We cannot Pareto rank bundle Z versus bundle X or bundle Y (even though “bundle Z is bigger”). 1.3. Basics of Consumer Behaviour Consumer behaviour is best understood in three distinct steps: Consumer preferences Budget constraints Consumer choices Consumer preferences: This describes the reasons people might prefer one good to another. Budget constraints: Consumers have limited incomes which restrict the quantities of goods they can purchase. That is, in the basic problem of consumer’s choice, not all consumption bundles are affordable, because a consumer is constrained by wealth (income). Consumer choices: Given their preferences and limited incomes, consumers choose to buy combinations of goods that maximise their satisfaction. These combinations will depend on the prices of various goods. Understanding consumer choice will help understand demand; that is, how the quantity of a good that consumer chooses to purchase depends on its price. Section 1 Problem Set 1. Describe the theory of consumer behaviour. 2. Explain the three distinct steps of consumer behaviour. 3. Describe consumption bundle. 4. Give the assumptions underlying consumption bundle. 5. Describe Pareto preferred consumption bundle. 2 | e lawluvi – intermediate microeconomics  Unit 1: Theory of Consumer Behaviour Section 2: Consumer Preferences 2.0. Introduction There are many possible goods consumers can afford to consume. Consumer preferences determine which of these goods will be consumed. Preferences refer to a consumer’s ability to compare or rank one commodity bundle over another. For any two consumption bundles, ( x1 , x2 ) and ( y1 , y 2 ), the consumer can rank these bundles as to their desirability. The consumer can determine that one of the bundles is strictly better than the other, or decide that she is indifferent between the two bundles. Preferences ignore costs. 2.1. Preference Relations Preference relation specifies the consumer’s tastes or satisfactions for the different goods of choice. It is generally assumed that the consumer seeks to identify and select an available alternative that is most preferred in the light of his personal tastes/interests. For our discussion of consumer preferences, we will use the notation to indicate preference relation, and this can be in the forms of strict preference, indifference, and weak preference. Strict preference relation. The preference relation indicates that one bundle is strictly preferred to another, so that ( x1 , x2 ) ( y1 , y 2 ) is interpreted as the consumer strictly prefers ( x1 , x2 ) to ( y1 , y 2 ) in the sense that the consumer definitely wants the x-bundle instead of the y-bundle. If the consumer prefers one bundle to another, it means that the consumer would choose one over the other, given the opportunity. The idea of preference is based on the consumer’s behaviour. In order to tell whether one bundle is preferred to another, we see how the consumer behaves in choice situations involving the two bundles. If the consumer always chooses ( x1 , x2 ) when ( y1 , y 2 ) is available, then it is natural to say that this consumer strictly prefers ( x1 , x2 ) to ( y1 , y 2 ). Indifference relation. The preference relation ~ means the consumer is indifferent between two bundles of goods, so that ( x1 , x2 )  ( y1 , y 2 ) can be interpreted as the consumer is equally satisfied with either consuming bundle ( x1 , x2 ) or bundle ( y1 , y 2 ). Weak preference relation. The preference relation means the consumer weakly prefers one bundle to another, so that ( x1 , x2 ) ( y1 , y 2 ) can be interpreted as the consumer is indifferent between the two bundles or strictly prefers ( x1 , x2 ) to ( y1 , y 2 ). These relations are related. For example, if ( x1 , x2 ) ( y1 , y 2 ) and ( y1 , y 2 ) ( x1 , x2 ), we can conclude that ( x1 , x2 )  ( y1 , y 2 ). If the consumer thinks that ( x1 , x2 ) is at least as good as ( y1 , y 2 ) and that ( y1 , y 2 ) is at least as good as ( x1 , x2 ), then the consumer must be indifferent between the two bundles of goods. Monotonicity of Preferences If one bundle contains at least as much of every good as another bundle, then it is at least as good as the other. Strict monotonicity means that having more of at least one good makes the consumer strictly better off. Monotonicity of preferences means individuals’ desires for 3 | e lawluvi – intermediate microeconomics goods: the more, the better. However, the consumer may dislike other goods (that is, “bad”), such as garbage or pollution, thus preferring bundles with the smallest possible amount of those goods. 2.2. Assumptions about Preferences The economic theory of consumer behaviour assumes that consumers’ preferences are consistent and make sense. Consumer’s preference relation is assumed to satisfy basic axioms: completeness, transitivity, and reflexive. These axioms form the basis of consumer theory about people’s preferences for one bundle versus another. These axioms of consumer choice are intended to give formal mathematical expression to fundamental aspects of consumer behaviour and attitudes towards the objects of choice. Complete. Consumers can compare and rank all possible bundles. For any two different bundles, ( x1 , x2 ) ( y1 , y 2 ), or ( y1 , y 2 ) ( x1 , x2 ), or ( x1 , x2 )  ( y1 , y 2 ). A consumer will prefer X to Y , will prefer Y to X , or will be indifferent between the two. Transitivity. Preferences are transitive. If ( x1 , x2 ) ( y1 , y 2 ) and ( y1 , y 2 ) ( z1 , z 2 ), then ( x1 , x2 ) ( z1 , z 2 ). If a consumer prefers bundle X to Y and prefers Y to Z , then the consumer also prefers bundle X to Z. Transitivity is necessary for consumer consistency. Reflexive. Any bundle is assumed at least as good itself; ( x1 , x2 ) ( x1 , x2 ). If bundle X has at least as much of every good as bundle Y and more of some good, then bundle X is preferred to bundle Y. Goods are assumed to be desirable, so consumers always prefer more of any good to less. Consumers are never satisfied or satiated. The completeness assumption says that any two bundles can be compared; the transitivity assumption requires the consumer’s choice be consistent; and the reflexive assumption says that every consumption bundle is good as itself. A preference relation that satisfies these three properties is called a preference ordering. Thus, a preference ordering enables the consumer to rank different consumption bundles in order of preference. 2.3. Indifference Curves Consumer preferences can be graphically illustrated with the use of indifference curves. An indifference curve represents all combinations of bundles that give a consumer with the same level of satisfaction. The consumer is indifferent among the bundles represented by the points located on the curve. The indifference curve through a consumption bundle consists of all bundles of goods that leave the consumer indifferent to the given bundle. In Figure 2.1, movement along the curve, the consumer feels neither better nor worse off in giving up x1 units of good 1 to obtain x2 additional units of good 2. Graph containing a set of indifference curves showing the bundles among which a consumer is indifferent is called an indifference map; illustrated in Figure 2.2. Every bundle on indifference curve III is preferred to those on curve II, and every bundle on indifference curve II is preferred to those on curve I. The three indifference curves are convex and do not cross. Curves farther from the origin imply higher levels of satisfaction than curves closer to the origin. 4 | e lawluvi – intermediate microeconomics  Figure 2.1. Indifference curve Figure 2.2. Indifference map 2.4. Marginal Rate of Substitution We refer to the magnitude (that is, the absolute value) of the slope of an indifference curve at a point as the marginal rate of substitution. Marginal rate of substitution is the maximum amount of a good that a consumer is willing to give up in order to obtain an additional unit of another good and still maintain the same satisfaction level. Specifically, the marginal rate of substitution of good 1 for good 2 ( MRS x1x2 or just MRS ) is the maximum units of good 2 that a consumer is willing to give up to obtain an additional unit of good 1, holding utility constant. Suppose the MRS is 3; it means that the consumer will give up 3 units of good 2 to obtain 1 additional unit of good 1 and still maintain the same utility. To describe the MRS, we must be clear about which good we are giving up and which we are getting more of. To be consistent, we will define the MRS in terms of the amount of the good on the vertical axis that the consumer is willing to give up in order to obtain an additional unit of the good on the horizontal axis. The MRS can be stated as: x2 MRS x1x2 = − x1 5 | e lawluvi – intermediate microeconomics The indifference curve depicts diminishing marginal rate of substitution. As a consumer obtains more units of good 1, the units of good 2 the consumer is willing to give up to obtain another unit of good 1 decrease. This assumption implies that indifference curves are convex from the origin. 2.5. Properties of Indifference Curves A consumer prefers a higher indifference curve to a lower one. Higher indifference curves contain more goods, and because the consumer gets more satisfaction from consuming more, a higher indifference curve represents bundles of goods that provide the consumer with more satisfaction than the bundles on lower indifference curves. Without constraints, it simply does not make any sense for the consumer to pick a bundle on a lower indifference curve when he could pick a bundle with more goods, and more satisfaction. Indifference curves do not cross. Each indifference curve represents all the bundles that provide the consumer with a certain level of utility. Suppose that one curve represents bundles with utility of 10 and a second curve is bundles with utility of 13. If these curves crossed, the bundle at the point of intersection would simultaneously have utilities of 10 and 13. Indifference curves usually slope downward. If both goods are desirable, then the consumer would always prefer more of each good to less of it. A downward-sloping indifference curve – one with a negative slope – shows that when a consumer gets less of one good, his utility decreases and he requires more of the other good in order to compensate. Indifference curves are usually convex or bowed inward. This inward curve follows from the property of diminishing marginal utility. The term convex means that the slope of the indifference curve falls in magnitude as we move down along the curve. Thus, an indifference curve is convex if the MRS diminishes along the curve. At the top of the curve, the slope is steep, because the consumer has a lot of the good (good 2) and is willing to trade more of it for even a little of the other good (good 1). As the curve goes downward, the consumer has less of good 2 and requires more and more of good 1 to compensate for the loss of good 2, and the slope flattens out. 2.6. Examples of Preferences The shape of indifference curve depends on consumer’s preferences; it describes the willingness of a consumer to substitute one good for another. An indifference curve with a different shape implies a different willingness to substitute. Two goods are perfect substitutes when the marginal rate of substitution of one for the other is a constant. Indifference curves describing the trade-off between the consumption of perfect substitutes are straight lines. Suppose a consumer’s preferences for blue pen and black pen, and the consumer is entirely indifferent between having a blue or black pen. These goods are perfect substitutes. The consumer is always willing to trade 1 pen of one colour for 1 pen of the other colour. For a bundle, say (6, 6), this consumer sees any other bundle that has 12 pens in it as just good as (6, 6); any bundle ( x1 , x2 ) such that x1 + x2 = 12 will be on this consumer’s indifference curve through (6, 6). The indifference curves for this consumer are all parallel straight lines with a slope of –1. Bundles with more total pens are preferred to bundles with fewer total pens, so the direction of increasing preference is up and to the right. 