Econ 101 Winter 2010 Supply and Demand Functions

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Unformatted text preview: Economics 101 Section 400 Principles of Microeconomics Demand and Supply Functions University of Michigan Winter 2010 Econ 101.400 Winter 2010 1. Introduction The simple supply and demand model is a powerful tool. With little other than the conviction that supply curves slope upward and demand curves slope downward, we can generate a great many insights into changes in prices and sales volumes in competitive markets. For example, if I know that beer and wine are substitute goods, and that the price of beer rises, then the supply and demand model allows me to predict that wine prices will increase and that wine consumption will increase. But what if I wish to know more than simply the direction of the changes in market price and quantity traded? What if I wish to know by how much the price of wine rises when the price of beer increases? Or how much more wine will be consumed? To generate quantitative results like these, I need to have more information about the shape of the supply and demand curves. This information is most easily captured in the mathematical representation of the curves: the supply and demand functions. These are equations that capture the relationship between the quantity demanded or supplied and the price of the good (and any other explanatory variables). 2. The Demand Function In its most basic form, the demand function expresses the quantity of a good demanded as a function of the good’s price. Consider the discussion from class, relating to the quota imposed on sugar imports into the United States. In that scenario, we were informed that 20.3 billion pounds of sugar are demanded when the price of sugar is $0.21/pound, and that 24.3 billion pounds would be demanded when the price of sugar is $0.08/pound. A demand function that reflects that information is the following: Qdsugar = 26.76 – 30.77 Psugar where Psugar is the price per pound of sugar and Qdsugar is the quantity of sugar demanded, measured in billions of pounds. We can check to see whether this demand function is consistent with what we know about demand for sugar. By substituting a price of P = $0.08/pound of sugar into this expression, we find that the quantity of sugar demanded is (approximately) Qdsugar = 24.3 billion pounds. Similarly, if P = $0.21/pound of sugar, then the quantity of sugar demanded is (approximately) Qdsugar = 20.3 billion pounds. Therefore, the graph of this function will pass through both of the points we have been given from the domestic demand curve for sugar (marked A and B in Figure 1 below). Econ 101.400 Winter 2010 $/lb $0.21 A B $0.08 Domestic Demand 20.3 24.3 Lbs (billions) Figure 1: A quota limiting sugar imports to 3.5 billion pounds is imposed in the US market for sugar. Notice, we have information about two points that lie on the demand function (A and B), and have found a function associated with a curve that goes through those points. But a little reflection should convince you that there are many curves that pass through points A and B, and therefore many functions that could represent demand for sugar. The particular demand function we have generated here is just one of these possible demand functions. It has a specific characteristic: its graph is a straight line. Is there any reason to think that the demand curve is a straight line? Absolutely none. The actual demand curve that passes through points A and B could have just about any shape. The only thing we are reasonably sure about is that the demand curve is always negatively sloped. However, there are several reasons why we might be interested in imagining that the demand curve is a straight line. We will discuss these reasons as we proceed through these notes. But for the moment, we will simply note the first of the reasons. Using linear demand (and supply) curves will keep our analysis simple. While it may not be completely accurate, the simplicity we gain in dealing with only linear functions can be a real boon. Econ 101.400 Winter 2010 Linear Demand Functions A linear demand function is one that takes the following form: Qd(P) = a + b P d where Q is the quantity demanded, specified as a function of the price, P. The two parameters, a and b, are constants. To appreciate what this demand curve tells us, we need to understand what the constants a and b mean. Suppose that the price of the good were P = 0. The linear demand function tells us that Qd(P=0) = a + b (0) = a. Thus, the constant a identifies the quantity of the good demanded when the price is zero. In other words, this is the maximum quantity that will be demanded at any price. Graphically, we can also identify this as the value of Qd where the demand curve intercepts the Qd‐axis. We will refer to a as the demand function’s intercept. P Slope = 1/b < 0 P0+1 1 P0 b Q0+b Q0 a Q We will call the constant b the price coefficient of the demand function. It will be closely related to the slope of the demand curve. The fact that the demand curve slopes downward tells us that the quantity demanded falls as the price increases. This implies that the coefficient b should be negative. The magnitude of b reveals how much Qd falls Figure 2: A generic linear demand curve, with intercept a and price coefficient b < 0. The demand function is Qd(P)=a + b P. Econ 101.400 Winter 2010 as P rises. For every one‐unit increase in P, the quantity demanded falls by |b| units. This makes it clear how the price coefficient is related to the slope of the demand curve. If we try to calculate the slope of the demand curve, we look for the ratio of the “rise” to the “run” between two points on the curve. As we described, a one‐unit increase in P (i.e. a one‐unit “rise”) is associated with a |b|‐unit decrease in Qd (i.e. a b‐unit “run”, where b is a negative number). The slope of the demand curve, then, is 1/b (which is negative, since b < 0). When b is relatively large in magnitude, this tells us that the quantity demanded responds quite sensitively to changes in price. This tends to make the demand curve’s slope relatively low (in absolute value), which means the curve is relatively flat. In contrast, a low value of b implies that Qd is insensitive to changes in P, and that the demand curve is relatively steep. Given any two points on a linear demand curve, we can determine the function describing that curve. Take the two points A and B on the demand curve for sugar (see Figure 1). Imagine moving from point A to point B ‐ we can calculate the “rise” and the “run” between these two points. The “rise” is ($0.08 ‐ $0.21) = ‐$0.13. The “run” is (24.3 – 20.3) = 4 (billion pounds). The slope of the demand curve is –0.13/4 = ‐0.0325. The defining characteristic of a straight line is that the slope between any two points on the curve will be exactly the same. Therefore, the slope between the point (24.3, $0.08) and any other point (Qd, P) on the demand curve must also be ‐0.0325. We can express this in the following way: P − 0.08 = −0.0325 . Q d − 24.3 Rearranging this expression gives: [ P − 0.08] Q d = 24.3 − = 26.76 − 30.77 P 0.0325 which is exactly the demand function initially presented. It