Modelling Economic Decisions EC2065 Lecture 7 PDF

Modelling economic decisions  We now have the skills that we need to understand the construction of economic models. Identification of objectives Mathematically we know how to maximise or minimise the objective function. We may need to solve these first order conditions. We may nee...

Modelling economic decisions  We now have the skills that we need to understand the construction of economic models. Identification of objectives Mathematically we know how to maximise or minimise the objective function. We may need to solve these first order conditions. We may need to check that the solution is a maximum. We understand both decisions under certainty and those under uncertainty. EUT: Background It is possible that any uncertain decision can be analysed using expected utility theory. Marriage ( https://www.livemint.com/Sundayapp/Hn2CFTCtgFCb0O 9iTvzCSL/The-economics-of-marriage.html ) Divorce https://www.parisschoolofeconomics.eu/IMG/pdf/Magali. pdf Suicide https://www.gwern.net/docs/psychology/2004-becker.pdf  One of the first papers to utilise this framework was Becker’s model of crime https://www.jstor.org/stable/1830482 Or you can find it here https://www.nber.org/system/files/chapters/c3625/c3625.pdf This paper contains much more than the simple model that we will consider here. A Model of Crime Begin by assuming that everyone could be a criminal. Each person is an expected utility maximiser. That means that given an uncertain situation, the individual chooses the action that maximises his or her expected utility. Associated with any action we have a payoff. This payoff may be certain, we know with certainty what we are going to receive, or it may be uncertain (we have an idea or expectation of what we will receive but do not know this with certainty). Crime If someone commits a crime, then they will receive some payoff, B, and this occurs with probability p. If the crime is unsuccessful a fine F is imposed. This happens with probability (1-p) There may be uncertainty over any aspect of this decision. Probabilities may be subjective, rather than objective. Expected utility associated with crime Assuming that the assumptions regarding EUT are satisfied, we can write the expected utility associated with committing a crime as The utility received if one does not commit a crime is the certain utility. Commit a crime? The condition for committing a crime is Commit if, We can write the (utility) gain from crime as It is this function that has been referred to as the rational theory of crime. Comparative statics In simple terms this just entails taking the partial derivative of the gain with respect to each of the arguments of the function B This is greater than zero as both marginal utility and the probability of not being caught are greater than zero  F This is less than zero as both the probability of being caught and marginal utility are both positive. Increasing the fine will reduce the probability of crime.  p- probability of getting away with the crime As we assume that utility is increasing in income U(w+B)>u(w-F) so this derivative is positive. An increase in the probability of getting away with the crime, increases the gain and therefore the amount of crime. Of course the key here is to invert the above. If we reduce this probability, that is increase the probability of detection, then the gain will reduce What about Wealth? We can write the (utility) gain from crime as Wealth-w This is difficult to sign. We have a linear combination of marginal utilities compared to a given marginal utility. Cases: marginal utility constant, risk neutral. In this case we have a linear utility function, U=a+bw. The marginal utility at any wealth is given by b, in which case we have Wealth has no effect on the probability of crime  Marginal utility is falling, risk aversion In this case we have Risk aversion is equivalent to diminishing marginal utility of income, thus we know that So the first term is negative. As we also know that w>(w-F), then. The second part is positive It is tempting to say that this will be negative unambiguously, that risk averse individuals will be less likely to commit crime and risk loving individuals will be more likely but as we can see, the result is ambiguous. Another way to look at this The gain is given by If this is positive, crime is committed. Re-writing as (notice that these are the same), we get Factorising gives Multiplying term 1 by B/B and term two by F/F gives  or So we can write If this is greater than zero, we will commit a crime.  Since by concavity, then we must have that the expected gain pB must be greater than the expected fine (1-p)F as follows The greater the degree of risk aversion, the greater the difference in MU and the greater the difference between Bp and (1-p)F has to be. Basic takeaways from the rational choice model To Reduce crime Increase the punishment/fine Increase the probability of being caught, more police/surveillance etc. Reduce inequaility? Reduce the gains from crime Testing the model Microeconomic data: We would need data on people who had committed a crime and those who had not. Ideally this data would be available from before the time that they had committed a crime and was collected after the crime had been committed Life cycle data, for example the National Child Development Survey wave 8, 2004 (46 years old) included questions related to crime. Problem, those who commit crime are less likely to be involved in the panel at that age Testing the model Macrodata: this is a more common approach. There has been some support for the Becker model. An example of such a paper, available on canvas is, Testing the Theory of Rational Crime with United States Data, 1994-2002 ICPSR Bulletin (Inter-University Consortium for Political and Social Research), Vol. 27, No. 3, 2007 This is an undergraduate Princeton dissertation Extending the model Whilst the model as written , with monetary gains and losses, would appear to be a model of theft or financial crime, the basic model could be applied to any crime. The key is that a crime will be committed, if, at that moment, the utility gain is greater than the expected loss. It does not mean that the decision has to be time consistent or that ex ante and ex post rationality have to coincide. Utility does not have to be defined over just materialistic items.  Economics as a ‘magpie’ science  Economics as a science progresses by a continual process of model building and testing. When models are not supported by the data, we need to alter the model or begin again. Sometimes this means looking outside the discipline into other areas, Psychology, Sociology As such economists could be accused of acquiring ideas from other areas. This is a strength of the discipline.  