I need help explaining what is being shown in the figures I have attached. Can you help me break them down and understand what they are showing?

v Demand curves are usually graphed on logarith- mic axes because it facilitates a visual and quantitative Demand 1,000 100 Consumption - o Inelastic |Elastic Demand |Demand | I 1 10 100 1,000 10,000 Price FIGURE 8.1. Diagrammatic demand curve showing the usual shape and increasing elasticity across the demand curve. The vertical line marks the point of unit elasticity (slope = 1), which is the transition from inelastic to elastic demand. The level of demand is denoted as the y-intercept or the quantity consumed at zero priceQ,. 1072 100,000 Food 1,000 A Saccharin 10,000 100 D Total Responses per Day (Food Pellets or Squirts) Daily Consumption ADDA A A A 44 1,000 10 AL A DA AA 100 10 Pmax (sacc) 100 max (food) 1,000 10 Pmax (sacc) 100 max (food) 1,000 Price Price FIGURE 8.2. Left: Two demand curves for rhesus monkeys working for either food or saccharin- sweetened water. The demand functions show the total number of reinforcers consumed each day under a series of fixed-ratio schedules (prices) that ranged from fixed-ratio 10 to fixed-ratio 372. Data from Hursh and Silberberg (2008). Right: Daily output of responding that accompanied the levels of consumption shown in the left panel. The curves were fit with an exponential equation.The Translational Utility of Behavioral Economics Most studies of choice with animals have arranged 1,000 Earned Only 2.3e-006 for the alternative behaviors to provide the same, A 1 Hr Postfood 5.7e-006 perfectly substitutable reinforcer, usually food, which yields a specific kind of environment- behavior interaction in which the amount of behav- 100 ior allocated to each source of reinforcement Total Earned Food Consumption roughly matches the relative rate of reinforcement received from the source (the matching law; see Volume 1, Chapter 14, this handbook, or Chapter 7, 10 this volume). These studies do not measure total consumption or essential value; however, the P max response rate and response allocation data are sug- TTT TTTTTT gestive of the effects on measures of essential value. 10 100 1,000 For example, as noted earlier, responses allocated to Price obtaining Reinforcer A decline when a perfect sub- FIGURE 8.5. Two demand curves produced by stitute, B, is available at a lower price (e.g., Green & a rhesus monkey responding for food during a Rachlin, 1991). Likewise, periodically providing free 12- hour work period, either with no other source access to a perfect substitute will decreases instru- of food or with a 1-hour period of fixed-ratio (FR) 1 food reinforcement made available immediately mental responding for food (e.g., Rachlin & Baum, after the work period. Consumption is shown as 1972). Bickel, Madden, and DeGrandpre (1997) a function of the FR schedule that ranged from reported that periodically providing free cigarette FR 10 to FR 372. Alpha values were derived using puffs during a session in which puffs were earned Equation 1. Data from Hursh (2000). decreased consumption and generally increased price ican Psychological Association. Not for further distribution. elasticity of demand for earned puffs. Many choices in the open economy, reflecting reduced defense of made in the natural economy are between commodi- within-session consumption that is no longer essen- ties that are not perfect substitutes but fall into one of tial to survival. Similar differences have been docu- the other interactions depicted in Figure 8.4. mented in several studies comparing demand in In their everyday lives, people rarely choose open and closed economies (Bauman, 1991; Collier, between perfect substitutes. Instead, they choose to 1983; Collier, Johnson, Hill, & Kaufman, 1986; allocate their resources between different reinforcers Foster, Blackman, & Temple, 1997; Greenwald & that share some, but not all, characteristics. For Steinmiller, 2009; Hall & Lattal, 1990; Hursh, 1978, example, Kagel et al. (1995) examined rats' con- 1984, 1993; Hursh & Natelson, 1981; Hursh, Ras- sumption of equally priced root beer and Tomfurther distribution. Not for Copyright American Psychological Association. Commodities that are most vociferously defended are the most essential. Larger values of o reflect steeper demand curves and less essential value, and small a values come from shallow demand curves. Thus, a values are inversely related to value. A perhaps more intuitive measure of essential value, and one that may be derived from a, is Py, Py, is inversely propor- tional to a and can be found using the following approximation:? 