tells us that, if sugar were to be given away for free, then around 26.76 billion pounds would be demanded in the market. Furthermore, for every dollar increase in the price of sugar, quantity demanded will fall by 30.77 billion pounds. Econ 101.400 Winter 2010 2.2 Interpreting the slope coefficient Consider again the generic demand function: Qd(P) = a + b P. As we have seen, the parameters of the function, a and b, have very specific interpretations. The constant a is the intercept of the demand curve on the Qd‐axis. The coefficient b (which is negative) is the inverse of the slope of the demand curve. There is also a more direct interpretation of the slope parameter, b. It reflects the sensitivity of the quantity demanded to changes in the price. Imagine that some price P0 prevailed, so that the quantity demanded can be computed to be Q0 = a + b P0. Then suppose that the price increased to P1 = P0 + 1 (i.e. price increases by $1). As a result, quantity demanded falls to Q1 = a + b(P0 + 1) = a + b P0 + b. The net change in quantity demanded is Q1 – Q0 = (a + b P0 + b) – (a + b P0) = b. Remember, b < 0, so this is a reduction in quantity demanded. Notice that the change in quantity demanded does not depend on the original price or the original quantity. This tells us that a $1 increase in price will always lead to a reduction in quantity demanded equal to |b|. This is a consequence of insisting on a linear demand specification. More importantly, it makes it clear that the magnitude of the coefficient b conveys valuable information. It reflects the sensitivity of the quantity demanded to changes in price. If |b| is small, then changing price will have little effect on quantity demanded. Relatively high values of |b| will imply that the quantity demanded will be far more sensitive to changes in price. Obviously, these variations in the sensitivity of the quantity demanded to changes in price will be reflected in the slope of the demand curve. When |b| is large, the demand curve is relatively flat. A glance at figure 3(a) confirms that this implies that a relatively small change in price will have a relatively large impact on quantity demanded. In contrast, figure 3(b) shows a steep demand curve. This is associated with a small value of |b|, which implies that the quantity demanded will not respond significantly to changes in price. Figures 3(c) and 3(d) show extreme cases of demand sensitivity. In figure 3(c), the horizontal demand curve is generated if one were to allow the demand curve to get flatter and flatter and flatter. It reflects an overwhelming sensitivity to price. At any price above P0, the quantity demanded falls to zero. At any price below P0, the quantity demanded is arbitrarily high. While the price is precisely P0, however, consumers are indifferent between consuming a lot or a little of this good. While the slope of this Econ 101.400 Winter 2010 demand curve is clearly zero, this makes it difficult to identify the value of the slope coefficient b. That coefficient should be the reciprocal of the slope; but as the reciprocal of zero is undefined, we cannot generate a value. Intuitively, this demand curve reflects an infinite sensitivity to price changes, so one may be tempted to identify the slope parameter as ‐∞. Mathematically, it is more precise to observe that the demand function itself is not defined. In figure 3(d) we have the extreme opposite case. Here there is absolutely no sensitivity to changes in price. The demand function is simply Qd(P) = Q0, so that the coefficient b is equal to zero. P (a) P1 P0 P P1 (b) P0 Q1 Q0 Q P P1 Q1 Q0 Q P (c) (d) P0 P0 Q1 Q0 Q Q0 Q One of the major determinants of the slope of the demand curve will be the availability of close substitute goods. When there are very close substitutes for one particular product, we will imagine that the demand curve will be very sensitive to changes in price. For example, if the price of one brand of rice rises even a little, we might imagine Figure 3: Varying degrees of demand sensitivity: (a) demand curve that is relatively sensitive to price changes, or price elastic; (b) demand curve that is relatively insensitive to price changes, or price inelastic; (c) an infinitely elastic demand curve; and (d) a perfectly inelastic demand curve. Econ 101.400 Winter 2010 that most consumers would choose to buy a competing brand that is now cheaper than the other brand simply because there is little or no perceived difference in the products. On the other hand, if the price of insulin were to rise significantly, we might predict that there would be little change in the quantity of insulin demanded, simply because there are no alternative products that can satisfy the specific needs of diabetics. As a final comment in this section, we note that economists often refer to the sensitivity of one variable to changes in another by using the term elasticity. In this case, we have been talking about the sensitivity of demand to changes in price, so we might wish to refer to the price elasticity of demand. When demand is relatively sensitive to changes in the price we will say that demand is relatively price elastic, and when demand is relatively insensitive to changes in price we will say that demand is relatively price inelastic. At the extremes, we will refer to infinitely elastic demand when the demand curve is horizontal and to perfectly inelastic demand when the demand curve is horizontal. 2.3 Linear demand functions with many variables Adopting the linear formulation of demand described in Section 2.1 makes it simple for us to draw the demand curve accurately in our supply‐and‐demand diagram. But it is an oversimplification of the determination of quantity demanded as it suggests that the good’s own price is the only determinant of the quantity demanded. Our earlier discussion of demand argued that many other things might influence the quantity of a good demanded – things such as: Consumers’ incomes; Prices of other goods (substitutes or complements); Environmental factors; Advertising; Population; Etc. If we wished to describe how the quantity demanded depends upon these sorts of variables, we should be able to include that information in the functional expression of the demand relationship. This means we should be able to build a model of demand for a product where quantity demanded (Qd) is expressed as a function of many variables. Econ 101.400 Winter 2010 As an example, suppose the following are the determinants of demand for vanilla ice cream in a local market: P – the price of vanilla ice cream Pc – the price of chocolate ice cream (a substitute) Ps – the price of strawberry ice cream (a substitute) Pcs – the price of chocolate syrup (a complement) Y – average family income (in thousands of dollars per annum) Pop – population (in thousands of people) Adv – advertising expenditure (in thousands of dollars per annum) Summer – takes a value of 1 when we are in summer, 0 otherwise Spring – takes a value of 1 when we are in Spring, 0 otherwise Autumn – takes a value of 1 when we are in Autumn, 0 otherwise Then at any given time, the quantity of vanilla ice cream demanded in the market would be a function of all of those variables: Qd = Qd(P, Pc, Ps, Pcs, Y, Pop, Adv, Summer, Spring, Autumn). If we make the same simplifying assumption that we made in the much simpler model above (i.e. that the demand function is linear), then we can write the demand function in the following way: Qd = b0 + b1 P + b2 Pc + b3 Ps + b4 Pcs + b5 Y + b6 Pop + b7 Adv + b8 Spring + b9 Summer + b10 Autumn. We say that this function is linear in each of its arguments since a one‐unit change in any independent variable always has the same impact upon quantity demanded, regardless of the initial value of that variable. For example, if P rises by $1 then quantity demanded will change by b1 units, no matter whether P is high or low. Just as we needed to know the values of the parameters a and b in the simple demand functions in the previous section to make sense of the demand curve, knowing the full list of parameters (b0, b1, b2…,b10) in this more complicated model will tell us everything we need to know about demand for the good. What follows is an interpretation of these parameters, or coefficients. b0: We will label this the intercept again, as it is analogous to the intercept term, a, in our previous discussion. In this case, b0, represents the quantity of vanilla ice cream demanded if all of the explanatory variables take a value of 0 (i.e. if the prices of vanilla ice cream, chocolate ice Econ 101.400 Winter 2010 cream, strawberry ice cream and chocolate syrup were all zero, if there was no population, if advertising expenditures were $0 and the season was not summer, spring or autumn). b1: As was the case previously, this coefficient captures the sensitivity of quantity demanded to changes in the good’s own price. Note: quantity demanded should fall as P increases, so the coefficient b1 should be negative. b2 & b3: These coefficients capture the sensitivity of quantity demanded to changes in the price of the related goods (chocolate and strawberry ice cream). As these goods are substitutes, an increase in either Ps or Pc will increase the quantity of vanilla ice cream demanded. Therefore, these coefficients should be positive. b4: This coefficient captures the sensitivity of quantity demanded to changes in the price of the related good (chocolate syrup). As this is a complement, an increase in Pcs will decrease the quantity of vanilla ice cream demanded. Therefore, this coefficient should be negative. b5: This coefficient captures the sensitivity of quantity demanded to changes in average income. If vanilla ice cream is a normal good, then increasing Y will increase quantity demanded. In that case, this coefficient would be positive. On the other hand, if vanilla ice cream is inferior then this coefficient will be negative. This coefficient captures the sensitivity of quantity demanded to changes b6: in population. It would be reasonable to presume that this coefficient will be positive, showing that increasing the number of consumers will increase the quantity demanded. This coefficient captures the sensitivity of quantity demanded to changes b7: in advertising expenditures. Advertising executives will be hoping that this coefficient is positive, or they will quickly find themselves out of jobs! b8, b9, b10: These coefficients capture the effects of changing seasons. Notice, there is no variable to represent the winter month. This means that winter can be viewed as the status quo. When the three variables Spring, Summer and Autumn are all set equal to zero (i.e. during the winter), only the first seven determinants of demand are relevant. In spring, however, the Spring variable takes on a value of 1, so that there will be an additional Econ 101.400 Winter 2010 quantity demanded equal to the coefficient b8. When the season changes again, the Spring variable becomes 0, and the Summer variable will become 1. Now the quantity demanded exceeds that in the winter by an amount given by the coefficient b9. Similarly, b10 is a measure of how much more vanilla ice cream is demanded in the autumn than in the winter. If we imagine that winter is the season where vanilla ice cream demand is lowest, it would be sensible to suppose that b8, b9 and b10 are all positive. Suppose that the demand relationship involves the following values of the coefficients: b0 = 100 b1 = ‐125 b2 = 40 b3 = 20 b4 = ‐50 b5 = 20 b6 = 0.25 b7 = 2 b8 = 400 b9 = 1000 b10 = 600 Then the demand function can be written: Qd =100 ‐ 125 P + 40 Pc + 20 Ps – 50 Pcs + 20 Y + 0.25 Pop + 2 Adv + 400 Spring + 1000 Summer + 600 Autumn. We will interpret this function as stating the quantity of vanilla ice cream demanded per week, measured in thousands of pints. Prices of vanilla, chocolate and strawberry ice cream, as well as chocolate syrup, are measured in dollars per pint. Average income is measured in thousands of dollars per family per annum. Population is measured in thousands of people. Advertising expenditure is measured in thousands of dollars per annum. With this demand function, we can determine the quantity demanded in any one‐week period simply by substituting the values of the explanatory variables into the function. For example, if the season is summer, the prices of vanilla, chocolate and strawberry ice cream are all $4 per pint, the price of chocolate syrup is $2/pint, average income is $40,000, the population is 500,000 and advertising expenditures are $50,000 per annum, then we can calculate the quantity demanded to be: Econ 101.400 Winter 2010 Qd = 100 ‐ 125 (4) + 40 (4) + 20 (4) – 50 (2) + 20 (40) + 0.25 (500) + 2 (50) + 1000 = 1,725 (thousand pints). If the price of vanilla ice cream fell to $3.50, then quantity demanded would rise by 62.5 (thousand pints) to Qd = 1,787.50 (thousand pints). If no other independent variable changes, but the season changes from summer to autumn, then the Summer variable changes to 0 (reflected in a reduction in quantity demanded of 1000) and the Autumn variable changes to 1 (reflected in an increase in quantity demanded of 600). The net effect is for quantity demanded to decrease by 400 (thousand pints) to Qd = 1387.50 (thousand pints). If the price of chocolate syrup then falls to $1, then the quantity of vanilla ice cream, demanded will increase by 50 (thousand pints), to Qd = 1437.50 (thousand pints). Notice, the first change in quantity demanded we considered was a response to a reduction in the price of vanilla ice cream. In terms of our standard demand curve diagram, this is depicted as a movement along the demand curve. The latter two changes, however, were responses to a change in the season and a change in the price of chocolate syrup respectively. These generated changes in the quantity of vanilla ice cream demanded even while the price of vanilla ice cream remained unchanged. In terms of our demand curve diagram, then, these must be describing shifts in the demand curve. These three effects are depicted in Figure 4 below. In the case of the simple linear demand function discussed in the previous section, we were able to see specifically how the parameters in the function (the coefficients) affected the demand curve. We can do the same with the more complicated demand function with many explanatory variables. Econ 101.400 Winter 2010 P (a) P (b) 4 3.50 A