6 | e lawluvi – intermediate microeconomics  Figure 2.3. Perfect substitutes Two goods are perfect complements when the marginal rate of substitution of one for the other is zero or infinite. These goods are always consumed together in fixed proportions. Example is a consumer’s preferences for left shoes and right shoes. For the consumer, the two goods are perfect complements because a left shoe will not increase his satisfaction unless he can obtain the matching right shoe. The consumer will not give up any left (right) shoes to get additional right (left) shoes. Or, say the consumer’s bundle is (8, 8). One more right shoe, we have (9, 8) or 1 more left shoe, we have (8, 9). Either case leaves the consumer indifferent to the original position (8, 8). The indifference curves are shaped as right angles (L-shaped) as depicted in Figure 2.4, and the MRS is either zero or infinite. Increasing both the number of right shoes and the number of left shoes at the same time will move the consumer to a more preferred position, so the direction of increasing preference is up and to the right. Figure 2.4. Perfect complements A good is a neutral good if the consumer does not care about it. Suppose a bundle ( x1 , x2 ) where the consumer only cares about the amount of x1 and does not care at all about the units of x2 consumed. The more the units of x1 , the better. But adding more units of x2 does not affect the consumer one way or other. In this case the indifference curves will be vertical lines as depicted in Figure 2.5. A bad is a good that the consumer does not like. For some goods, more is worse, such as air pollution or trash. Suppose a bundle ( x1 , x2 ) where the consumer likes good 1 ( x1 ) and dislikes good 2 ( x2 ). This consumer will have indifference curves that slope up and to the right as depicted in Figure 2.6. The direction of increasing preference is down and to the right – that is, toward the direction of decreased good 2 consumption and increased good 1 consumption, just as the arrows in the diagram illustrate. 7 | e lawluvi – intermediate microeconomics  Figure 2.5. Neutral good Figure 2.6. Bads Section 2 Problem Set 1. Describe consumer preference relation. 2. Explain the following preference relations: a. Strict preference relation b. Indifference relation c. Weak preference relation 3. Give the assumptions underlying the consumer preference relation. 4. Define preference ordering. 5. Ama indicates that she prefers going to the zoo over the park and prefers going to the beach over the zoo. When given the choice between the park and the beach, she chooses the park. Which of the assumptions of consumer preference theory is she most likely violating? Explain. 6. Describe marginal rate of substitution. 7. Ed’s marginal rate of substitution of good X for good Y ( MRS XY ) is 2. Describe what this means. 8. Ki and Ed have identical baskets containing books (B) and compact discs (D). Ki’s MRSBD equals 0.8, and Ed’s MRSBD equals 1.25. State and justify whether Ki or Ed has a stronger preference for books. 9. Ed currently has 50 units of good 1 and 20 units of good 2. His marginal rate of substitution of good 1 for good 2 is 4, and his indifference curve is strictly convex. a. Determine whether Ed would be willing to trade at the rate of three of his good 2 in exchange for one more unit of good 1. b. Suppose that Ed is indifferent between his current bundle and one containing 40 units of good 1 and 25 units of good 2. Describe Ed’s MRS evaluated at the new bundle. 8 | e lawluvi – intermediate microeconomics  10. For a consumption bundle ( x1 , x2 ), where x1 is units of good 1 consumed and x2 is that of good 2. Illustrate the indifference curve if the goods are perfect substitutes. 11. For a consumption bundle ( x1 , x2 ), where x1 is units of good 1 consumed and x2 is that of good 2. Illustrate the indifference curve if the goods are perfect complements. 12. For a consumption bundle ( x1 , x2 ), where x1 is units of good 1 consumed and x2 is that of good 2. Illustrate the indifference curve if good 1 is a neutral good. 13. If good 1 is a “neutral”, what is its marginal rate of substitution for good 2? Give reason. 9 | e lawluvi – intermediate microeconomics  Unit 1: Theory of Consumer Behaviour Section 3: Budget Constraints 3.0. Introduction Consumers have limited incomes which restrict the amounts of goods they can purchase. This means that in making decisions, consumers face budget constraints. Simply stated, budget constraints are constraints that consumers face as a result of limited incomes. This section looks at the budget set and the properties of the budget set taking into consideration the effects of tax, subsidy, and rationing on the slope of the budget line. 3.1. Budget Line Suppose a consumer has a fixed income, m , that can be spent on the consumption bundle ( x1 , x2 ) where x1 is the units of Good 1 and x2 is the units of Good 2. Let p1 be the price of Good 1 and p2 be the price of Good 2, then the budget constraint (also called the budget set) of the consumer can be stated as p1 x1 + p2 x2  m. In this expression, p1 x1 is the amount of money spent on Good 1 and p2 x2 is the amount of money spent on Good 2. The budget constraint of the consumer requires that the amount spent on the two goods be no more than the total amount the consumer has to spend. The consumer’s affordable consumption bundles are those that do not cost any more than m. This set of affordable consumption bundles at prices ( p1 , p2 ) and income ( m ) is called the budget set of the consumer. Budget set is the bundles of goods a consumer can afford. A budget line shows the bundle of two goods a consumer can buy given her income and prices of the goods in question. The budget line indicates all combinations of x1 and x2 for which the total amount of money spent is equal to income. The budget line states that the expenditure by the consumer on the two goods should be less or equal to the individual’s income. If the consumer spends entire income on the two goods, the equation holds with equality. This relation is called the budget line: p1 x1 + p2 x2 = m The budget line is the set of bundles that cost exactly m. It defines the bundles of goods that exhaust a consumer’s income. The consumer’s budget set is depicted in Figure 3.1. The heavy line is the budget line – the bundles that cost exactly m – and the bundles below this line are those that cost strictly less than m. In particular, any combination of goods x1 and x2 below the budget line represents an affordable consumption bundle. The budget set comprises of the budget line and the bundles below the budget line. Any point above the budget line represents a bundle of goods that is unaffordable. 10 | e lawluvi – intermediate microeconomics  m The horizontal intercept, , measures the maximum units of Good 1 that the consumer can p1 m purchase if all income is spent on Good 1. The vertical intercept, , measures the maximum p2 units of Good 2 that the consumer can purchase if all income is spent on Good 2. x2 m p2 p1 Budget line; slope = − p2 m p1 x1 Figure 3.1. The budget set p  The slope of the budget line is −  1 . It is the negative of the ratio of the prices of the two  p2  goods (the ratio of the price of the good on the horizontal axis and that on the vertical axis). The magnitude of the slope indicates the rate at which the two goods can be substituted for each other without changing the total amount of money spent. The budget line equation can be restated to obtain an alternative expression for the budget constraint in slope-intercept form as: m p1 x2 = − x1. p2 p2 m p and a slope of − 1. It This is an expression for a straight line with a vertical intercept of p2 p2 gives how many units of Good 2 the consumer needs to consume in order to just satisfy the budget constraint if she is consuming x1 units of Good 1. The slope of the budget line is called the market rate of substitution between Good 1 and Good 2. Market rate of substitution is the rate at which one good may be traded for another good in the market. Suppose a consumer increases the consumption of Good 1 by x1. Then we need to find out how much the consumption of Good 2 will have to change in order to satisfy budget constraint. Let x2 indicates the change in the consumption of Good 2. If the 11 | e lawluvi – intermediate microeconomics consumer satisfies budget constraint before and after making the change, then the consumer must satisfy the following equations p1 x1 + p2 x2 = m p1 ( x1 + x1 ) + p2 ( x2 + x2 ) = m These two equations reduce to p1x1 + p2x2 = 0. x2 Solving for , the rate at which Good 2 can be substituted for Good 1, gives x1 x2 p =− 1. x1 p2 This is the slope of the budget line. The negative sign is there because x1 and x2 must always have opposite signs. If a consumer consumes more of good 1 (good 2), the consumer will have to consume less of good 2 (good 1) to satisfy the budget constraint. 3.2. Effects of Changes in Income and Prices Consumers’ behaviour is constrained by their income and the prices they must pay for the goods they consume. The budget line depends on income ( m ) and on the prices of the goods ( p1 , p2 ). If one or more of these parameters change, consumers are likely to change their consumption behaviour (the set of goods that a consumer can afford changes). 