EC 2065 Lecture 8 PDF

EC 2065 Lecture 8 Ex ante and ex post optimality We have been discussing models under uncertainty and one thing that we might want to consider is optimality. When we have full information (certainty) then something that is optimal today will also be optimal tomorrow. However when we ha...

EC 2065 Lecture 8 Ex ante and ex post optimality We have been discussing models under uncertainty and one thing that we might want to consider is optimality. When we have full information (certainty) then something that is optimal today will also be optimal tomorrow. However when we have uncertainty, this need not be the case. A decision taken today, in an uncertain environment, can be regretted tomorrow when we know more about the states of the world. This is where a decision that is ex ante rational (optimal Examples When you leave education, you will move into the world of work. You will do so based on the information that you have on your skills, attributes of the job, and various other factors. You will choose the job that you believe, ex ante, best suits you. That is, the job that maximises your utility. Once you take the job, information is revealed, both about your suitability for the particular role but also you will find how well matched you are to the job in question. On the basis of that you will decide whether you would like to stay in the job or not, that is whether what was ex ante optimal is also ex post optimal. Approaches to economic analysis This links to the material that you were taught in last year’s approaches module. We use the same techniques to decide which job is most suitable as the physicists use to put a man on the moon. The difference is that in the physics setting, parameters are known. The moon will be where we expect it to be. In economics, parameters are not known or we have best guesses as to what they might be. Sequentially, what is optimal or seems optimal today, may turn out to be not so in the future. A stylised education model with stigma Staying on in education confers benefits in terms of higher expected future wages. However if one is breaking a social norm, that leaving school is what happens at 16, then staying on in education is associated with a stigma effect from one’s peers. We can model this using a hybrid of the models that we have considered thus far.  Suppose that the wage associated with leaving school is known and is associated with a lifetime income stream, y. This stream will vary according to the rate at which we discount the future. The utility to leaving school can then be written as If you stay on in school, then there is a random income stream associated with this (you may be lucky and get a very high paying job, or unlucky and not), y+z, where z is a random variable with mean value. However staying on results in a direct cost, C, and a stigma cost, S. The stigma cost could be positive, if the social norm is to leave school, or negative, if the social norm is to stay on. Note that the cost of education/stigma etc. can be made functions of ability and other factors.  The random utility associated with staying on is One stays on if Where is the known utility associated with leaving A second order Taylor series expansion around the mean return, , gives  You stay on if Taking expectations of the left hand side yields as. It is possible that different pupils have different information sets, possible that  Where I have used as the subjective expectation of the education premium. Break this down. The first two terms are the simple difference in utility streams. This is similar to the basic deterministic model. You invest in education if the return is positive. The difference is that here we have the difference in utility. Rates of time preference will be a factor here.  If we are using objective/mathematical expectations, then this value would be zero. However, if expectations are subjective, then some will be greater than the average and others will be less than. As the marginal utility is positive, expectations below the average will reduce the likelihood of staying on in school, whereas those above will increase the likelihood. This means that how expectations are formed and the source of information for those expectations is especially important. Pupils with no experience of post-compulsory education may have less reliable information. How will this affect the probability of staying on?  This term has two parts, the second derivative of the utility function and the expected variance of the return to schooling. Risk averse individuals will be less likely to stay on, the more risk averse the less likely. The larger the expected variance, the less likely, again if the individual is risk averse. Ex post and ex ante optimality again Under perfect certainty, individuals would invest in education early in the life cycle. Why? But we observe significant proportions of the population who return to full-time education having left prior. Why? This is an example of the divergence between ex ante and ex post optimality. The role of peers and externalities The fact that peers affect decisions in this setting is an example of the possible strength of extenalities. We will consider some further examples of this in the following. In the next example we will look at the work decision of two workers that share their output. The intuition is obvious. If you can rely on the other person to work, why would you supply effort. Alternatively, if the proceeds of your effort is going to be taken by somebody else, then why would you do it. The issue is that these kinds of effects can result in sub-optimal outcomes. An Example Suppose that production is a function of the effort exerted by two workers, A and B. A provides effort and B provides effort. Production is given by The agent’s utility function is given by , for both agents. De-centralised outcome In this outcome each agent maximises their utility independently. Of the jointly produced output, A gets share= and B gets share= (1- ) The first order condition for a maximum is or upon re-arranging , the effort of A is positively related to the effort of B but at a decreasing rate.  For B we have To solve this we simply solve this set of simultaneous equations. Substituting for in the equation above gives  Rearranging or or and As Note that since is a share, it is less than 1 and both and will be less than 1. Equal shares In this case we will have Which gives Centralised outcome (externality internalised) In this outcome we solve the problem for both agents. Substituting the constraint into the objective function, this simplifies to FOCs - - (1)/(2) yields = Substituting into either FOC gives -  Once the externality is internalised, the agents provide the amount of effort that yields the greatest benefit. To see this, compute utility in both cases. For agent B we will have Economy wide we have Decentralised welfare To calculate this we need to plug the values into the utility functions. Whilst this is a little messy (and is left for you as a homework problem) we find Which gives. Note that this is maximised at which gives More General The graph shows the privately rational decision as a function of the(expected) decisions of others. If all others are providing high effort, then it is optimal for the individual to follow suit. Similarly if low effort is provided it is optimal to do likewise Note that the middle equilibrium is unstable. If we perturb this, then individuals will choose either to reduce effort (shown) or increase it. This will eventually settle at one of the stable equilibria Real world examples The Big Push: Rosenstein-Rodan, Paul N. "Problems of Industrialisation of Eastern and South-eastern Europe." Econ. J. 53 (June-September 1943): 202- 11. Problem of lack of demand Not profitable for a single entrepreneur to invest as would make a loss But if all invest then demand will increase and this allows for development Reading: Industrialization and the Big Push Author(s): Kevin M. Murphy, Andrei Shleifer and Robert W. Vishny Source: Journal of Political Economy , Oct., 1989, Vol. 97, No. 5 (Oct., 1989), pp. 1003- 1026  The key in these models is that there is an externality. The original big push idea was discussed in terms of a shoe factory. It would be profitable only if all the workers bought shoes but that would not be the case, there were demand spillovers. So need a variety of manufacturers to invest simultaneously, to internalise the externality.  Key is Is there opportunity for strategic behaviour? Are there spillovers? Externalities MSV model 2 sectors, productive sector with Increasing returns to scale and competitive sector with constant returns to scale. Returns to scale are modelled by labour productivity. In competitive sector 1unit of labour produces 1 unit of output. In the productive sector, 1 unit of labour produces units of output  With no investment, workers work in the competitive fringe. Price and wage are set at 1. Investment incurs a fixed cost F, (e.g. takes some labour to build a machine) Firm faces a downward sloping demand curve. Profit is given by Where is labour productivity in productive sector, Y is revenue and is labour cost. Wage =1  This gives profit as If n firms invest, aggregate profit is National income , since w=1, wage income is given by the size of the labour force.  Using the value for aggregate profit , where L is total labour force and nF is the labour used to make machines. L-nF is the effective labour force. , a unit increase in Y increases profit by a. In aggregate profit increases by na. (1-na) represents a multiplier effect. In this framework there is no need for a big push.  When will there be a co-ordination failure? Suppose workers have to be paid more to work in the productive sector, that is suppose that. In this case investment will be zero if The condition for investment to take place is that profit is greater than zero And for no investment since Y=L in this case  For given values of it is possible for both conditions to hold. We need both If =12 and v=0.2, then the first condition is equivalent to F>10 and the second to F