B 3.50 C B 1725 1787.5 Q 1387.5 1787.5 Q P (c) 3.50 D C 1387.5 1437.5 Q Figure 4: Three changes in quantity demanded: (a) a reduction in P from $4 to $3.50; (b) a seasonal change, from summer to autumn; and (c) a reduction in the price of a complementary good. First, the slope of the demand function is found in exactly the same way as before. It is the reciprocal of the price coefficient in the demand function. As before, that coefficient will be negative, which reflects the fact that the demand curve is downward sloping. When the coefficient is relatively large, then it suggests that the quantity demanded is very sensitive to changes in the price. This results in a relatively flat demand curve. On the other hand, when b1 is relatively small, it shows that quantity demanded is relatively insensitive to changes in price, which implies that the demand curve is relatively steep. As we observed above, changes in the value of any explanatory variable other than the good’s own price will result in a shift in the demand curve. Sometimes, we might even refer to these other variables as demand shifters, or shift variables. With this linear specification of demand, changes in these variables do not affect the sensitivity of quantity demanded to changes in the good’s own price (this is fully described by the coefficient b1). Therefore, changes in these variables will generate parallel shifts in the Econ 101.400 Winter 2010 demand curve. This implies that these changes are really only changing the intercept of the demand curve (labeled in the previous section as a). In fact, we could rewrite the demand function in the following way: Qd(P) = a + b1 P where a = b0 + b2 Pc + b3 Ps + b4 Pcs + b5 Y + b6 Pop + b7 Adv + b8 Spring + b9 Summer + b10 Autumn. In terms of our demand curve diagram, the value of a determines where the demand curve intersects the Qd‐axis. This makes it clear that changes in any of the variables other than P will result in a change in the horizontal intercept of the demand curve, and therefore will shift the curve left or right. Moreover, the sign of the coefficient attached to that variable will determine the direction of the shift of the demand curve. Consider, for example, the coefficient b2. It refers to how the quantity of vanilla ice cream demanded changes in response to changes in the price of chocolate ice cream. As explained earlier, this coefficient is positive reflecting the fact that chocolate ice cream and vanilla ice cream are substitutes. An increase in the price Pc, then, will increase the value of the horizontal intercept. This confirms that an increase in the price of the substitute good will shift the demand curve to the right. Similar observations can be made about the coefficients in front of the other shift variables. If the coefficients are positive, increasing the associated variable will shift the demand curve to the right. If the coefficients are negative, then increasing the associated variable will shift the demand curve to the left. Furthermore, the size of the shift in the demand curve will larger when the magnitude of the coefficient is larger. Econ 101.400 Winter 2010 3 Estimating demand functions The discussion above gives some insight into how to interpret a linear demand function, and how information in that demand function is reflected in the demand curve diagram. However, this begs all sorts of questions: where do these demand functions come from? How on Earth do we know what the price coefficient is in the demand curve for vanilla ice cream? How do we know the value of the income coefficient in the demand curve for cigarettes? How do we know the value of the coefficient for the price of gas when we talk about demand for automobiles? This kind of information doesn’t simply materialize the way it seems to in a text book. Instead, it is gleaned only through the application of statistical estimation techniques. The following section gives a very brief introduction to how these estimates are derived. 3.1 The single variable model To keep the discussion simple, we will return to the very simple model of demand originally introduced in section 2.1, where demand is a function of only one variable: price. Furthermore, we will retain our simplifying assumption that the relationship between quantity demanded and price is linear, so that the demand function has the form: Qd(P) = a + b P, where a > 0 and b < 0. If the relationship between price and quantity demanded were truly of this form, it would be a simple matter to deduce the values of a and b. All you would need to do is observe the quantity that is actually demanded at two different prices. This means that you can observe two points that lie on the demand curve. Immediately this allows you to calculate the slope of the straight line (i.e. 1/b), and then to infer the value of the horizontal intercept (a). This is precisely the process we went through in determining the demand function associated with the linear demand curve for sugar, in Section 2.1. Unfortunately, the world never provides such neatly manageable problems. Instead, there are all sorts of weird and wonderful things that influence the decisions people make that could never be predicted; influences that do not have consistent, systematic effects on behavior. These capricious influences show up as random variations in the decisions that people make, and serve to introduce noise into the economist’s attempts to discern the true nature of relationships such as that embodied in the demand function. Econ 101.400 Winter 2010 To account for these capricious influences, suppose that quantity demanded is specified in the following way: Qd(P) = a + b P + ε, d where a, b, Q and P have the same interpretations as before and ε is a purely random term. Being random, ε simply cannot be predicted. The best guess we can make is that ε will be zero, and, on average, this guess will be correct. So, if we wanted to make predictions about the quantity of a good demanded at any particular price, the natural prediction for us to make is that the quantity demanded will be a + b P units (i.e. simply ignore the random component of demand). For an economist interested in making use of the demand function, the important problem to be solved is the determination of the values of the parameters a and b. To determine these values, we must observe actual demand decisions made and use statistical techniques to estimate those values. When observing demand decisions over time, for example, we might collect data that look like those shown in Figure 5 below. P P1 1 P2 P3 P4 2 3 4 P5 5 Q1 Q2 Q3 Q4 Q5 Q Figure 5: Plotting price-quantity observations. Each point represents a price‐quantity pair, (P, Q), that should be interpreted to mean “when a price of P was quoted in the market, a quantity of Q was demanded.” Here we have five of these observations. Econ 101.400 Winter 2010 If the five observations are associated with a demand model where price is truly the only determinant of the quantity demanded, and where there is not any sort of random variation in the quantity of the good demanded, then it must be the case the all five points lie on the demand curve. This implies that the demand function is