3.2.1. Income Changes Change in income alters the horizontal and vertical intercepts of the budget line but does not change the slope (because the price of neither good changed). If income is increased, the budget line shifts outward, and if income is cut, the budget line shifts inward. A change in income (with prices unchanged) causes the budget line to shift parallel to the initial budget line. Thus, when a consumer’s income increases, more bundles of goods become affordable. When income decreases, fewer bundles of goods are affordable. Given the assumption that prices remain unchanged, an increase in income by k; the entire budget line shifts out by the equivalent of k, maintaining the same slope as it did before. This is shown in Figure 3.2. Figure 3.2. Change in income 12 | e lawluvi – intermediate microeconomics 3.2.2. Price Changes Decline in the price of Good 1 (Good 2) means the maximum units of Good 1 (Good 2) that can be purchased increases. Price decrease causes the budget line to rotate outward. Increase in the price of Good 1 (Good 2) means the maximum units of Good 1 (Good 2) that can be purchased decreases. Price increase causes the budget line to rotate inward. A change in the price of one good rotates the budget line. Change in Price of Good 1 Suppose consumer’s income remains fixed at m , but the price of Good 1 decreases to p1i (  p1 ). Further, suppose the price of Good 2 ( p2 ) remains unchanged. Given the slope of the p budget line is − 1 , the reduction in the price of Good 1 changes the slope, making it flatter p2 m than before. The maximum amount of Good 2 that can be purchased is , so a reduction in p2 the price of Good 1 does not change the vertical intercept of the budget line. But the maximum amount of Good 1 that can be purchased at the lower price (the horizontal intercept m m of the budget line) is i which is greater than. The ultimate effect of a reduction in the p1 p1 price of Good 1 is to rotate the budget line outward (see Figure 3.3). Figure 3.3. Decrease in price of Good 1 Suppose consumer’s income remains fixed at m , but the price of Good 1 increases to p1j (  p1 ) and the price of Good 2 ( p2 ) remains unchanged. The slope of the budget line is given p by − 1 , the increase in the price of Good 1 changes the slope, making it steeper than before. p2 The maximum amount of Good 1 that can be purchased at the higher price (the horizontal m m intercept of the budget line) is j which is less than. An increase in the price of Good 1 p1 p1 leads to an inward rotation of the budget line. An increase in the price of Good 1 makes the budget line steeper (see Figure 3.4). 13 | e lawluvi – intermediate microeconomics  Figure 3.4. Increase in price of Good 1 Change in Price of Good 2 Suppose consumer’s income remains fixed at m , but the price of Good 2 decreases to p2i (  p2 ) and the price of Good 1 ( p1 ) remains unchanged. The slope of the budget line is given p by − 1 , the reduction in the price of Good 2 changes the slope, making it steeper than p2 m before. The maximum amount of Good 1 that can be purchased is , so a reduction in the p1 price of Good 2 does not change the horizontal intercept of the budget line. But the maximum amount of Good 1 that can be purchased at the lower price (the vertical intercept of the budget m m line) is i which is greater than. The ultimate effect of a reduction in the price of Good p2 p2 2 is to rotate the budget line outward, as illustrated in Figure 3.5. Figure 3.5. Decrease in price of Good 2 Suppose consumer’s income remains fixed at m , but the price of Good 2 increases to p2j (  p2 ) and the price of Good 1 ( p1 ) remains unchanged. The slope of the initial budget line is p given by − 1 , so increase in the price of Good 2 changes the slope, making it flatter than p2 before. The maximum amount of Good 2 that can be purchased at the higher price (the 14 | e lawluvi – intermediate microeconomics  m m horizontal intercept of the budget line) is j which is less than. An increase in the price p2 p2 of Good 2 leads to an inward rotation of the budget line. An increase in the price of Good 2 makes the budget line flatter. This is illustrated in Figure 3.6. Figure 3.6. Increase in price of Good 2 Simultaneous Price Change in Good 1 and Good 2 Let us see what happens to the budget line when the prices of Good 1 and Good 2 change at the same time. Suppose the original budget line, stated as: p1 x1 + p2 x2 = m. Now suppose that both prices become  times as large, then we get  p1 x1 +  p2 x2 = m This equation can be restated as m p1 x1 + p2 x2 =.  This equation implies that multiplying both prices by a constant amount  is equivalent to 1 dividing income by the same constant . This causes the budget line to shift by. For  example, if the prices of both goods doubled (  = 2 ), both the horizontal and vertical 1 intercepts shift inward by. Equally, if the prices of both goods are halved (  = 0.5 ), both 2 the horizontal and vertical intercepts shift outward by 2. If both prices ( p1 , p2 ), and income ( m ) are multiplied by  , the budget line is unchanged. If m m m decreases and both prices ( p1 , p2 ) increase, then the intercepts and must both p1 p2 decrease. This means that the budget line will shift inward. If p2 increases more than p1 , so 15 | e lawluvi – intermediate microeconomics  p1 that − decreases (in absolute value), then the budget line will be flatter; if p2 increases p2 p less than p1 , so that − 1 increases (in absolute value), the budget line will be steeper. p2 3.3. Taxes, Subsidies, and Rationing Economic policy uses tools that affect a consumer’s budget constraint, such as taxes and subsidies. If government imposes a quantity tax, this means that a consumer has to pay a certain amount to the government for each unit of the good purchased. This tax affects the budget line of a consumer. For the consumer, the tax is like a higher price. A quantity tax of t cedis per unit of Good 1 changes the price of Good 1 from p1 to p1 + t. This makes the consumer to buy less of Good 1 given her income hence rotating the budget line inward (on p +t the horizontal axis). The budget line becomes steeper and the slope is given as − 1. p2 Another kind of tax is a value (ad valorem) tax. This is a tax on the value (the price) of a good. It is usually expressed in percentage terms. If price of Good 1 is subject to a sales tax at a rate  , then the actual price of Good 1 facing the consumer is (1 +  ) p1. The consumer has to pay p1 to the supplier and  p1 to the government for each unit of the good so the total cost of the good to the consumer is (1 +  ) p1. This makes the consumer to buy less of Good 1 given her income hence rotating the budget line inward (on the horizontal axis). The budget line becomes steeper and the slope is given as − (1 +  ) p1. p2 A subsidy is the opposite of a tax. For a quantity subsidy, the government gives an amount to the consumer that depends on the amount of the good purchased. If the subsidy is s cedis per unit of consumption of Good 1, then the price to the consumer would be p1 − s. An ad valorem subsidy is a subsidy based on the price of the good being subsidised. If price of Good 1 is subject to an ad valorem subsidy at a rate  , then the actual price of Good 1 facing the consumer is (1 −  ) p1. These two cases make the budget line flatter (the budget line p1 − s rotates outward on the horizontal axis). For quantity subsidy, the slope is given as − , p2 and ad valorem subsidy, the slope is given as − (1 −  ) p1 p2 Government might also use a lump-sum tax or subsidy. In the case of a tax, the government takes away some fixed amount of money, so a lump-sum tax means that the budget line shifts inward because the consumer’s income has been reduced. For a lump-sum subsidy, the budget line shifts outward. Quantity taxes and value taxes rotates the budget line one way or the other depending on which good is being taxed, but a lump-sum tax shifts the budget line inward. Sometimes, governments also impose rationing constraints. This means that the level of consumption of some good is fixed to be no larger than some amount. For instance, in situations where there is a shortage of petrol the government can authorise petrol stations not 16 | e lawluvi – intermediate microeconomics to sell petrol above some certain gallons to consumers. Suppose that good 1 is rationed so that no more than x1 could be consumed by a given consumer. The budget set for the consumer is depicted in Figure 3.7; it would be the old budget set with a piece cut off. If good 1 is rationed, the section of the budget set beyond the rationed quantity will be cut off ( x1  x1 ). Figure 3.7. Budget set with rationing Let us consider taxes and rationing combined. Suppose a situation where a consumer can consume good at a price of p1 up to some level x1 and then had to pay a tax t on all consumption in excess of x1. The budget set for this consumer is depicted in Figure 3.8. The p p +t budget line has a slope of − 1 to the left of x1 , and a slope of − 1 to the right of x1. p2 p2 Figure 3.8. Taxing consumption greater than x1. Section 3 Problem Set 1. A consumer has income of ¢100 to spend on good 1 and good 2, and faces prices of p1 = ¢1 for good 1 and p2 = ¢5 for good 2. a. Derive the consumer’s budget constraint. b. Graph the budget line for the consumer. 17 | e lawluvi – intermediate microeconomics  c. Show how the budget line changes when the price of Good 1 increases to ¢5. 2. El has a total budget of ¢128 per week to spend on cod and lamb. The price of cod is ¢10 per kilogram and the price of lamb is ¢16 per kilogram. a. Calculate El’s budget constraint. b. Construct a diagram of El’s budget constraint. c. Determine the slope of El’s budget constraint. 3. Eli, a college student, has an income of ¢100, which he spends on coke and pizza. The price of coke is ¢1 per can and the price of pizza is ¢2 per slice. a. Find the equation of his budget line. Graph it. b. Show how Eli’s budget set change when his income increases to ¢150. c. Suppose Eli’s college subsidises pizza, decreasing its price by ¢1. Find the new budget set of Eli. 4. A consumer has ¢400 to spend on good 1 and good 2. The market price of good 1 is ¢10 and of good 2 is ¢40. a. Find the market rate of substitution between goods 1 and 2. b. Illustrate the consumer’s budget set in a carefully labelled diagram. c. Show how the consumer’s budget set changes if income increases by ¢400. How does ¢400 increase in income alter the market rate of substitution between goods 1 and 2? 5. A consumer has ¢250 to spend on good 1 and good 2. The relevant market prices are p1 = ¢5 and p2 = ¢10. a. Write the equation for the consumer’s budget line. b. Illustrate the consumer’s budget set in a carefully labelled diagram. c. Show how the consumer’s budget set changes when the price of good 1 rises to ¢10. How does this change alter the market rate of substitution between goods 1 and 2? 6. The budget line of a consumer is stated as p1 x1 + p2 x2 = m. Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income. 7. Examine what happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant. 8. Suppose the price of good 1 doubles and the price of good 2 triples, does the budget line become flatter or steeper? Illustrate. 