Behavioural economics rejects the assumption that economic agents are atomistic. My decisions are affected by your decisions and vice versa. When this is included in the model, it can lead to complexity. We will consider two elements, the first is hassle and the second is stigma effects. We will consider these in a relatively general way; hassle is a common issue in many situations as is stigma. Hassle model Hassle is any cost (usually a psychic cost) that is associated with the hassle involved in making a decision. Examples would be form filling that could result in a positive monetary reward. True story: Recently I was told that I had overpaid national insurance (two years ago). To claim this back I needed; 1. Two p60s, one from my current employer, one from my previous. 2. To fill in an online form As yet, still to be completed. Why? Marginal utility of the extra income is less than the (perceived) hassle associated with the application.  This could also be an example of uncertainty aversion. I know that I will get some refund but I do not now how much it will be. My subjective belief is that it will not be very much but that is not based in fact. The uncertainty over the refund reduces its value and thus reduces the probability of claiming. See also the advert for ‘SMART INSURANCE’ on TV. Sells policy on the basis of ‘no complicated forms’. Model: We will consider the problem of a student that is considering whether to apply for a bursary or not. The probability that the application will be successful is p, so unsuccessful=(1-p) If you are successful you gain monetary reward, B The application involves a utility loss, hassle (H), which you bear whether you are successful or not. Where x is current income. You will apply if  The student applies if , expand the brackets and –U(x) , cancel U(x) and factorise , multiply by B/B Distribute the B/B Consider. This is the change in utility of B, divided by B, which is approximately the marginal utility of the extra income B,  So we will apply if Rearranging That is, apply if pB, the expected monetary gain is greater than the monetary cost of the hassle. An interesting question is whether hassle is experienced differently by different groups of students.  Hassle could be related to the time and effort involved in collating documents It could be related to the language that is used in the forms. Suppose that a student’s background, proxied by income, x, affects these parameters. We could write that indifference would be defined by , is defined to be the income at which there is indifference. The question as to the effect of income on claiming will depend upon its effect on the left hand side and right hand side of the equation.  For example if the subjective probability of success is increasing in income (greater familiarity with language used in forms, supporting statements, for example), then this will make students from income groups more likely to apply. It may also be the case that hassle costs are reducing in income, for the same reason as above, this would again explain greater applications. However, it is also possible to argue that the reverse could be true, which would explain more applications from lower income backgrounds. Diagrammatically it depends whether the LHS and RHS are increasing or decreasing in income Case (i) benefit is increasing in income, Costs are falling with income, income greater than apply. Case (ii) the opposite, benefit is decreasing and costs are increasing, now income less than apply Case (iii) more complex. Benefit increasing first and then falling, costs increasing. Allows more complex outcome. Stigma model: Union membership/asking questions/benefit take up The stigma model is similar to the hassle cost model. Can be used to investigate benefit take-up, union membership, education. Associated with an action is a stigma cost. For example, being labelled a benefit scrounger or free- rider. The key difference is that stigma is endogenous, it is related to the actions of all. If everyone is claiming benefit, there is no stigma If no-one is a member of the union, there is no stigma. Model Let income be x and the associated utility be u(x). There is some benefit that is associated with an action, B. Let us suppose that this is a model of union membership but this could easily be any other decision which involves stigma, for example the decision to run a red light. One may join a union at a cost or one may choose to free-ride on union wages (higher on average). If one free rides, you receive the union wage premium, B but there is a probability (q) that you will identified as a free rider. If this is the case, then you incur a stigma cost, S.  The expected utility of the free rider is given by This should be compared with the non-union wage U(x). Proceeding as we did for the hassle model, we again derive a result that we will free ride if The utility gain from free riding is greater than the expected stigma. If this is not the case, then the person joins the union.  Once again we can think of an income level where this will hold with equality, the individual will be indifferent. However in this case the outcome is somewhat different to the case with hassle. Stigma is a function of the number of people who are engaging in the stigmatised action. If one extra person engages in the action, then the stigma associated with the action falls. If one extra person does not engage, then the stigma of being caught rises. The solution with indifference is possible but unstable A diagram will help to see this clearly  In the diagram, the horizontal line represents the income associated with indifference. However this point is not stable. If one U() extra person joins the union, then expected stigma increases and this causes the indifferent person to join. This increases expected U() stigma again and another joins and so on, we end at U(x+B). If however we have an extra free-rider, this U(x) reduces expected stigma and results in an unravelling in the opposite direction, no- one joins the union. In this case the outcome is that there is no union wage premium. Proportion of union members Revisiting Crime There is a large literature in the economics of crime that seeks to identify peer effects as a explanator of crime. A review of this literature can be found here, https://www.elgaronline.com/edcollchap/book/97817899 09333/book-part-9781789909333-14.xml. How would we model this? Adding stigma to the crime model The gain from crime, including a stigma effect would be The effect of stigma depends upon the sign of the stigma effect. For individuals breaking a social norm of criminal behaviour, the effect would be negative and would make crime more likely. Individuals for whom the social norm is no crime would receive positive stigma from committing a crime, so the social norm would tend to reduce crime in this case.