non‐linear. Mathematically, there are difficulties searching for non‐linear functions. They are potentially complicated functions that are difficult to manipulate. But more importantly, there are many different non‐linear functions that will pass through all five of those observations. An example is depicted in Figure 6 below. If we are interested in keeping our analysis simple, then we will avoid searching for non‐linear functions that are consistent with these observations. P P1 1 P2 P3 P4 2 3 4 P5 5 Q1 Q2 Q3 Q4 Q5 Q Figure 6: Fitting a non-linear curve to the price-quantity observations. Instead, we might imagine that the five observations are associated with a linear demand function plus some random element, as described previously: Q(Pi) = a + b Pi + εi. This scenario is depicted in Figure 7. The blue line reflects the predictable portion of the demand model (i.e. the portion that varies systematically with the change in P). The aqua colored dots mark the quantities demanded at various prices as specified by the non‐random or deterministic portion of the demand model, Q(Pi) = a + b Pi. As Econ 101.400 Winter 2010 discussed earlier, thehorizontal intercept of this line is the value a and the slope of the line is 1/b. P P1 ε1 1 P2 P3 P4 2 ε2 3 ε4 4 P5 5 ε5 Q Figure 7: The blue line shows the deterministic variation in quantity demanded. The actual price-quantity observations differ from this due to random or unpredictable influences on quantity demanded. In addition to the deterministic portion of demand, for any particular observation of P there is also some random element to demand. This means that for every observation of price, Pi, there is a different realization of the random variable, εi. For example, in the case of observation 3, where the actual observation falls on the blue line, there is no difference between the quantity demanded according to the deterministic model and the actual quantity demanded. In other words, the random variable ε3 has a value of 0. At other times, the actual observation of quantity demanded is greater than that suggested by the deterministic model (observations 1 and 4). This implies that the random variable in each of those cases, ε1 and ε4, takes a positive value. In the remaining cases, (observations 2 and 5), the actual quantity demanded is less than that suggested by the deterministic model. This implies that the random variable in each of those cases, ε2 and ε5, takes a negative value. While the random variable, εi, may take positive or negative values, on average it will be equal to zero. Therefore, if we know the price in the market is P, our best guess as to the quantity demanded is a + b P. The blue line in Figure 7 not only represents the deterministic portion of demand, but also reveals our ideal prediction of quantity Econ 101.400 Winter 2010 demanded for any given price. Of course, this observation is only useful if we happen to know the value of the parameters a and b. If we do not know those values, then we need to estimate them, using the data on hand. 3.2 Ordinary least squares (OLS) In the single variable world, estimation requires us to search for a demand function that explains the greatest proportion of observed variation in quantity demanded purely through variations in price. Graphically, this process identifies a line of best fit through the observed data points. ˆ ˆ Suppose we were simply to choose values of the parameters ( a > 0 and b < 0 ) at random, and then used these to form an estimate of the demand function: ˆ ˆˆ Q( P) = a + b P . This estimate tries to represent all of the variation in quantity demanded as a result of variations in price. It is bound to do a bad job for two reasons. First, it ignores that there is a second source of variation in quantity demanded (the random effect represented in our model by the random variable ε). Second, it almost certainly uses ˆ ˆ the wrong values of the parameters, a and b rather than a and b. We can measure just how badly this estimate performs. For any price we have observed in the market, we can compare the observed quantity demanded with the predicted quantity demanded at the same price. For example, suppose we have five observations: (P1, Q1), (P2, Q2), (P3, Q3), (P4, Q4) and (P5, Q5). For any of these observations, we know that Qi = Qi(Pi) = a + b Pi + εi. Our estimated demand function, ˆ ˆˆ however, predicts that the quantity demanded would have been Qi ( Pi ) = a + b Pi . The ˆ difference between Qi(Pi) and Q ( P ) is a measure of howinaccurate the estimated i i demand function is in this instance. We will give a name to the difference between the predicted quantity demanded and the observed quantity demanded: the residual. We will denote the residual in the following way: ˆ ei = Qi − Qi ( Pi ) . We will have a different measured residual for every different observation in our data set. So, in our example with five different observations, we will be able to calculate five different residuals (e1, e2, e3, e4 and e5). A graphical interpretation of this is presented below in Figure 8. Econ 101.400 Winter 2010 P P1 1 e1 1 P2 P3 P4 2 3 e3 3 4 4 e4 P5 5 Slope = 1 ˆ b Q Figure 8: The blue line is an estimated demand curve. The measured differences between the observed quantities demanded and the predicted quantities demanded are the residuals. ˆ a The blue line in Figure 8 is some estimated demand curve. It has a horizontal intercept ˆ ˆ of a > 0 and a slope of 1 b < 0 . The red dots are the actual observations of price and quantity demanded. The aqua dots all lie on the estimated demanded curve, and represent the estimates of quantity demanded at the specific prices that have been observed in the market. In the cases of prices P2 and P5, the actual observations also lie on the estimated demand curve. In other words, the estimated quantity demanded at each of those prices is exactly the same as the observed quantity demanded, so that the residual in those cases is equal to zero. In the other cases, the estimated demand curve is inaccurate. The residuals are marked on the diagram, and reflect the difference between the quantity truly demanded and the predicted quantity demanded. In each instance, the observed quantity demanded exceeds the predicted quantity demanded, so that the residual takes a positive value. The residual values, however, could easily take negative values if the predicted quantity demanded exceeded the observed quantity demanded. ˆ ˆ Obviously, we aren’t interested in randomly choosing values of a and b to generate an estimated demand curve. Instead, we want to choose values that do a good job at reflecting how observed quantity demanded responds to variations in price. In other words, we want to choose values of the estimated parameters that make the associated residual measurements small. Our example in Figure 8 shows that it is not easy to Econ 101.400 Winter 2010 determine exactly what that goal means. For the particular estimated values used in that diagram, it makes some of the residual values very small. In fact, e2 and e5 are both equal to zero. But at the same time, other residuals are made larger than they need to ˆ be. For example, if we had chosen a slightly smaller value for