9. Suppose that the government puts a tax of ¢0.20 a gallon on petrol and then later decides to put a subsidy on petrol at a rate of ¢0.11 a gallon. Find the net tax in this combination. 10. Suppose that a budget equation is given by p1 x1 + p2 x2 = m. The government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, and a quantity subsidy on good 2 of s. Give an expression for the new budget line. 18 | e lawluvi – intermediate microeconomics  11. Suppose the budget equation of a consumer is given as p1 x1 + p2 x2 = m. If government decides to impose an ad valorem tax on good 1 of  , and a quantity subsidy on good 2 of . Give an expression for the new budget line. Find the new market rate of substitution between good 1 and good 2. 12. Ed has a budget of ¢250 to spend on good 1 and good 2. The market price of good 1 is ¢5 and of good 2 is ¢10. a. Find the market rate of substitution between goods 1 and 2. b. Write the equation for Ed’s budget line. Illustrate his budget set. c. The government decides to impose a lump-sum tax of ¢10, ad-valorem tax of 20% on good 1, and a quantity subsidy of ¢2 on good 2. Give an equation for the new budget line of Ed. How does these events alter the market rate of substitution between goods 1 and 2? 13. Suppose the budget equation of a consumer is given as p1 x1 + p2 x2 = m , where p1 is price of good 1, p2 is price of good 2 and m is income of the consumer. a. Find the market rate of substitution between goods 1 and 2. b. Suppose the government decides to impose an ad-valorem tax on good 1 of 10% and an ad-valorem subsidy on good 2 of 20%. Give an expression for the new budget constraint. Find the new market rate of substitution between good 1 and good 2. 19 | e lawluvi – intermediate microeconomics  Unit 1: Theory of Consumer Behaviour Section 4: Utility 4.0. Introduction Consumers are assumed to be rational. Given income and market prices of the various goods, consumers plan the spending of their incomes so as to attain the highest possible satisfaction or utility. This is known as the axiom of utility maximisation. In the traditional theory, the consumer is assumed to have full knowledge of all the information relevant to his decision, that is, he has complete knowledge of all the available goods, their prices and his income. In order to attain this objective, the consumer must be able to compare the utility (satisfaction) of the various ‘baskets of goods’ which he can buy with his income. 4.1. Utility The indifference curves describe consumer preferences graphically, on the assumption that consumers can rank alternatives. We can also assign numerical values to bundles. Thus, we use utility functions to mathematically represent consumer preferences. Utility is a measure of the level of satisfaction that a consumer derives from the consumption bundle of goods. Utility simplifies the ranking of bundles such that if bundle ( x1 , x2 ) is preferred to bundle ( y1 , y 2 ) then the x-bundle had a higher utility level than the y-bundle. 4.2. Utility Functions Suppose the consumer’s purchases are limited to two goods. His ordinal utility function is u = f ( x1 , x2 ) where x1 and x2 are the quantities of the two goods X 1 and X 2 which he consumes. It is assumed that f ( x1 , x2 ) is continuous and has continuous first- and second-order partial derivatives. The consumer’s utility function is not unique. In general, any single valued increasing function of x1 and x2 can serve as a utility function. The utility number u 0 assigned to any particular good combination indicates that it is preferable or superior to all combinations with lower numbers and inferior to those with higher numbers. A utility function is a formula for calculating the total utility that a particular person derives from the consumption bundle of goods. Utility function is a way of quantitatively describing preferences; it is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles. This means ( x1 , x2 ) ( y1 , y 2 ) if and only if u ( x1 , x2 )  u ( y1 , y2 ). If a consumer prefers bundle ( x1 , x2 ) to bundle ( y1 , y 2 ), we would say that the utility assigned to x-bundle, denoted by u ( x1 , x2 ) , exceeds the utility assigned to y-bundle, u ( y1 , y2 ). The utility function is defined with reference to consumption during a specified period of time. The level of satisfaction that the consumer derives from a particular bundle depends upon the length of the period during which he consumes it. Different levels of satisfaction are derived from consuming 10 cones of ice cream within one hour and within one month. There 20 | e lawluvi – intermediate microeconomics is no unique time period for which the utility function should be defined. But the utility function must not be defined for a period so short that the desire for variety cannot be satisfied. On the other hand, tastes (the shape of the function) may change if it is defined for too long a period. Any intermediate period is satisfactory for the static theory of consumer behaviour. The present theory is static in the sense that the utility function is defined with reference to a single time period, and the consumer’s optimal expenditure pattern is analysed only with respect to this period. No account is taken of the possibility of transferring consumption expenditures from one period to another. Suppose that a consumer’s utility function for good 1 and good 2 is u ( x1 , x2 ) = x1 + 3x2. Then the bundle (8, 3) gives a utility of 17. The consumer will be, thus, indifferent between this bundle and a bundle (5, 4). Either bundle is preferred to a third bundle (9, 2). This is because this third bundle has a utility level of 15. The utility measurements that go into a utility function are relative, not absolute. The utility number assigned to any particular bundle indicates that it is preferable or superior to all bundles with lower numbers and inferior to those with higher numbers. For a bundle that gives a utility of 17, the only implication is that the consumer values the bundle more highly than a bundle with a utility of 16 and less than a bundle with utility of 18. These numbers are arbitrary; they do not mean anything except an ordering for bundles the consumer likes more or less. Thus, utility function orders consumer choices in terms of levels of satisfaction. Utility function that ranks bundles in order of most to least preferred is called an ordinal utility. Ordinal utility function simply ranks bundles; the magnitude of the utility difference between any two bundles is not important. This is because we have no means of objectively measuring a person’s satisfaction from the consumption of a bundle. We can only infer that one bundle is better than other; we do not, however, know by how much one is preferred to the other. Ordinal utility does not indicate by how much one is preferred to another. It suffices for the consumers to be able to rank the various baskets of goods according to the satisfaction that each bundle gives them. That is, a consumer must be able to determine his order of preference among the different bundles of goods. The main ordinal theories are the indifference-curves approach and the revealed preference hypothesis. Geometrically, a utility function is a way to label indifference curves. The locus of all bundles from which the consumer derives the same level of satisfaction forms an indifference curve. An indifference map is a collection of indifference curves corresponding to different levels of satisfaction. Every bundle on an indifference curve must have the same utility; therefore, 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. A movement to a higher indifference curve would increase the consumption of both x1 and x2. 4.3. Revealed Preference Theory Utility is hard to measure. But we can observe what people actually do, and assume that, as rational individuals, they are doing what gives them the most utility. Economists call this idea revealed preference; the idea that people’s preferences can be determined by observing their choices and behaviour. According to the theory of revealed preference, a consumer’s preferences can be inferred from a sufficient number of observed choices or purchases in the marketplace. We can tell what maximises other people’s utility by observing their behaviour. 21 | e lawluvi – intermediate microeconomics The fact that someone chose to do something “reveals” that he preferred it to the other available options. This inference is specific to a particular person and situation. If a consumer chooses a basket of goods over another, given budget constraint, then the consumer has revealed his/her preference for that commodity. The chosen basket of goods maximises the utility of the consumer. For example, if a consumer is observed to purchase basket A rather than basket B, and A is not cheaper than B (or A has same cost as B), then for this consumer, basket A must be superior to basket B. This means the consumer reveals his/her preference for basket A. However, if the two baskets have different costs and the consumer chooses basket A over B because basket A is cheaper, then his preference is not revealed because he can choose basket B another time when it becomes cheaper. Revealed preference axiom. Revealed preference rests on the following assumptions: The individual’s tastes do not change over the period considered. Rationality. The consumer is assumed to be rational. The consumer will always prefer more quantities of a good than less. Consistency. The consumer is consistent in making decision. If the consumer chooses basket X over Y then he will not choose bundle Y in another situation when both are available. If X  Y then Y  X. Simply put, if the consumer is observed to prefer basket X to basket Y , then this consumer will never prefer Y to X. Transitivity. The consumer’s preferences are transitive; that is, if X  Y and Y  Z then X  Z. The consumer can be induced to purchase any basket of goods if its price is made sufficiently attractive. 