# The Role of Money Illusion in Nominal Price Adjustment

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1 The Role of Money Illusion in Nominal Price Adjustment LUBA PETERSEN AND ABEL WINN Online Appendices 1

2 Appendix A. Adaptive Best Responding in FT s Experiments and Our Revised Experiments Section 1. Statistical Analysis of Adaptive Best Response and Adjustment Asymmetries in FT s Experiments At the aggregate level participants prices are well described by adaptive best responding (ABR) in FT s original experiments. After the first post-shock period, 45 percent of all prices in the NH are exactly equal to the ABR price, and 77 percent deviated by no more than one price increment. For the RH treatment, 40 percent of prices equaled the ABR and 77 percent were within one price increment. 1 We test the hypothesis that participants played their ABR more formally with the following random effects regression model: (A1) where NH equals one for observations in the NH and zero otherwise, is the ABR price of participant i in period t and is assumed to be distributed normally with zero mean and constant variance. We clustered the standard errors by group because participants within a group were responding to one another s prices. If participants systematically followed the ABR strategy in both treatments, we would expect to find coefficient estimates of 1 and 0. These expectations are upheld for all three beta coefficients, and the estimate is very close to (but statistically distinct from) our expectation for. The regression estimates a coefficient of (p < 0.001), with a 95 percent confidence interval from the estimate (see Table A1). Notice that 1 lies within this interval. The estimated constant of (p = 0.016) is statistically significant, but close to zero. Overall, the regression results indicate that participants in the RH set their prices slightly above the ABR. Nevertheless, the ABR strategy comes very close to describing participant behavior. Moreover, the coefficients for NH and NH are statistically insignificant, implying that adjustment behavior was identical across the RH and NH treatments after the initial post-shock period. 1 These percentages exclude observations for which a participant was in equilibrium in the previous period, because in equilibrium firms play their ABR by definition. Including these observations would have biased the results in favor of the ABR hypothesis. We also exclude these observations from our regression models in this section and Section 2. 2

3 Regressing model (A1) on the price data from the last T-1 periods of the positive shock experiments we find that participants also followed the ABR strategy in FT s RH + and NH + (see Table A2). The coefficients for NH + and are statistically insignificant at the 5 percent level, implying that participants adjustment behavior was identical in the NH + and RH + treatments. The coefficient for is (p < 0.001) and not significantly different from 1 at the 5 percent level. Finally, the estimated constant is statistically insignificant. The adjustment processes in the NH and NH + were so similar that it is hard to believe that the asymmetry in adjustment speed was primarily due to the natures of the monetary shocks. We look instead at the underlying economic environments in these treatments. Figure A1a shows the best responses of type x and type y firms to the average price of other firms in the post-shock phase of the negative shock experiments. For any, the vertical distance between a best response function and the 45-degree line indicates the amount by which a firm of that type should submit a price above (or below) the average price of his opponents. These vertical distances drive the dynamics of price adjustment. To see this, assume four firms who employ the ABR strategy and, in period, each face their own,. By definition, the average price of the group in that period,, is equal to 1 4,. Let represent the difference between, and the best response of firm to, (i.e., the vertical distance between the 45-degree line and firm s best response function). In period 1 each firm will submit a price equal to,, so that 1 4,. It follows that the difference in average prices between periods and 1 will be equal to, the average of the. Consequently, asymmetries in the absolute values of the will determine the rate at which a group of firms will reach the equilibrium if they follow the ABR strategy. When the are symmetrical (i.e., 0) the firms have reached the Nash equilibrium. For the majority of the set of in the post-shock phase of the negative shock experiments, the best response of type x firms is given by 5, while for type y firms it is 3. The net effect, in this range of, is that the average price will fall by one increment per period, provided participants are playing their ABR. This is in stark contrast to the best response functions in the positive shock experiments (see Figure A1b). In those experiments, for 3