b , the estimated demand curve would have been steeper, and the residuals e1, e3 and e4 would all have been smaller. So we are left with an important question. If we aim to choose the parameter estimates in order to make the residuals small, which residuals should we care about most? The easiest answer to this question is that all residuals should be equally important. In that case, we could simply aim to choose the parameters so that the sum of all residuals is minimized. But this approach has some serious problems. To see those problems, consider the following very simple example. Suppose we had three observations: (P1, Q1) = (10, 150) (P2, Q2) = (20, 100) (P3, Q3) = (30, 50) Now compare the following two estimated demand functions: ˆ (A) Q A ( P) = 300 − 10 P ; and ˆ (B) Q ( P) = 200 − 5P . B We can calculate the residuals for each of these estimated demand curves, ˆ remembering that the residual is given by ei = Qi − Qi ( Pi ) . In case (A), the residuals are: (1) e1A = Q1 – [300 – 10 P1] = 150 – [300 – 100] = ‐50 (2) e2A = Q2 – [300 – 10 P2] = 100 – [300 – 200] = 0 (3) e3A = Q3 – [300 – 10 P3] = 50 – [300 – 300] = 50. The sum of all these residuals is exactly zero. In case (B), the residuals are: (1) e1B = Q1 – [200 – 5 P1] = 150 – [200 – 50] = 0 (2) e2B = Q2 – [200 – 5 P2] = 100 – [200 – 100] = 0 (3) e3B = Q3 – [200 – 5 P3] = 50 – [200 – 150] = 0. Econ 101.400 Winter 2010 In this case, the sum of all residuals is also zero. ˆ ˆ If we try to choose the values of a and b to minimize the sum of all residuals, then we would be unable to choose between the two estimated demand curves above. However, it is clear that estimate (B) is superior to (A). In every instance, function (B) generates residuals of zero. This means that this estimated demand curve perfectly explains all the variation in quantities demanded. On the other hand, estimate (B) misestimates the quantity demanded two out of three times. This is shown in Figure 9 below. P eA3 = 50 30 3 20 2 10 1 eA1 = - 50 50 100 150 200 300 Q Figure 9: Two estimates of the demand function. At prices of 10, 20 and 30, the yellow estimate perfectly describes how quantity demanded varies. The blue estimate makes errors. The red dots in Figure 9 represent the three observations. The gold line is the demand curve generated by the estimated demand function (B). It passes through each of the three observations, and so all three of its residuals are calculated to be zero. Function (B) explains all observed variation in quantity demanded. The blue line is the curve associated with estimated demand function (A). Using that function to estimate demand, we find that estimated quantity demanded is 200 when P = 10, while the quantity demanded that is actually observed is 150. Hence the residual of e1A = ‐ 50. Similarly, at a price of P = 30, we can calculate the residual to be e3A = 50. Econ 101.400 Winter 2010 Given that estimate (B) so clearly outperforms estimate (A), why is it that the sum of residuals in the both cases is equal to zero? The problem stems from the fact that some residuals are negative. If we simply examine the sum of all residuals, a negative residual will compensate for positive residuals. As a result, we simply cannot use a simple sum‐ of‐residuals metric to determine the best estimated demand function. Instead, we could try to find a metric to express the accuracy of a particular estimated demand curve that would first convert all residuals, whether positive or negative, into positive numbers before taking their sum. Two methods readily spring to mind that would accomplish this task. (i) Take the absolute value of each residual, and sum those absolute values. We would then want to find the estimated demand curve that minimizes the sum of absolute residuals. (ii) Square each residual, and take the sum of all squared residuals. We would then want to find the estimated demand curve that minimizes the sum of squared residuals. While the former approach might seem the most obvious one to adopt, there are various reasons why we might prefer to adopt the latter. In fact, the most significant factor weighing in favor of the sum of squared residuals approach is that the estimated demand function generated using this approach can be shown (probabilistically) to be “best” amongst all linear estimators of the demand function. The details of this argument are left for your future statistics and econometrics classes. For the moment, it will suffice to observe that this is generally the preferred approach for making linear estimates of the demand function. The method we will consider for estimating the demand function, then, is to choose ˆ ˆ values of a and b that will minimize the sum of all squared residuals. As we observed before, the task of “making the residuals small” is not a uniquely defined idea. In this case, when minimizing the sum of squared residuals, we do not weigh each residual equally. Instead, we weigh large residuals significantly more than small residuals. This process will suggest that an estimated demand function that generates some small and some large residuals is less desirable than an estimated demand function that generates only residuals of moderate size. This is a direct result of assessing the estimates on the basis of the sum of the squared residuals. To see how the process works, consider the following extremely simple example. Suppose that the following demand function truly describes demand decisions: Econ 101.400 Winter 2010 Qi(Pi) = 50 – Pi + εi but that model is not known to us. What we do know is the following three price‐ quantity observations: (P1, Q1) = (10, 43) (P2, Q2) = (20, 25) (P3, Q3) = (30, 24) As we believe these observations to have been generated by a demand model of the form: Qi(Pi) = a + b Pi + εi. we will look to generate an estimated demand curve of the form: ˆ ˆˆ Q( P) = a + b P . ˆ ˆ In particular, we will search for the values a and b that minimize the sum of squared residuals. ˆ ˆ We can calculate each of the residuals for arbitrary values of a and b . ˆ ˆˆ ˆ e1 = a + bP1 − Q1 = a + 10 b − 43 ˆ ˆˆ ˆ e = a + bP − Q = a + 20 b − 25 2 2 2 ˆ ˆˆ ˆ e3 = a + bP3 − Q3 = a + 30 b − 24 . Squaring each of these residuals, we get: ˆ ˆ ˆ ˆ ˆˆ (e1 ) 2 = a 2 − 86 a + 20 a b + 100 b 2 − 860 b + 1849 ˆ ˆ ˆ ˆ ˆˆ (e ) 2 = a 2 − 50 a + 40 a b + 400 b 2 − 1000 b + 625 1 ˆ ˆ ˆ ˆ ˆˆ (e1 ) 2 = a 2 − 48 a + 60 a b + 900 b 2 − 1440 b + 576 . Now summing the squares of the residuals we get: ˆ ˆ ˆ ˆ ˆˆ Sum of squared residuals = 3 a 2 − 184 a + 120 a b + 1400 b 2 − 3300 b + 3050 . Econ 101.400 Winter 2010 ˆ The value of this expression obviously depends upon the particular values of a and ˆ ˆ ˆ b chosen. Remember, we are interested in choosing the values of a and b that minimize the sum of squared residuals. Fortunately, this minimization process isn’t too ˆ ˆ tricky, since the sum of squared residuals expression is quadratic in a and b . The values that minimize the sum of squared residuals are ˆ a = 49.67 ˆ = −0.95 . b If you like, you can use Excel on a computer, or use a graphing calculator, or (if you have some experience with the techniques) you can use some simple calculus to verify that these values do in factminimize the sum of