4.4. Cardinal Utility We can tell if a person prefers one bundle of goods to another by simply offering him a choice between the two bundles and see which one is chosen. Thus, we know how to assign an ordinal utility to the two bundles of goods: we just assign a higher utility to the chosen bundle than to the rejected bundle. Any assignment that does this will be a utility function. We have an operational criterion for determining if one bundle has a higher utility than another bundle for some individual. Unfortunately, we have no way of telling whether a person gets twice as much satisfaction from one bundle as from another. Nor do we know whether one person gets twice as much satisfaction as another from consuming the same bundle. There are some theories of utility that attach significance to the magnitude of utility. Utility function that describes by how much one bundle is preferred to another is known as a cardinal utility. In a theory of cardinal utility, the size of the utility difference between two bundles of goods is supposed to have some sort of significance. Cardinal utility function attaches to bundles numerical values that cannot arbitrarily be doubled or tripled without altering the differences between the values of various bundles. Knowing how much larger of one bundle to another bundle is unimportant to our description of choice. Our goal is to understand consumer behaviour, so what is relevant is to know how consumers rank different bundles. Therefore, we will work only with ordinal utility functions. This approach is sufficient for understanding both how individual consumer decisions are made and what this knowledge implies about the characteristics of consumer demand. Simply put, the numbers which the utility function assigns to the alternative bundles need not have cardinal significance; they need only serve as an index of the consumer’s satisfaction. 22 | e lawluvi – intermediate microeconomics 4.5. Monotonic Transformation It is only the ranking of consumption bundles that matters, so there is no unique way to assign utilities to bundles of goods. If we can find one way to assign utility numbers to bundles of goods, we can find an infinite number of ways to do it. One such way is monotonic transformation; transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers. A monotonic transformation of a utility function is a utility function that represents the same preferences as the original utility function. If a particular set of numbers associated with various combinations of x1 and x2 is a utility index, any monotonic transformation of it is also a utility index. We typically represent a monotonic transformation by a function f ( u ) that transforms each number u into some other number f ( u ) , in a way that preserves the order of the numbers in the sense that u1  u2 implies f ( u1 )  f ( u2 ). Examples of monotonic transformations are multiplication by a positive number (such as, f ( u ) = 2u ), adding any number (such as, f ( u ) = u + 13 ), raising u to an odd power (such as, f ( u ) = u 3 ), and so on. 4.6. Marginal Rate of Substitution The marginal rate of substitution of good 1 for good 2 is the rate at which a consumer is willing to give up units of good 2 to obtain an additional unit of good 1, in order to keep his utility constant. The MRS of good 1 for good 2 is given by the ratio of the marginal utilities: MU x1 MRS x1x2 = MU x2 The marginal utility of a good is the rate at which utility changes as consumption of a good increases. Intuitively, marginal utility indicates how much better off a consumer becomes by consuming 1 more unit of good 1 (or good 2). Mathematically, we measure the marginal utility of good 1, MU x1 , by partially differentiating the utility function u ( x1 , x2 ) with respect to good 1 ( x1 ): u ( x1 , x2 ) MU x1 = x1 u ( x1 , x2 ) and similarly for the marginal utility of good 2 ( x2 ), MU x2 =. x2 4.7. Utility Functions for Specific Preferences In Section 2 we described some examples of preferences and the indifference curves that represented them. We can also represent these preferences by utility functions. Given a utility function, u ( x1 , x2 ) , it is relatively easy to draw the indifference curves: we just plot all the points ( x1 , x2 ) such that u ( x1 , x2 ) equals a constant. Each different value of the constant 23 | e lawluvi – intermediate microeconomics results in a different indifference curve. Suppose the utility function, u ( x1 , x2 ) = x1 x2 ; the indifference curves associated with this utility function is depicted in Figure 4.1. Suppose another utility function is given as v ( x1 , x2 ) = x12 x22. This utility function v ( x1 , x2 ) is just the square of the utility function u ( x1 , x2 ). The utility function u ( x1 , x2 ) cannot be negative, so it follows that v ( x1 , x2 ) is a monotonic transformation of the utility function, u ( x1 , x2 ). This means that the utility function v ( x1 , x2 ) = x12 x22 has to have exactly the same shaped indifference curves as those depicted in Figure 1. Thus, the utility function is invariant to positive monotonic transforms. Cobb-Douglas Utility Function A utility function that is used frequently for illustrative and empirical purposes is the Cobb- Douglas utility function, given as u ( x1 , x2 ) = x11 x2 2 with i  0, i = 1, 2 and 1 +  2 = 1. The exponents,  1 and  2 , describe the preferences of the consumer. The Cobb-Douglas utility function can be expressed in a variety of ways. For example, it can be stated as u ( x1 , x2 ) = x11 x12−1. Linear Utility Function The utility function that describes perfect substitute between goods is the linear utility function of the form u ( x1 , x2 ) = 1 x1 +  2 x2 where  1 and  2 are some positive numbers that measure the ''value'' of goods 1 and 2 to the 1 consumer. The slope of a typical indifference curve is given by −. 2 Leontief Utility Function The utility function that describes perfect complement between goods is the Leontief utility function, given as 24 | e lawluvi – intermediate microeconomics  u ( x1 , x2 ) = min 1 x1 ,  2 x2  where  1 and  2 are positive numbers that indicate the proportions in which the goods are consumed. This utility function means increasing the amount of one of the goods alone does not yield utility gains, as the consumer needs to enjoy both goods in fixed proportions. In other words, if the consumer increases the amount of good 1 ( x1 ) by one unit without increasing the amount of good 2 ( x2 ), her utility does not necessarily increase. To illustrate this point, suppose a consumer has a utility function u ( x1 , x2 ) = min 2 x1 , 2 x2  and 5 units of each good, yielding a utility level of 10: u ( 5,5 ) = min 2  5, 2  5 = min 10,10 = 10. If x1 is increased from 5 to 6 units, but x2 is unaffected, the utility of the consumer remains at the same level because u ( 6,5 ) = min 2  6, 2  5 = min 12,10 = 10. Section 4 Problem Set 1. Explain the concept of utility. 2. Examine the two basic approaches to the problem of the comparison of utilities. 3. Describe the revealed preference axiom. 4. Give the assumptions of the revealed preference approach. 5. Consider the utility function u ( x1 , x2 ) = 2 x1 + 3 x2 , and the bundles A = ( 40, 20 ) and B = ( 30,30 ). a. What kind of preferences does the utility represent? b. Does the consumer prefer bundle A, or B, or is she indifferent between them? c. Find her preferences if the utility function is represented by v ( x1 , x2 ) = 4 x1 + 6 x2. d. Is the function v ( x1 , x2 ) a monotonic transformation of u ( x1 , x2 ) ? 6. Consider the utility function u ( x1 , x2 ) = min 2 x1 ,3x2  , and the bundles A = ( 40, 20 ) and B = ( 30,30 ). a. What kind of preferences does the utility represent? b. Does the consumer prefer bundle A, or B, or is he indifferent between them? c. Find his preferences if the utility function is represented by v ( x1 , x2 ) = min 4 x1 , 6 x2 . d. Is the function v ( x1 , x2 ) a monotonic transformation of u ( x1 , x2 ) ? 7. Consider the utility function u ( x1 , x2 ) = x1 x2. 25 | e lawluvi – intermediate microeconomics  a. What kind of preferences does it represent? b. Is the function v ( x1 , x2 ) = x12 x2 a monotonic transformation of u ( x1 , x2 ) ? c. Is the function w ( x1 , x2 ) = x12 x22 a monotonic transformation of u ( x1 , x2 ) ? 8. For each utility function defined over goods x1 and x2 , calculate the marginal rate of substitution for positive levels of both goods. a. u ( x1 , x2 ) = ( x1 + 2 x2 ) 2 b. u ( x1 , x2 ) = 2 x12 + x23 c. u ( x1 , x2 ) = 2 ln x1 + 3ln x2 d. u ( x1 , x2 ) = x1 + 2 x2 9. For each utility function, determine whether the marginal rate of substitution is increasing, decreasing or constant in x1. a. u ( x1 , x2 ) = 3 x1 + x2 b. u ( x1 , x2 ) = 2 x12 + x23 c. u ( x1 , x2 ) = x11/ 2 x1/2 2 d. u ( x1 , x2 ) = x11/ 3 x22 / 3 10. The utility function of a consumer is given as u ( x1 , x2 ) = x11/ 2 x1/2 2. Graph the indifference curve for a utility level, u = 6. 11. The utility function of a consumer is given as u ( x1 , x2 ) = x11/ 3 x22 / 3. Graph the indifference curves for utility levels, u = 9 and u = 16. 12. The utility function of a consumer is given as u ( x1 , x2 ) = 5 x11/ 2 x1/2 4. Graph the indifference curves for utility levels, u = 10 and u = 20. 13. The utility function of a consumer is u ( x1 , x2 ) = 2 x1 + 3 x2. Graph his indifference curve if he seeks to reach a utility level u = 21. 14. Naa’s utility function is u ( x1 , x2 ) = 3 x1 + 2 x2. Graph her indifference curves for utility levels u = 10 and u = 20. 15. Ro’s utility function is given by u ( x1 , x2 ) = min 3x1 , 6 x2 . Graph his indifference curves for utility levels u = 12 and u = 24. 16. Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (that is, whether the marginal rate of substitution declines as x1 increases). a. u ( x1 , x2 ) = 3 x1 + x2 b. u ( x1 , x2 ) = x1 x2 26 | e lawluvi – intermediate microeconomics  c. u ( x1 , x2 ) = x1 + x2 d. u ( x1 , x2 ) = x12 − x22 17. The preference for a consumer for good 1 ( x1 ) and good 2 ( x2 ) given by u ( x1 , x2 ) = x1 x2. a. If x1 = 5 and x2 = 80 , what will utility be? b. If x1 = 10 , what value for x2 will provide the same utility as in part a? c. If x1 = 20 , what value for x2 will provide the same utility as in parts a and b? d. Graph the indifference curve implied by parts a through c. e. This consumer has ¢8 to spend on good 1 and good 2, and good 1 cost ¢0.40 each and good 2 cost ¢0.10. Graph this budget constraint on the same graph as in part d. Which of the points identified in parts a through c can be bought by this consumer? 18. Ed derives utility only from good 1 and good 2. His utility function is u ( x1 , x2 ) = x1 x2. a. Sketch Ed’s indifference curve for u = 5 , u = 10 , and u = 20. b. Suppose that Ed has ¢200 to spend and that good 1 cost ¢5 and good 2 cost ¢20. Draw Ed’s budget constraint on the same graph as his indifference curves. 27 | e lawluvi – intermediate microeconomics  Unit 1: Theory of Consumer Behaviour Section 5: Choice 5.0. Introduction Given consumers’ preferences and budget constraints, how individual consumers choose how much of each good to buy can be determined. Consumers are assumed to make this choice in a rational way; consumers choose goods to maximise the satisfaction they can achieve, given the limited budget available to them. Understanding consumer choice helps explain demand (how the quantity of a good that consumers choose to purchase depends on its price). 