4 most the best responses for the type x and type y firms are 1 and 7 respectively. This implies an adjustment rate of three price increments per period. Closer to the equilibrium the best responses are 2 and 6 for an adjustment rate of two increments per period. Thus, we would expect participants playing their ABR to converge to the equilibrium at a rate of two to three times that in the negative shock treatments. Section 2. Statistical Analysis of Adaptive Best Response in Our Revised Experiments Examination of the participant-level data suggested two main pricing strategies in the post-shock phase of our experiments. The most common was to roughly follow the ABR strategy, but a minority of participants (9 of 128) repeatedly chose their equilibrium price, even when doing so was not the best response to the average price in the prior period. (Of these nine participants, only one participated in the RH treatment; the remaining eight were divided evenly between the NH and NH +.) We refer to this practice as anchoring on the equilibrium. We categorize a participant as anchoring if, for a majority of the periods 1 in which his group was not in equilibrium in period 1, the participant set his price equal to the equilibrium when it was not the ABR to do so. We fit the data from our three treatments with human opponents to regression model A1, adding a third dummy variable, NH +, so that all three treatments may be analyzed simultaneously. The pricing data from those who anchored on the equilibrium was largely invariant with respect to the ABR. This tends to inflate the constant terms and depress the slope coefficients, despite the fact that only about 7 percent of participants could be described as anchoring. As a result, we report results from two models, the first using data from all participants (the full sample model) and the second excluding observations from participants who anchored on the equilibrium (the restricted sample model). In both models we exclude three observations in which the participant chose no price by the end of the period and a random price was generated for him. Table A3 displays the estimates. Both models fit the data extremely well, with R 2 statistics exceeding 0.88 in both cases. The hypothesis that participants in the RH followed the ABR strategy is well supported. When all data is included the constant term is statistically significant (p = 0.048) but very small. Excluding data from the anchoring participants renders the constant statistically insignificant (p 4

5 = 0.715). The estimated coefficient for is not significantly different from one whether we use the full sample ( 0.934, p < 0.001) or the restricted sample ( 1.001, p < 0.001). None of the interaction variables are statistically significant in either model, and except for the coefficient of NH + in the full sample model ( 3.280, p = 0.001), no dummy variables in either model are significant. Moreover, the significant coefficient for NH + in the full sample model is due to the fact that participants had different equilibrium prices to anchor on in the NH + than in the RH and NH. As a result, including data from the anchoring participants exaggerates differences in prices between the positive shock and negative shock treatments. Consequently, we are satisfied that price adjustment behavior was the same across all three treatments with human opponents. 5

6 Table A1. Results of Regression Model Comparing Actual Prices to the Adaptive Best Response in the Post-Shock Phase of FT s RH and NH treatments. Standard errors have been clustered by group Regressor Coefficient (Std. Err.) Periods percent Confidence Interval * (0.811) *** (0.120) NH (1.010) NH* (0.137) Obs. 739 Wald R * indicates significance at the 5 percent level ** indicates significance at the 1 percent level *** indicates significance at the 0.1 percent level 6

7 Table A2. Results of Regression Model Comparing Actual Prices to the Adaptive Best Response in the Post-Shock Phase of FT s RH + and NH + treatments. Standard errors have been clustered by group. Regressor Coefficient (Std. Err.) Periods percent Confidence Interval (1.50) ** (0.061) NH (1.924) NH + * (0.076) Obs. 298 Wald R * indicates significance at the 5 percent level ** indicates significance at the 1 percent level *** indicates significance at the 0.1 percent level 7

8 Table A3. Results of Regression Model Comparing Actual Prices to the Adaptive Best Response in the Post-Shock Phase of our revised RH, NH and NH + treatments. Standard errors have been clustered by group. Periods Data from all participants Anchoring participants excluded Regressor Coefficient (Std. Err.) 95 Percent Confidence Interval Coefficient (Std. Err.) 95 Percent Confidence Interval * ± ±0.338 (0.188) (0.173) *** ± *** ±0.040 (0.036) (0.021) NH ± ±1.084 (0.576) (0.553) * (0.073) (0.062) NH *** (1.008) (1.357) NH + * (0.072) (0.075) Obs Wald 36, , R * indicates significance at the 5 percent level ** indicates significance at the 1 percent level *** indicates significance at the 0.1 percent level 8