065 max Q, e 2) As noted earlier, P,,, is the price (in units of C) at which the slope of the demand curve (i.e., point- price elasticity) is 1. P, is also the price at which peak responding is achieved; at higher prices, responding declines along the descending limb of the inverted u-shaped response output function. Higher P,,, values reflect a greater expenditure of resources in defense of consumption. As an example of how these techniques may be used to compare the value of different reinforcers, we summarize the results of a comparison of two different drug reinforcers (alfentanil, a potent opi- ate, and methohexital, a short-acting anesthetic) self-administered in different sessions by monkeys (Hursh & Winger, 1995). Figure 8.3 shows the across-subjects average consumption (number of drug deliveries) of each drug at the prices (number of lever presses per injection) shown on the x-axis; the demand curves were fit using Equation 1. The first thing to note is that within-drug doses varied by three orders of magnitude. Because Equation 1 accounts for scalar differences by expressing price as the cost of achieving 9y, the o value obtained at every dose of the same drug was the same (see the bottom of Figure 8.3). Because scalar differences in The Translational Utility of Behavioral Economics Alfentanil Demand __ 1000 e (.01 mg/kg Alfentanil 8 4 0.003 mg/kg Alfentanil _g Y 0.001 mg/kg Affentanil g 100 v 0.0003 mg/kg Alfentanil T 2 o 10 = w S o, = 1.248e-005 o 10 100 1,000 Price (FR) Methohexital Demand = 1,000 A 1.0 mg/kg Methohesital 2 O 0.32 mg/kg Methohexital 3 0 0.1 mg/kg Methohexital 100 = = g- 10 A 2 5 o = 3.183e-005 o 10 100 1,000 Price (FR) Pmax k o (normalized) R2 Alfentanil 1.893|1.25E-05 244 0.98 Methohexital | 1.893|3.18E-05 96 0.96 FIGURE 8.3. Exponential demand curves fit to average consumption of two drugs self-administered by rhesus monkeys. The drugs were alfentanil and methohexital. Parameters of the demand curves and P, values are shown at the bottom of the figure. The a value for methohexital was 2.5 times greater than the value for alfentanil. P,,,, is shown in normalized units of price, which is the result of Equation 2 multi- plied by Q; and divided by 100. The values are con- stant across doses of each drug, and the k values were constant for all curves. The global R? for each drug is also shown. FR = fixed ratio. condition, methadone and a different opiate, hydro- shoe to one right shoe). A complementary relation morphone (0.15 milligrams per delivery), were con- between reinforcers is demonstrated if the price of currently available, but hydromorphone was one reinforcer increases (e.g., the price of guacamole available at a constant price (FR 32). At the lowest doubles) and consumption of both reinforcers (chips price of methadone (FR 32), very little hydromor- and guacamole) decreases, despite no increase in the phone was consumed. As the price of methadone price of chips. Such a decrease reveals the tendency increased and methadone consumption decreased of the two commodities to be consumed in a con- consumption of hydromorphone increased, reveal- stant ratio. further distribution. ing a substitute relation between these two drugs. A complementary relation is illustrated in Figure This is called a cross-price change in demand for 8.7. In this study conducted by Spiga, Wilson, and hydromorphone, because consumption changed in Martinetti (2011), humans smoked cigarettes and response to the price of another good (discussed in drank ethanol-containing beverages available at a more detail later). Of further note is the change in range of prices. In the left panel, cigarette consump- elasticity of methadone (a) when hydromorphone tion was measured across a wide range of cigarettes was available as a substitute. Sensitivity to metha- prices (FR 10-512). In one condition, cigarettes done price more than doubled when hydromor- were the only commodity available for purchase phone was available and peak consumption (20) was (closed circles), whereas in a separate condition reduced by a third (80 vs. 120). Similar findings cigarettes (open circles) could be purchased at this range of prices, and ethanol (closed diamonds) was concurrently available at a fixed price (FR 32). Two