5.1. Optimal Choice Figure 5.1 graphically illustrates the solution to the consumer’s choice problem of two goods, good 1 and good 2. The maximising consumption bundle must satisfy two conditions. The bundle must be located on the budget line. From Figure 5.1, we see that any bundle to the left of and below the budget line leaves some income unallocated, and income if spent, could increase consumer’s satisfaction. We also see that any bundle to the right of and above the budget line cannot be purchased with available income. The only rational and feasible choice is a bundle on the budget line. Second, the bundle must give the consumer the most preferred combination of goods. Figure 5.1. Maximising consumer satisfaction Consider bundle B in Figure 5.1. This combination of good 1 and good 2 lies on the budget line, so the cost of bundle B completely exhausts the consumer’s income. But, bundle B on indifference curve U1 is not the most preferred choice, because given income and prices corresponding to the budget line, the consumer can do better – thus, the consumer can achieve a higher satisfaction. By moving to point A, the consumer spends the same amount of money and achieves the increased level of satisfaction associated with indifference curve U 2. In addition, bundles located to the right and above indifference curve U 2 , like the bundle associated with D on indifference curve U 3 , achieve a higher level of satisfaction but cannot be purchased with the available income. Therefore, point A maximises the consumer’s satisfaction; bundle A represents the consumer's equilibrium choice ( x1* , x2* ). The term 28 | e lawluvi – intermediate microeconomics equilibrium refers to the fact that the consumer has no incentive to change to a different affordable bundle once this point is reached. But, in general when prices and income change, the consumer’s optimal choice will change. We see from this analysis that the bundle which maximises satisfaction must lie on the highest indifference curve that touches the budget line. Point A is the point of tangency between indifference curve U 2 and the budget line; at this point, the budget line and indifference curve U 2 are tangent, and no higher level of satisfaction can be attained. At point A (the point of maximisation), the slope of the budget line is exactly equal to the slope of the indifference curve. The MRS (the slope of the indifference curve) between the two goods equals the price ratio (the slope of the budget line). This is stated as: p1 MRS = p2 This result means that satisfaction is maximised when the marginal rate of substitution (of good 1 for good 2) is equal to the ratio of the prices (of good 1 to good 2). The consumer can obtain maximum satisfaction by adjusting his consumption of goods 1 and 2 so that the MRS equals the price ratio. This equation also illustrates that satisfaction is maximised when the marginal benefit (the benefit associated with the consumption of one additional unit of good 1) is equal to the marginal cost (the cost of the additional unit of good 1). The marginal benefit is measured by the MRS, and the marginal cost is measured by the magnitude of the slope of the budget line. If this condition did not hold, the personal rate at which the consumer is willing to substitute between goods 1 and 2 would differ from the market rate at which he is able to substitute between the goods. For example, at point B in Figure 5.1, the slope of the indifference curve is steeper than the slope of the budget line. This means the consumer is willing to give up more of good 2 to get an additional unit of good 1 than he actually has to give up based on market prices. Consequently, it is in the consumer's interest to consume less of good 2 and more of good 1. This substitution continues until ultimately the consumer is at a point such as A in Figure 5.1, where the MRS is equal to the ratio of prices. 5.2. Price Changes and Consumer Behaviour Changes in prices of goods affect the market rate at which a consumer can substitute among various goods. Therefore, changes in prices will change the behaviour of consumers. A change in the price of a good leads to a change in the equilibrium consumption bundle. The new equilibrium point after a price change depends on consumer preferences. It is useful to recall the definitions of substitutes and complements. Goods 1 and 2 are called substitutes if an increase (decrease) in the price of Good 1 leads to an increase (decrease) in the consumption of Good 2. If Goods 1 and 2 are substitutes, a reduction in the price of Good 1 would lead the consumer to move from point A in Figure 5.2 to a point such as B, where less of Good 2 is consumed than at point A. Figure 5.2 illustrates that when the price of Good 1 falls, the consumption of a substitute good (Good 2) falls. 29 | e lawluvi – intermediate microeconomics  Figure 5.2. Change in consumer equilibrium due to fall in price of good 1 Goods 1 and 2 are called complements if an increase (decrease) in the price of Good 1 leads to a decrease (increase) in the consumption of Good 2. If goods 1 and 2 are complements, a reduction in the price of good 1 would lead the consumer to move from point A in Figure 5.3 to a point such as B, where more of good 2 is consumed than before. Ultimately, price changes alter consumer incentives to buy different goods, thereby changing the mix of goods they purchase in equilibrium. Figure 5.3 illustrates that when the price of Good 1 falls, the consumption of a complementary good (Good 2) rises. Figure 5.3. Change in consumer equilibrium due to fall in price of good 1 5.3. Income Changes and Consumer Behaviour A change in income either expands or contracts the consumer's budget constraint, and the consumer therefore finds it optimal to choose a new equilibrium bundle. The new equilibrium point will depend on consumer preferences. Let us recall the definitions of normal and inferior goods. Good 1 is a normal good if an increase (decrease) in income leads to an increase (decrease) in the consumption of good 1. In Figure 5.4, the consumption of both 30 | e lawluvi – intermediate microeconomics goods 1 and 2 increased due to the increase in consumer income. Thus, the consumer views good 1 and good 2 as normal goods. Figure 5.4. Change in consumer equilibrium due to increase in income Good 1 is an inferior good if an increase (decrease) in income leads to a decrease (increase) in the consumption of good 1. Figure 5.5 depicts the effect of an increase in income for the case when good 1 is an inferior good. When income increases, the consumer moves from point A to point B to maximise his satisfaction given the higher income. Given that at point B the consumer consumes more of good 2 than at point A, then good 2 is a normal good. At point B less of good 1 is consumed than at point A, so the consumer views good 1 as an inferior good. Figure 5.5. Change in consumer equilibrium due to increase in income 5.4. Marginal Utility and Consumer Choice We showed graphically how a consumer can maximise his satisfaction, given a budget constraint. We do this by finding the highest indifference curve that can be reached, given that budget constraint. Because the highest indifference curve also has the highest attainable level 31 | e lawluvi – intermediate microeconomics of utility, it is natural to restate the consumer's problem as one of maximising utility subject to a budget constraint. Marginal utility measures the additional satisfaction obtained from consuming one more unit of a good. The marginal utility associated with a consumption increase from 0 to 1 unit of good 1 might be 7; from 1 to 2, it might be 5; from 2 to 3, it might be 3. These numbers imply that the consumer has diminishing marginal utility: as more and more units of a good are consumed, consuming additional amounts will yield smaller additions to utility. The concept of marginal utility can be related to consumer’s utility-maximisation problem. Consider a small movement down an indifference curve (in Figure 2.1). The additional consumption of good 1, x1 , will generate marginal utility MU1. This shift results in a total increase in utility of MU1  x1. At the same time, the reduced consumption of good 2, x2 , will lower utility per unit by MU 2 , resulting in a total loss of MU 2  x2. All points on an indifference curve generate the same level of utility, so the total gain in utility associated with the increase in good 1 must balance the loss due to the lower consumption of good 2. This gives 0 = MU1  x1 + MU 2  x2 This equation can be rearranged as x2 MU1 − = x1 MU 2 x2 Given that − is the MRS of good 1 for good 2, it follows that x1 MU1 MRS = MU 2 This equation indicates the MRS is the ratio of the marginal utility of good 1 to the marginal utility of good 2. As the consumer gives up more and more of good 2 to obtain more of good 1, the marginal utility of good 1 falls and that of good 2 increases, so MRS decreases. Also, we have seen that when consumers maximise their satisfaction, the MRS of good 1 for good 2 is equal to the ratio of the prices of the two goods: p1 MRS = p2 Because the MRS is also equal to the ratio of the marginal utilities of consuming good 1 and good 2, it follows that MU1 p1 = MU 2 p2 or MU1 MU 2 = p1 p2 32 | e lawluvi – intermediate microeconomics This equation means that utility maximisation is achieved when the budget is allocated so that the marginal utility per cedi of expenditure is the same for each good. Marginal utility divided by price must be the same for all goods. This ratio gives the rate at which satisfaction would increase if an additional cedi were spent on a particular good. To see why this principle must hold, suppose that a person gets more utility from spending an additional cedi on good 1 than on good 2. In this case, her utility will be increased by spending more on good 1. As long as the marginal utility of spending an extra cedi on good 1 exceeds the marginal utility of spending an extra cedi on good 2, she can increase her utility by shifting her budget toward good 1 and away from good 2. Eventually, the marginal utility of good 1 will decrease (because there is diminishing marginal utility in its consumption) and the marginal utility of good 2 will increase (for the same reason). Only when the consumer has satisfied the equal marginal principle – that is, has equalised the marginal utility per cedi of expenditure across all goods – will she have maximised utility. Section 5 Problem Set 1. Ed allocates his entire budget between two goods, x1 and x2. Illustrate the different possible cases of the change in Ed’s new equilibrium following a rise in price of x1. 2. Ed allocates his entire budget between two goods, x1 and x2. Illustrate the different possible cases of the change in Ed’s new equilibrium following a fall in price of x1. 3. Suppose that a consumer allocates her entire budget between two goods, x1 and x2. Illustrate the different possible cases of a change in the consumer’s new equilibrium following an increase in income. 