9 Figure A1. a) Best Response Functions in FT s Experiments with a Negative Monetary Shock, b) Best Response Functions in FT s Experiments with a Positive Monetary Shock a) 30 Best Response Firm Type x Firm Type y Average Price of Other Firms b) 30 Best Response Firm Type x Firm Type y Average Price of Other Firms 9

10 Appendix B. Software and Procedures Participants were seated in a computer laboratory and given a set of written instructions explaining the rules of the experiment, as well as a pencil and scratch paper. An experimenter read the instructions aloud, while screenshots highlighting the functions of the computer interface were shown on all participants computer screens. The experimenter paused at several pre-determined points in the instructions to answer questions. We used a computer interface to display participants income tables and allow them to enter their decisions. Features of the interface common to all treatments were as follows. The income table for each firm type was shown on a separate tab of the display window, and participants could switch between tabs to compare them. Payoffs in the income tables were designated by a white background for the table of a given participant s firm type, and a green background for the table of the opposite firm type. The prices a participant could charge were designated with a grey background in the first column of the table, and the 30 possible average prices of the other firms, which were also given a grey background, were displayed in the top row of the table. We provided participants in the nominal payoff treatments with an income converter on their computer displays. If a participant entered a hypothetical and, the income converter would display the real payoff from the income table that was currently displayed. This established a sort of parity in the difficulty of deflating nominal payoffs between FT s experiments and our revised versions, which employed a more complex nominal mapping. In the original study participants could find the real payoff by entering two numbers into an ordinary calculator: the nominal payoff and the average price of other firms. In our experiments participants also had to enter two numbers into the income converter to find the real payoff: their own price and the expected average price of the other three firms. Section 1. Computer Interface for Experiments with Human Opponents Participants selected a price by clicking on one of the prices in the first column of their own income table, which also highlighted the payoffs in the corresponding row in blue. They were allowed to switch prices as often as they liked within a period before finalizing their decision. The computer interface showed each participant the average price of the other firms in 10

12 At the outset, the experimenter explained that the experiment would consist of two phases, each of which would last for T periods and use a distinct set of income tables. Participants completed a practice period using the pre-shock tables before commencing the first of the 2T periods for which they were paid. Each period lasted up to 2 minutes with the exception of the practice period, which lasted up to 5 minutes. A period ended when all participants had submitted their prices or when the time ran out. If any participant had not selected a price prior to the end of the period, the computer software randomly chose a number from a discrete uniform distribution with support 1,,30 and submitted that as the participant s price for the period. After period T, the income tables on participants screens were populated with the post-shock payoffs. They were given 10 minutes to examine the new tables prior to the start of period T+1. A button on their computer display allowed participants to toggle between the pre- and post-shock tables in order to compare them. This button was disabled prior to period T+1. Section 5. Procedures for Experiments with Self as Opponent The procedures for our self-opponent experiments were the same as above with three exceptions. First, there were only two periods (one pre-shock and one post-shock), each of which allowed participants 15 minutes to select their prices. Second, the practice period was replaced with an instructions comprehension task that required the participants to calculate the, real income and (if appropriate) nominal income for a set of pre-determined prices from four firms. This task employed a novel set of income tables distinct from the pre- and post-shock tables used in the experiment. Finally, the time limit was not enforced by submitting random prices. Participants who had not submitted their prices within 15 minutes simply received a reminder on their screens to finalize their prices immediately. Shortly thereafter a lab monitor observed each computer terminal to ensure that each participant had chosen his prices. Section 6. Screenshots of the Computer Interface 12

13 Figure B1. Participant Interface in Experiments with Real Payoff Framing and Human Opponents 13

14 Figure B2. Price Submitted, Average Price of others Revealed in Experiments with Real Payoff Framing and Human Opponents 14

15 Figure B3. Participant Interface in Experiments with Nominal Payoff Framing and Computerized Opponents (Average Price of Opponents Conditional on Participant s Price Highlighted in Yellow) 15