Substitution Copyright American Psychological Association. Not for f interesting effects were observed. First, when ciga- 1,000 . Methadone alone a = 3.9e-006 rette prices increased, consumption of cigarettes and O Methadone concurrent a = 8.8e-006 Hydromorphone cross-price I = -3, B = 0.032 ethanol decreased. The latter decrease, despite no change in the price of ethanol, reveals a complemen Q ary relation between cigarettes and ethanol. Second, 100 N 80 - making a complement (ethanol) available increased cigarette consumption and increased the essential (reinforcers per day) Consumption of A or B value of cigarettes (i.e., lower value of a). A similar relation is shown in the right panel of Figure 7. 10 When the price of ethanol increased, ethanol and cigarette consumption both declined, and the essen- tial value of ethanol increased when cigarettes were concurrently available. THT 10 100 1,000 10,000 Reinforcer interactions from substitutes to com- FR for Methadone (A) plements may be quantified using cross-price elas- icity of demand-the slope of the demand function FIGURE 8.6. Mean daily consumption by human sub- jects of methadone and hydromorphone as a function of relating consumption of a fixed-price reinforcer to the price (fixed-ratio [FR] schedule) of methadone, in the changes in price of an alternative commodity. log-log coordinates. Data from Spiga (2006). As noted earlier, if this function has a positive slope, 200The Translational Utility of Behavioral Economics Cigarette Puffs With Ethanol Ethanol With Cigarette Puffs 100 7 100 7 Mean Ethanol Deliveries Mean Cigarette Puffs 10 10 Cross-Price Parameters Cross-Price Parameters I = 0.4640 1 = 0.6252 B = 0.007384 3 = 0.004301 Cigarette puffs alone ( = 5.50 e -5 . Ethanol alone 0 = 3.97 e -5 o . Cigarette puffs concurrent o = 1.75 e -5 Ethanol concurrent o = 1.54 e -5 1 - * Ethanol concurrent (Qa) Cigarette puffs concurrent (Qa) TITT 10 100 1,000 10 100 1,000 Cost (FR) Cost (FR) FIGURE 8.7. For human subjects, consumption of cigarette puffs or ethanol drinks. Left: The price of ciga- rettes was manipulated with and without ethanol concurrently available at a fixed price. Right: The price of ethanol was changed with and without cigarettes concurrently available at a fixed price. FR = fixed ratio. then the second commodity is a substitute for the indicating a parallel or complementary relation first (Figure 8.6); if the slope is negative, then the between consumptions of ethanol and cigarettes. 5 second commodity is a complement of the first To summarize, essential value may be dramatically (Figure 8.7); if the slope is zero, they are indepen affected by the availability of alternative reinforcers. dent (i.e., no interaction between the reinforcers). When substitutes are available, the essential value of a Copyright American Psychological Association. Not for further distribution. An extension of exponential demand was used to reinforcer declines relative to when no other source of fit the cross-price demand curves in Figure 8.6 for reinforcement is available. Low-priced concurrently hydromorpone (substitute for methadone) and in available perfect substitutes produce large decreases in Figure 8.7 for ethanol and cigarettes (complements essential value, with imperfect substitutes (Kagel et al., to each other): 1995) and delayed alternatives (e.g., Roane et al., 2005) producing more modest declines in essential 2 = log (alone ) + Ie BC, (3 ) value. At the other end of the continuum, concur- rently available complements increase the essential where Qalone is peak consumption of the fixed-price value of a reinforcer. These reinforcer interactions are reinforcer at the lowest price of the other reinforcer, I not traditionally incorporated into prominent models is the interaction constant, B is sensitivity of con- of decision making such as Herrnstein's (1970) match- sumption of the fixed-price reinforcer to changes in ing law, although interested readers should see Green the price of the other reinforcer, and C is the cost of and Rachlin (1991) or Herrnstein and Prelec (1991).further distribution. Not for Copyright American Psychological Association. economy study conducted by Madden et al. (2005), pigeons pecked a key to obtained three 45-milligram food pellet reinforcers in two separate conditions. In one condition, the cost of food was set by an FR schedule, whereas in the other condition, costs were programmed according to a random-ratio (RR) schedule (each response has a constant probability of leading to reinforcer). Because the reinforcer was identical across conditions, the essential value of food should be unaffected by how cost was opera- tionalized. Despite the effort costs being identical across conditions (e.g., 48 pecks per reinforcer or an average of 48 pecks per reinforcer), demand for food proved to be substantially more elastic when the FR schedule controlled the delivery of food. This difference is reflected in the values shown in Figure 8.8. 