4. Suppose that a consumer allocates her entire budget between two goods, x1 and x2. Illustrate the different possible cases of a change in the consumer’s new equilibrium following a decrease in income. 5. The utility function for a consumer for purchasing good 1 ( x1 ) and good 2 ( x2 ) is given by u ( x1 , x2 ) = x1 x2. Her income is ¢80, and faces market price of good 1 at ¢2 per unit and that of good 2 at ¢4 per unit. a. Find the marginal utility of each good. b. Using the marginal utilities, find the MRS. c. Find the consumption bundle that maximises the consumer’s utility. 6. Ed’s utility function for buying good 1 ( x1 ) and good 2 ( x2 ) is given by u ( x1 , x2 ) = x1 x2. His income is ¢120, and faces market prices p1 = ¢4 and p2 = ¢10. Find Ed’s optimal consumption bundle. 7. Ed’s utility function is u ( x1 , x2 ) = x11/ 2 x1/2 2. His income is ¢60, and faces market prices p1 = ¢1 and p2 = ¢3. Find Ed’s optimal consumption bundle. 33 | e lawluvi – intermediate microeconomics  8. The utility function of Ki for consuming good 1 ( x1 ) and good 2 ( x2 ) is given by u ( x1 , x2 ) = x11/ 2 x1/2 4. Her income is ¢17, and faces market price of good 1 at ¢3 per unit and that of good 2 is at ¢2 per unit. Find Ki’s optimal consumption bundle. 9. Lo has an income of ¢100 and a utility function of the type u ( x1 , x2 ) = x10.6 x20.4. The price for good x1 is ¢3, and the price for good x2 is ¢1. Find his optimal consumption bundle. 10. Ed’s utility function for buying good 1 ( x1 ) and good 2 ( x2 ) is given by u ( x1 , x2 ) = x11/ 3 x22 / 3. His income is ¢120, and faces market prices p1 = ¢10 and p2 = ¢20. Find Ed’s optimal consumption bundle. 11. For each of the following utility functions, calculate the demand functions for each good, h1 = ( p1 , p2 , m ) and h2 = ( p1 , p2 , m ) , as functions of the prices and income. a. u ( x1 , x2 ) = x1 x2 b. u ( x1 , x2 ) = x1 x2 c. u ( x1 , x2 ) = x11/ 2 x1/2 4 d. u ( x1 , x2 ) = x10.4 x20.6 e. u ( x1 , x2 ) = 2 ln x1 + 3ln x2 12. El’s utility function is u ( x1 , x2 ) = x11/ 3 x22 / 3. She faces market prices p1 = ¢10 and p2 = ¢20 , and has a utility target of u = 6. Find El’s optimal consumption bundle. 13. Ed’s utility function is u ( x1 , x2 ) = x1 x2. He faces market prices p1 = ¢1 and p2 = ¢4 , and has a utility target of 20 utils. Find Ed’s optimal consumption bundle. 14. Ki consumes both sorbet (S) and gelato (G) each week. Her marginal rate of substitution of gelato for sorbet is 0.75, that is, MRSGS = 0.75. The price of gelato is ¢1 per scoop and the price of sorbet is ¢1.25 per scoop. a. Determine whether Ki is currently optimising her budget over these two desserts. Justify your answer. b. Explain whether Ki should buy more sorbet or more gelato, given that she is not currently optimising her budget. 34 | e lawluvi – intermediate microeconomics  Unit 1: Theory of Consumer Behaviour Section 6: Individual Demand 6.0. Introduction This section illustrates how the demand curve of an individual consumer follows from the consumption choices that the consumer makes. The consumption choices generally depend on three parameters: the prices of the two goods and income. Changing any one of these at a time enables us to trace the path of utility-maximising bundles in the commodity space. Changing one of the prices yields a price consumption curve and changing income yields an income consumption curve. The analysis will rely on the utility-maximisation approach; this approach explains how a consumer chooses the bundle on the highest indifference curve that touches the budget line. 6.1. Price Changes Figure 6.1(a) shows the consumption choices that a person will make when allocating a fixed amount of income between two goods. Suppose, initially, the price of good 1 is ¢1, the price of good 2 is ¢2, and the consumer’s income is ¢20. The utility-maximising consumption choice is at point B in Figure 6.1(a). The consumer buys 12 units of good 1 and 4 units of good 2, achieving the utility level associated with indifference curve U 2. Figure 6.1(b) shows the relationship between the price of good 1 and the quantity demanded of it. The horizontal axis measures the quantity of good 1 consumed, as in Figure 6.1(a), but 35 | e lawluvi – intermediate microeconomics the vertical axis now measures the price of good 1. Point G in Figure 6.1(b) corresponds to point B in Figure 1 (a). At G, the price of good 1 is ¢1, and the consumer buys 12 units of good 1. Suppose the price of Good 1 increases to ¢2. This causes the budget line in Figure 6.1(a) to rotate inward about the vertical intercept. The consumer now achieves maximum utility at A, which is found on a lower indifference curve, U1. Because the price of good 1 has risen, the consumer's purchasing power – and thus attainable utility – has fallen. At A, the consumer chooses 4 units of good 1 and 6 units of good 2. In Figure 6.1(b), this modified consumption choice is at E, which shows that at a price of ¢2, 4 units of good 1 are demanded. Suppose the price of Good 1 decreases to ¢0.50. This causes the budget line to rotate outward, and the consumer can achieve the higher level of utility associated with indifference curve U 3 in Figure 6.1(a) by selecting D, with 20 units of good 1 and 5 units of good 2. Point H in Figure 6.1(b) shows the price of ¢0.50 and the quantity demanded of 20 units of good 1. In Figure 6.1(a), the price-consumption curve traces the utility-maximising combinations of good 1 and good 2 associated with every possible price of good 1. Price-consumption curve is curve tracing the optimal consumption bundles of two goods as the price of one good changes. We see that as the price of Good 1 falls, attainable utility increases and the consumer buys more units of good 1. This pattern of increasing consumption of a good in response to a decrease in price almost always holds. Let see what happens to the consumption of good 2 as the price of Good 1 falls. As Figure 6.1(a) shows, the consumption of good 2 may either increase or decrease. The consumption of both good 1 and good 2 can increase because the decrease in the price of good 1 has increased the consumer’s ability to purchase both goods. 6.2. Income Changes Figure 6.2(a) shows the consumption choices that a consumer will make when allocating a fixed income to good 1 and good 2 when the price of good 1 is ¢1 and the price of good 2 is ¢2. Initially, the consumer's income is ¢10. The utility-maximising consumption choice is then at A, at which point she buys 4 units of good 1 and 3 units of good 2. This choice of 4 units of good 1 is also shown in Figure 6.2(b) as E on demand curve D1. Demand curve D1 is the curve that would be traced out if we held income fixed at ¢10 but varied the price of good 1. Because we are holding the price of good 1 constant, we will observe only a single point E on this demand curve. If the consumer’s income is increased to ¢20, Her budget line then shifts outward parallel to the original budget line, allowing her to attain the utility level associated with indifference curve U2. Her optimal consumption choice is now at B, where she buys 10 units of good 1 and 5 units of good 2. In Figure 6.2(b) her consumption of good 1 is shown as G on demand curve D2. D2 is the demand curve that would be traced out if we held income fixed at ¢20 but varied the price of good 1. If her income increases to ¢30, she chooses D, with a bundle containing 16 units of good 1 (and 7 units of good 2), represented by H in Figure 6.2(b). In Figure 6.2(a), the income- consumption curve traces out the utility-maximising combinations of good 1 and good 2 associated with every income level. Income-consumption curve is curve tracing the optimal consumption bundles of two goods as a consumer’s income changes. The income- consumption curve in Figure 6.2(a) slopes upward because the consumption of both good 1 36 | e lawluvi – intermediate microeconomics and good 2 increases as income increases. Each demand curve is measured for a particular level of income, so any change in income must lead to a shift in the demand curve itself. Thus, A on the income-consumption curve in Figure 6.2(a) corresponds to E on demand curve D1 in Figure 6.2(b); B corresponds to G on a different demand curve D2. The upward-sloping income-consumption curve implies that an increase in income causes a shift to the right in the demand curve – in this case from D1 to D2 to D3. 6.2. Income Offer Curves and Engel Curves An increase in income shifts the budget line outward in a parallel fashion. 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 the bundles of goods that are demanded at the different levels of income, as depicted in Figure 6.3a. The income offer curve is also known as the income expansion path. If both goods are normal goods, then the income expansion path will have a positive slope, as depicted in Figure 6.3a. For each level of income, M , there will be some optimal choice for each of the goods. Let us focus on good 1 and consider the optimal choice at each set of prices and income. This is simply the demand function for good 1. If we hold the prices of goods 1 and 2 fixed and examine how demand changes as income changes, we generate a curve known as the Engel curve (see Figure 6.3b). The Engel curve is a graph of the demand for one of the goods as a function of income, with all prices being held constant. Engel curve is a curve relating the quantity of a good consumed to an individual’s income. 37 | e lawluvi – intermediate microeconomics Section 6 Problem Set 1. With the aid of diagram, explain price consumption curve. 2. With the aid of diagram, explain income consumption curve. 3. With the aid of diagram, explain the Engel curves. 4. Consider a utility function u ( x1 , x2 ) = x1 x2 , income of ¢80, and market price of good 1 at ¢2 per unit and that of good 2 at ¢4 per unit. Suppose the price of good 1 increase to ¢4 per unit. Find the effect of this price change on the optimal bundle. 38 | e lawluvi – intermediate microeconomics  5. Ed’s utility function is given by u ( x1 , x2 ) = x11/ 3 x22 / 3. His income is ¢180, and faces market prices p1 = ¢3 and p2 = ¢1. If the price of good 1 decreases from p1 = ¢3 to p1 = ¢1, find the effect of this price change on Ed’s optimal consumption bundle. 6. Ed’s utility function is given by u ( x1 , x2 ) = x11/ 3 x22 / 3. His income is ¢120, and faces market prices p1 = ¢10 and p2 = ¢20. If Ed’s income increases to ¢132, find the effect of this income change on Ed’s optimal consumption bundle. 7. Consider a consumer with a utility function u ( x1 , x2 ) = x1 x2 , facing an income m = 110 and prices p1 and p2 for goods 1 and 2, respectively. a. Find the demand function for each good. b. Assume that the price of both goods increases by 10 percent. Find the new demand functions for each good. 8. Consider a consumer with a utility function u ( x1 , x2 ) = x12 x2 , facing income of ¢800 and prices ¢2 and ¢4 for goods 1 and 2, respectively. a. Find the consumer’s optimal consumption bundle. b. Assume that the price of good 2 decreases to ¢3. Find the consumer’s new optimal bundle. Identify the effect of the price change on the optimal bundle. 