squared residuals. Importantly, you should appreciate how accurately this method has allowed us to estimate the real demand function. The real demand model is Qi(Pi) = 50 – Pi + εi and the estimated model is ˆ Qi ( Pi ) = 49.67 − 0.95 Pi . This estimate was generated with only three observations. In general, estimates will become more accurate when they are based on more data. In the context of this course, the mathematical details of this estimation process are not of any great importance. In fact, it will never really be important for you to grind through these calculations as powerful software is easily available that will automate this whole estimation process. But it is important for you to understand what ordinary least squares (OLS) estimation is all about, and to appreciate roughly how we would generate demand functions if ever we wished to use on in a practical application 3.3 Demand functions with many variables As we discussed earlier, it is unlikely that price is the only variable that produces predictable variations in quantity demanded. Changes in incomes, changes in prices of related goods, changes in season, and changes in advertising efforts – all of these are likely to produce predictable changes in quantity demanded too. As a result, we will typically be interested in considering demand to be a function of many explanatory Econ 101.400 Winter 2010 variables. Estimating these functions can also be accomplished using the OLS procedure. To take a simple example, suppose that real demand responds to two variables in a predictable fashion: price (P) and income (Y). In addition, there are unpredictable (or random) variations in demand, so that our real demand function looks like this: Qi(Pi) = a + b Pi + c Yi + εi. There are three unknown parameters in this function: a, b and c. Our estimation process will have to estimate all three of them. For any estimated values of those parameters, we can describe an estimated demand function ˆ ˆˆ ˆ Qi ( Pi ) = a + b Pi + c Yi . To perform the estimation, we need to have data – observations of all (observable) variables. Each observation must now tell us three pieces of information: the quantity demanded, the price and the level of income in a particular market at a particular time. For each observation, then, we can calculate residuals associated with the given estimated demand function: ˆˆ ˆ ei = a + b Pi + c Yi − Qi . ˆ ˆˆ OLS demands that we choose the estimated parameters, a , b and c , to minimize the sum of the squares of these residuals. As there are more parameters to estimate than in the single variable case, we will also need more observations (data) to perform the estimation effectively. But the OLS method in the two scenarios is exactly the same. Econ 101.400 Winter 2010 4 Supply Functions Our discussion of the supply function will follow the demand function discussion closely. In fact, most of the things we had to say about demand functions will be true of supply functions. Hence, we will focus here upon the important differences between supply and demand functions. 4.1 Linear supply functions with one variable Just like the demand function, it will often be useful for us to think of the supply function specifying the quantity producers wish to supply as a function of a single variable: the price of the good. The form of that function will be just like the form of the demand function: Q(P) = a + b P. The parameters a and b have similar interpretations to those in the demand function. The parameter b is again a sensitivity parameter. It tells us how large a change in quantity supplied to expect when the price changes by one unit. Again, this will be reflected in the graph of the supply curve as the reciprocal of the curve’s slope. Importantly, we expect supply curves to be upward sloping (i.e. the slope of the curve is positive), so unlike the demand function, the coefficient b in the supply function should not be negative. Figure 10 below shows a variety of supply functions, each demonstrating a different degree of sensitivity to changes in price. We say that the horizontal supply function in Figure 10 (c) is infinitely elastic, reflecting the extreme sensitivity to changes in price. If the price is below P0, producers will supply nothing to the market. At prices higher than P0, producers will want to supply arbitrarily large quantities to the market. Very small changes in price, then, generate extreme responses in quantity supplied. At the price P0, producers are indifferent between producing and selling at this price or not producing at all. Figure 10 (d) reflects the other extreme. Here, the quantity supplied does not respond at all to changes in price. We say that this supply curve is perfectly inelastic. Figures 10 (a) and (b) show more moderate cases where the supply curve in 10 (a) is more sensitive to price changes than the supply curve in 10 (b), so we will say that the former is relatively price elastic and the latter is relatively price inelastic. Econ 101.400 Winter 2010 P (a) P1 P0 P P1 (b) P0 Q0 Q1 Q P P1 Q0 Q1 Q P (c) (d) P0 P0 Q1 Q0 Q Q0 Q The parameter a again specifies the intercept in the horizontal (or quantity) axis. This is verified by observing that the supply function tells us that the quantity supplied when P = 0 is Q(0) = a. In the case of the demand function, we were pretty comfortable insisting that the horizontal intercept should be positive (i.e. if the price of the good were zero, people would demand something). In the case of the supply function, we cannot impose the same constraint. If the price is zero, will producers wish to supply any goods to the market? Perhaps they will; perhaps they will not. If the price is very low, but positive, will producers wish to supply goods to the market? Possibly; but again it is possible that nothing is supplied. The uncertainty over these answers makes it impossible for us to constrain the sign of the intercept term, a, in the case of the supply function. Note though, if the intercept is negative, this should not be interpreted to mean that producers supply negative quantities at prices close to zero. Instead, it implies that nothing is supplied until the price reaches a sufficiently high price. This is demonstrated in Figure 11 below Figure 10: Varying degrees of supply sensitivity: (a) supply curve that is relatively sensitive to price changes, or price elastic; (b) supply curve that is relatively insensitive to price changes, or price inelastic; (c) an infinitely elastic supply curve; and (d) a perfectly inelastic supply curve. . Econ 101.400 Winter 2010 P (a) Slope = 1/b P (b) Slope = 1/b a/b a Q a Q Figure 11: (a) Shows linear supply curve with b > 0 and a > 0. (b) Shows linear supply curve with b > 0 and a < 0. Notice, the dotted line-segment is not part of the supply curve in this case.. Instead, at any price less than a/b, quantity supplied will be 0. Therefore, the two red line-segments make up the supply curve. 