1,000 w 2 & e S B s 2 a R? o _ u o B FR 3.9-006 .97 s O RR 8.0e-007 .98 ' 100 10 100 1,000 Ratio Value FIGURE 8.8. Pigeons' averaged food consump- tion under fixed-ratio (FR) and random-ratio (RR) schedules. The R? values suggest the data are well fit by Equation 1, but the different o values reveal that essential value is affected by schedule of reinforcement type. Data from Madden et al. (2005). 202 tions may prove useful if one's goal is to precisely quantify the essential value of a reinforcer across many different cost manipulations. However, if one's goal is merely to rank order the essential value of a variety of reinforcers, then Equation 1 may prove adequate. Evaluating this possibility will require experiments in which several different reinforcers are rank ordered on the basis of their essential values obtained under one cost manipulation (e.g., delay to reinforcer delivery) and then these rankings are redetermined under a different cost manipulation (e.g., effort expended). Such a study would evaluate the ordinal generalizability of essential value. DISCOUNTING THE VALUE OF DELAYED OUTCOMES A few studies have examined the effects of reinforcer delay as a cost factor affecting consumer demand (e.g., Bauman, 1991; Tsunematsu, 2000). Those studies that have been conducted have reported that when delay costs are increased, demand for the rein- forcer follows a positively decelerating demand function comparable with those in the preceding fig- ures in this chapter. To our knowledge, no studies have yet used Equation 1 to estimate reinforcer value when delay is the cost variable. Instead, the bulk of the behavioral economic research on the effects of delay on reinforcer value has been con- ducted in the delay discounting literature. These studies have used psychophysical procedures to esti- mate the value of delayed reinforcers. One widely used and illustrative procedure asks human or ani- mal subjects to make repeated choices between a smaller-sooner reinforcer (SSR) and a larger-later reinforcer (LLR). Depending on the choice made on Trial x, the amount of the SSR arranged on Trial x + 1 is either increased (the subject chose the LLR) or function is shown as the solid curve in the top panel a strict hyperbola). of Figure 8.9. This finding is remarkably consistent This finding has been of great interest to behav- across species and reinforcer types (see Madden & ioral economists because the hyperbolic shape of the Bickel, 2010, for a review) delay discounting curve is not predicted by norma- Across species, the hyperbolic discounting equa- tive economic theory. The latter holds that the value tion (Mazur, 1987) well describes steady-state choices: of a delayed outcome should decline exponentially, as shown by the dashed curve in the top panel of V = - A ( 4 ) Figure 8.9 (e.g., Samuelson, 1937). An exponential (1+kD) decay function is the outcome of devaluing a rein- In Equation 4, A is the amount of the LLR, D is forcer at a constant rate over time. For example, the delay to its delivery, and k is a free parameter in Figure 8.9 the delayed reward loses 39% of its that varies with the steepness of the discounting value when delayed by 1 second (discounted value = 61%). At a 2-second delay, the reward is 100 discounted by an additional 39% (discounted value Hyperbolic is 37% = 61 - [61 X 0.39]), and so on. 80- - Exponential Given the considerable body of empirical evidence 60 that human and nonhuman choice reflects a hyper- Copyright American Psychological Association. Not for further distribution. bolic rather than an exponential discounting process, % Discounted Value 40 several behavioral economists have hypothesized 20 - that hyperbolic discounting may be the product of two exponential discounting processes (e.g., 5 10 15 Mcclure, Laibson, Lowenstein, & Cohen, 2004). One Delay (sec) process is controlled by limbic brain structures that discount delayed rewards