16 Figure B4. Price Selected in Experiments with Nominal Payoff Framing and Computerized Opponents 16

17 Appendix C: Instructions for Experiments with Nominal Payoff Framing and Human Opponents General instructions for participants You are participating in a scientific experiment which is funded by Chapman University. The purpose of this experiment is to analyze decision making in experimental markets. If you read these instructions carefully and make appropriate decisions, you may earn a considerable amount of money. At the end of the experiment all the money you earned will be immediately paid out in cash. Each participant is paid \$7 for attending. During the experiment your income will not be calculated in dollars, but in points. The total amount of points you collect during the experiment will be converted into dollars by applying the following exchange rate: 50 points = \$1.00 During the experiment you are not allowed to communicate with any other participant. If you have any questions, the experimenter(s) will be glad to answer them. If you do not follow these instructions you will be excluded from the experiment and deprived of all payments aside from the minimum payment of \$7 for attending. [Questions?] Overview of the experiment The following is a brief description of the experiment. A more detailed description is given below. The experiment will last for a number of rounds. All participants are in the role of firms, selling some product. In this experiment, there are two types of firms: firms of type x and firms of type y. Each firm has to choose a selling price in every round. The income you earn depends on the price you choose and on the prices the other firms within your group choose. [Questions?] Detailed description of the experiment The image on your screen is a screenshot of the computer display you will use to make your pricing decisions. We will refer to this screenshot several times through the course of these instructions. The experiment is divided between two phases, the first of which consists of 15 rounds plus a practice round. You are not paid for the practice round. You should nevertheless take the practice round seriously since you may gain experience in this round. This experience helps you to make decisions in the other rounds in which you are paid. The second phase consists of an additional 15 rounds, but no practice round. 17

21 other three firms in her group was 17. Her nominal income of 476 points can be found in the green shaded cell, in row 20, column 17 of the white income table. [Questions?] Round information and the History Tab Your computer display will provide you with some important information throughout the experiment. The upper left portion of the display contains the following information: Round: The current round of the experiment. Note that Round 0 is the practice round. Time: The number of seconds remaining in the round. During the practice round you will have 300 seconds (5 minutes) to submit your price. During all other rounds you will have 120 seconds (2 minutes) to do so. You should be aware of two things regarding the time. First, when all participants have submitted their prices, the round will end regardless of how many seconds are remaining. Second, if time runs out before you have clicked the Submit Price button, the software will automatically submit the last price you clicked on during that round. If you have not clicked on any prices during that round, the software will choose a random number between 1 and 30, and submit that as your price. Round Income: The income, in real points, that you have earned in the current round. In the example on your screen, the participant s nominal point income is 476, and the average price of the other firms in her group is 17. The Round Income box displays her real income of 11 points; that is (476/17) 17. Total Income: The total income, in real points, that you have earned up to this point in the experiment. In the example on your screen, the Total Income box displays zero points. This is because the participant is in the practice round, Round 0, the results of which do not affect her earnings. [Questions?] In addition to this information, you can click on the History tab to find information from previous rounds. This includes the following: Round: The round in which you chose a price. Your Price: The price you set in that round. Average Price of Others: The average price set by the other three firms in your group in that round. Income: The income, in real points, that you earned in that round. 21

24 To return to the new income tables, click the same button (now labeled New Tables ) a second time. Keep in mind that in Phase 2 you cannot set your price using the original income table of your firm type. You must have the new table for your firm type visible in order to set the price for your firm. 24

25 Appendix D: Income Tables for All Firms in All Treatments Left Column Represents Firm s Price; Top Row Represents Average Price of Other Firms 25

26 Table D1. Negative Shock, Pre-Shock Phase, Real Frame, Type x Firm

27 Table D2. Negative Shock, Pre-Shock Phase, Real Frame, Type y Firm

28 Table D3. Negative Shock, Pre-Shock Phase, Nominal Frame, Type x Firm

29 Table D4. Negative Shock, Pre-Shock Phase, Nominal Frame, Type y Firm

30 Table D5. Negative Shock, Post-Shock Phase, Real Frame, Type x Firm

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