4.2 Linear supply functions with many variables Like the quantity demanded, predictable variations in the quantity supplied will rarely be determined only by changes in price. We have discussed this previously: prices of alternative products, prices of inputs into production, technology, number of producers, environmental factors and many other things can influence supply decisions. To the extent that these factors are quantifiable, they may also be included in a linear supply curve specification. Suppose, for example, that the quantity of inner bicycle tubes supplied to the market depends on the following variables: P: the price of inner bicycle tubes w: the wage paid to workers in the manufacturing plants Pbr: the price of butyl‐rubber from which the tubes are made Pbt: the price of bicycle tires (an alternative product) Pct: the price of car tires (an alternative product) Then a linear demand function would be of the form: Qs = b0+ b1 P + b2 w + b3 Pbr + b4 Pbt + b5 Pct. Econ 101.400 Winter 2010 Essentially, we have simply rewritten the supply curve in the simple, single variable case in the following way: Qs = a + b1 P where a = b0+ b2 w + b3 Pbr + b4 Pbt + b5 Pct, so that, again, changes in any variables other than the own price of the good will result in shifts of the demand curve. Moreover, the coefficients in front of the other variables measure the sensitivity of quantity supplied to changes in those variables. As a result, we get real information from the sign and magnitude of those coefficients. The b2 coefficient identifies the sensitivity of quantity supplied to changes in the wage. We expect this coefficient to be negative, as firms should reduce output in response to higher input prices. For a similar reason, b3 should also be negative. The remaining two coefficients, b4 and b5, are measures of the sensitivity of quantity supplied to changes in the price of alternative products. If car tires or bicycle types become more expensive, it will become more appealing to produce more of those tires rather than inner tubes. Therefore, we expect the quantity of inner tubes supplied to fall (at any given price). This means that b4 and b5 should also be negative. The magnitude of these coefficients will tell us the extent of supply sensitivity to changes in the respective variables. 4.3 Estimating supply functions with OLS The OLS process for estimating the supply function is identical to that for the demand function. If the true supply function is of the form: Qi = a + b Pi + εi where εi is a random, and therefore unpredictable, element, then we can estimate the supply function using ˆ ˆˆ Qi = a + b Pi ˆ ˆ where the parameters a and b minimize the sum of squared residuals. Obviously, the results we expect from this estimation process will look somewhat different from the Econ 101.400 Winter 2010 ˆ demand function. The estimated parameter b should be positive (rather than negative, ˆ as in the estimated demand function), while the estimated parameter a may be positive or negative (whereas it was necessarily positive in the demand function case). The OLS process also generalizes to the case where quantity supplied is a function of many variables (just as it generalized in the demand estimation problem). 5 Are demand and supply functions really linear? As we have discussed the specification and estimation of demand and supply functions, we have continued to insist on dealing with only linear functions. However, it seems extremely unlikely that any demand or supply function would ever truly be linear. Given this, why on earth would we impose this sort of constraint on our estimation process? There are many partial answers to this question. We will outline some of these answers, but will do so only superficially. For a more in‐depth treatment you will have to wait for a statistics or econometrics course. First, linear functions have one great trait: they are simple to use. This is an especially valuable characteristic when we just starting to learn about economic analysis. We can gain a great deal of insight without having to get too complex on a technical level. Don’t be misled by our discussion of linear estimation approaches. There is a whole world of estimation techniques beyond OLS, and state‐of‐the‐art estimation will make use of non‐linear techniques. Yet there is still a great deal to be learned by adopting linear estimation approaches such as those outlined above. Second, the linear formulation we have adopted is far more flexible that it may seem on the surface. It is a simple process to adopt a linear technique, such as OLS, after first transforming the independent or dependent variables (or both) and so generating a de facto non‐linear model. For example, if we thought that the true relationship between price and quantity demanded actually linked the natural logarithm of the price to the natural logarithm of quantity demanded, we might envisage a model of the following form: log Qdi = a + b log Pi + εi. Econ 101.400 Winter 2010 The relationship between Qdi and Pi is certainly non‐linear. However, we could transform this into a linear model simply by defining the following transformed variables: pi = log Pi and qi = log Qdi. Now, the model can be rewritten as qi = a + b pi + εi which is linear, and can be estimated using OLS. Notice, though, that this approach is only useful if we know (or have a strong belief) about the form that the non‐linearity takes. If we do not know what form some non‐linearity takes, then adopting a linear specification can be a safe alternative. But it can also be quite accurate especially if the observations of the various explanatory variables are grouped closely. For small changes in explanatory variables, non‐linear functions tend to look “roughly” linear. This is depicted in Figure 12 below. P Figure 12: The aqua dots are price-quantity observations used in a demand function estimation. The estimated linear function, shown by the red curve, provides a good approximation of the blue and lavender nonlinear functions in the region where most of the observations are found. Therefore, even if the blue or the lavender curve is the real demand curve, we lose little by adopting the linear estimate. Q Econ 101.400 Winter 2010 6 Conclusion In the earlier portion of the class, we concentrated on describing various market responses qualitatively. We were able to do so by adopting a graphical analysis, shifting supply and demand curves and observing whether equilibrium prices and quantities increased or fell in response to shocks to the market equilibrium. In order to be more precise, for example if we wished to determine the magnitude of changes in quantity traded or price, we need more explicit information about the supply and demand curves. This information is most conveniently presented mathematically, in functional form. In these notes we have learnt how to interpret linear demand and supply functions, and developed a basic understanding of how those functions can be estimated using the simple procedure of ordinary least squares estimation. In the remainder of the course, this will allow us conveniently to capture important demand or supply information in the form of demand and supply functions. ...
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This note was uploaded on 04/20/2010 for the course ECON 101 taught by Professor Gerson during the Winter '08 term at University of Michigan.

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