39 | e lawluvi – intermediate microeconomics  Unit 2: Theory of the Firm Section 7: Basic Concepts 7.0. Introduction The second important actor on the microeconomic stage is the individual firm. This unit begins with the examination of the aspects of production and cost that are common to all firms. Then we consider the behaviour of perfectly competitive firms. At its simplest, a firm is an entity created by individuals for some purpose. This entity will typically acquire inputs and combine them to produce output. Inputs are purchased on factor markets and these expenditures are the firm’s costs. Output is sold on product markets and the firm earns revenue from these sales. The theory of the firm describes how a firm makes cost-minimising production decisions and how the firm’s resulting cost varies with its output. Understanding of production and cost will help us understand the characteristics of market supply. It will also prove useful for dealing with problems that arise regularly in business. To see this, just consider some of the problems often faced by an automaker. How much assembly-line machinery and how much labour should it use in its new automobile plants? If it wants to increase production, should it hire more workers, construct new plants, or both? Does it make more sense for one automobile plant to produce different models, or should each model be manufactured in a separate plant? What should the automaker expect its costs to be during the coming year? How are these costs likely to change over time and be affected by the level of production? These questions apply not only to business firms but also to other producers of goods and services, such as governments and non-profit agencies. 7.1. Production Decisions of a Firm The production decisions of firms can be understood in three steps: Production technology. This describes how inputs (such as labour, capital, and raw materials) can be transformed into outputs. The firm can produce a particular level of output by using different combinations of inputs. Cost constraints. Firms take into account the prices of labour, capital, and other inputs. That is, the firm will be concerned about its cost of production. Input choices. Given its production technology and the prices of inputs, the firm must choose how much of each input to use in producing its output. The firm must take into account the prices of different inputs when deciding how much of each input to use. For example, if an electronics firm operates in a country with low wage rates, it may decide to produce televisions by using a large amount of labour, thereby using very little capital 40 | e lawluvi – intermediate microeconomics  Unit 2: Theory of the Firm Section 8: Production 8.0. Introduction Production is the transformation of inputs into outputs of goods and services. The firm is a technical unit in which inputs are transformed into output. The inputs are often classified into categories as land, raw materials, labour, capital, and entrepreneurship. Land may be described as a gift of nature (such as arable land for farming, or land for buildings). Raw materials include such natural resources as coal, oil, and trees. Capital consists of produced goods used as inputs for further production. Capital goods include tools, machinery, equipment, office buildings, and storage facilities. Labour is the physical and mental talents of individuals engaged in the production process. Entrepreneurship refers to the process whereby other inputs are combined and organised to produce final goods and services. 8.1. Production Function Production function is the technological relationship that describes the process whereby inputs are efficiently transformed into outputs. The production function specifies the maximum output that can be produced with a given quantity of inputs. In general, the production function can be expressed as: q = f ( k , l , m, ) (8.1) In Equation 8.1, q represents the firm’s output of a particular good during a period, k represents capital input, l represents labour input, m represents raw materials used, and the notation ( ) indicates the possibility of other variables affecting the production process. All variables (inputs and outputs) are flows, that is, they are measured per unit of time. Firms use a wide variety of inputs in the production process, but we keep the analysis simple by focusing on only two inputs, labour l and capital k. Equation 8.1 can be restated as q = f (k,l ) (8.2) The firm's production function might be q = l 2 + 2kl , where q equals quantity of output, l is quantity of labour, and k is quantity of capital. The firm can vary its output q by varying the amounts of either capital or labour inputs. It is very easy to vary the quantity of labour used in the production process at any given time but a fairly long period of time is required to vary the quantity of capital used. For example, changing the quantity or usage of capital, to install a new machine, requires time. The production function applies to a given technology – that is, to a given state of knowledge about the various methods that might be used to transform inputs into outputs. As the technology becomes more advanced and the production function changes, a firm can obtain more output for a given set of inputs. For example, a new, faster assembly line may allow a hardware manufacturer to produce more high-speed computers in a given period of time. Production functions describe what is technically feasible when the firm operates efficiently; when the firm uses each combination of inputs as effectively as possible. In other words, 41 | e lawluvi – intermediate microeconomics production function indicates maximum quantities that can be produced with each combination of inputs because it is assumed all inputs are utilised efficiently. 8.2. The Time Periods of Production There are two general categories of time periods involved in the production process: short run and long run. The short run is a period in which the quantity of at least one input is fixed and the quantities of the other inputs can be varied. The long run is a time in which all inputs are considered to be variable in amount. The distinction between the short-run and the long-run is based on the difference between fixed and variable inputs. Fixed input is an input whose quantity cannot be changed as output changes. Variable input is an input whose quantity the firm can change as output changes. Short-Run If any of the inputs of a firm is fixed, then the firm is producing in the short-run. Short-run is a time period in which at least one input in the production process remains fixed. Land and capital are usually regarded as fixed inputs as well as the usage of highly skilled labour. Suppose the production process with only two inputs – capital and labour, capital can be treated as the fixed input and labour as the variable input. This implies output becomes a function of the variable input (labour) given a fixed quantity of capital. That is, the short-run production function is essentially only a function of labour. If k * is the fixed level of capital, the short-run production function may be stated as q = f (l ) = f ( k * , l ) Given the available technology and the fixed level of capital, if the firm wishes to produce more units of output in the short run, more variable input (labour) is needed to produce more output, because increasing fixed input (capital) is not possible. The short run does not correspond to a specific number of periods. It varies from industry to industry and from time to time even within the same industry. In some industries, it may extend over many years; in others, it may be a matter of months or even weeks. For example, in the electric power industry, it takes years to acquire and install a steam turbine generator. In contrast, a machine shop can acquire new equipment in a few weeks. The length of the short- run is influenced by two conditions: technology, such as how quickly equipment can be manufactured or installed economic, such as the price the firm is willing to pay for the equipment Long-Run If all the inputs of a firm can be varied, then it is said to be producing in the long-run. In the long run, all inputs can be varied. Long-run is a period in which all inputs in the production process can be varied, at a given level of technology. It is the length of time over which all of the firm’s inputs can be varied, but its technology is fixed. It is an amount of time sufficiently large long that all input usage levels can be varied. This means there are no fixed inputs in the long-run. The long-run allows factor or input substitution. More capital and less labour or more labour and less capital can be used to produce a fixed amount of output. 42 | e lawluvi – intermediate microeconomics Output changes in response to input changes in either the short-run or the long-run. The short- run output changes reflect changes in the proportions in which inputs are combined, and the long-run output changes reflect changes in the entire scale of operation. In the short-run, the output changes are called returns to variable input(s), and in the long-run the output changes are called returns to scale. 8.3. Productivity Measures One important component of decision making in the firm is the determination of the productivity of inputs used in the production process. There are three important measures of productivity in the production process: total product, average product, and marginal product. Total Product. Total product is the total amount of output produced during a given time by all the inputs used. Total product ( q ) is the overall output that results from employing a specific quantity of inputs in a given production process. In the short run, the total product will change as more or less of the variable input is used in combination with the given amount of the fixed input. Its curve, the total product curve, is generally upward sloping, reflecting the fact that more quantities of output are produced as more variable inputs are used. Average Product. Firms are interested in the average productivity of an input; that is, the firm may want to know, on average, how much each input contributes to the total output of the firm. This is the economic concept of average product. Average product is a measure of the total output per unit of input used. The average product of an input is the total product divided by the quantity of input. In particular, the average product of labour is given as: q f (k, l ) APl = = l l Marginal Product. Firms are much interested in how much extra output can be produced by adding one more unit of an input to the production process. The marginal physical productivity or