according to a steep expo- 100 7 nential decay function when an immediate reward is available. The other process is controlled by frontal 80 cortex structures and discounts rewards in the upper 60 range of delays according to a more shallow exponen- % Discounted Value tial curve. The temporal proximity of the reward is 40 presumed to control the degree to which these two 20 processes are engaged, with more immediate rewards uniquely able to control limbic function. This model Delay of discounting has not been universally embraced (e.g., Ainslie, 2010), and empirical challenges exist FIGURE 8.9. Top: Illustrative hyperbolic (e.g., Peters, 2011). However, as discussed later in the and exponential delay discounting functions. chapter, the model has inspired a new executive func- Bottom: Irrational preference reversals pre- dicted by the hyperbolic shape of human and tion training approach to improving delay tolerance nonhuman discounting functions. T1 = Time in drug-using populations (Bickel, Yi, Landes, Hill, & 1; T2 = Time 2. Baxter, 2011). 203further distribution. for Not Copyright American Psychological Association. addiction. For example, a growing literature has indicated that for some commodities, extended exposure to the reinforcing properties of that commodity leads to progressive changes in demand (see escalation, e.g., Ahmed & Koob, 1998). In a recent experiment with rodents as subjects (Christensen, Silberberg, Hursh, Huntsberry, & Riley, 2008), demand curves for infusions of cocaine were determined after a brief familiarization with the drug and then after a 2-week history of infusions. Figure 8.10 illustrates the effect of the extended history of consuming cocaine. So as to focus on changes in elasticity, we used an exponential demand equation that sets Q, of both demand curves to 100% (see Hursh & Winger, 1995). The 2-week history of self-administering cocaine rendered drug demand more inelastic when o O Before History 8.2e-005 4 After History 4.1e-005 10{)*i 3 s ] o 2 E ' 3 ] z ' c \\ 8 ' = 10 '_]' ! @ u - 1 1 v A E ] Vo o . z b Lo 1- P,y befOrE | | P ey after T T T T T T 1 10 100 1,000 Normalized Price FIGURE 8.10. For the mean of a group of rodents, consumption of cocaine infusions as a function of increasing fixed-ratio schedules before and after a 2-week history of exposure to cocaine, in log-log coordinates. The shift in P, is shown as the two vertical dashed lines. Data from Christensen, Silberberg, Hursh, Huntsberry, and Riley (2008). 206 cess that may be quantified using the procedures outlined earlier. Building on these assumptions, some researchers have begun to ask whether characteristics of demand curves are correlated with intensity of drug dependence, affected by relapse cues, and whether these characteristics have predictive utility in deter- mining responsivity to treatment, relapse, and so forth. Several first steps have been taken in this emerging literature. Murphy, MacKillop, Tidey, Bra- zil, and Colby (2011), for example, used a simulated cigarette purchase task to quantify elasticity of demand for cigarettes among adolescent smokers. The adolescents were asked to report how many cigarettes they would purchase per day if cigarette prices varied across a wide range. So as to quantify demand in a simulated closed economy, participants were asked to imagine that no other source of cheaper cigarettes was available. Equation 1 pro- vided good fits of individual participants' simulated demand functions, with O, (i.e., peak spending) most consistently correlated with participants' level of nicotine dependence. A very similar methodology was used by Madden and Kalman (2010), who reported that therapy-related changes in essential value of cigarettes (o) were predictive of smoking cessation at 2-month follow-up. The simulated purchase task also proved useful in quantifying increases in the essential value of alcohol when heavy drinkers were exposed to the smell of their preferred alcoholic beverage (MacKillop et al., 2010). Procedures such as these hold the promise of integrating more ambiguously defined concepts like \"craving\" into a quantifiable behavioral economic model of factors affecting drug use and relapse. A second translational application of the demand curve procedures outlined earlier is in evaluating the efficacy and mechanisms of action of medica- tions designed to reduce drug use. One category of medications, agonists, has been discussed briefly in