Advanced Macroeconomics An Easy Guide Consumption under uncertainty and macro finance

CHAPTER 12 Consumption under uncertainty and macro In the previous chapter we discussed optimal consumption in a world with certainty . The results cally came down to having people choose a consumption path as stable as possible . To estimate this sustainable level of consumption they take into account future income , net of the they plan to hand over to their children . There are two dimensions in which this strong result may be challenged . One has to do with . Uncertainty may affect expected future income or the return of assets . The second is about preferences themselves . What happens if people have an unusually high preference for present sumption ?

We will discuss both problems in this chapter . We will see that uncertainty changes the conclusion in a fundamental way it tilts the path upwards . Faced with uncertainty , people tend to be more cautious and save more than the permanent income hypothesis would suggest . Present bias delivers the opposite result , that people tend to overconsume and enter time inconsistent tion paths . This rises a whole new set of policy implications . We end this chapter by introducing a whole new topic , here succinctly sketched to get a . In traditional , we typically study portfolio ( the realm of asset management ) or ( the realm of corporate ) based on asset prices . But these asset prices have to make sense given the desired consumption and saving decisions of the individuals in the economy The area of macro puts these two things together . Because asset demands derive directly from tion decisions , we can the problem and ask given the consumption decisions what are the asset prices ?

The area of macro has been a very fertile area of research in recent years . Consumption with uncertainty Consumption with uncertainty needs to deal with the uncertainty of future outcomes . The value tion ( will be a useful instrument to estimate optimal sumption paths . The analysis of consumption under uncertainty is analogous to that under certainty with the difference that now we will assume that consumers maximise expected utility rather than just plain utility As it How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . 12 . Consumption under uncertainty and macro , London Press . DOI License .

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE turns out , it is more convenient to analyse the case with uncertainty in discrete , rather than continuous , time . The utility that the consumer in this case is max ?

17 , The uncertainty comes from the fact that we now assume labour income , to be How do we model individual behaviour when facing such uncertainty ?

When we impose that individuals use the mathematical expectation to evaluate their utility we are assuming that they have rational expectations they understand the model that is behind the uncertainty in the economy , and make use of all the available information in making their forecasts . Or , at the very least , they don know any less than the economist who is modelling their behaviour . As we will see time and again , this will have very powerful implications . Let us start with a model , not unlike the one that we used when analysing the model . As you will recall and can easily verify , the looks like this ( This to the case of many periods , with exactly the same economic intuition ( This is our equation for optimal consumption . To see how this helps us find the consumption level in a framework , we use the tools of dynamic programming , which you can review in the math appendix at the end of the book . We show there that problems can be solved with the help of a Bellman equation . The Bellman equation rewrites the problem as the choice between current utility and future utility . Future utility , in turn , is condensed in the value function that gives the maximum attainable utility resulting from the decisions taken today In short , Max , QE , The condition of the Bellman equation ( maximise relative to , and use the budget straint ) is 145 , Ia , but remember that ' along the optimal path . The intuition is that when the value function is relative to consumption , the marginal value of the program along the path has to be the marginal utility of consumption ( see our mathematical appendix to refresh the intuition ) But then ( 126 ) becomes ( In a nutshell , the key intuition of dynamic programming , captured by the Bellman equation is that you can break a ( potentially infinite ) problem into a sequence of problems where you choose today , making sure that your decisions today make sense when measured against future utility , and then again all the way to eternity if necessary .

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE 173 The random walk hypothesis VVith quadratic utility we find that , the random walk hypothesis of consumption . Changes in consumption levels should be unpredictable . Suppose utility is , that is . Here things become a bit simpler as marginal utility is linear ( act . This implies that ( ac , If we keep assuming that as we ve done before , it follows that ac , ac , or , more simply , that , Equation ( can be depicted as the following stochastic process for consumption , where , is a random disturbance ( also called white noise ) A stochastic process that looks like this is called a random walk , for this reason this description of consumption ( due to Hall 1978 ) is called the random walk of consumption . It is avery strong statement saying that only unexpected events can change the consumption all information that is already known must have already been taken into consideration and therefore will not change consumption when it happens . This result , one of the early applications of the rational expectations assumption , is a powerful empirical implication that can easily be tested . Testing the random walk hypothesis Empirical evidence does not support fully the random walk hypothesis of consumption . A large number of papers have tried to assess the random walk hypothesis . One classical contribution is the Shea ( 1995 ) test on whether predictable changes in income are or are not related to predictable changes in consumption He looks into union contracts which specified in advance changes in wages . He then runs the consumption growth on the income growth . The theory suggests the should be zero , but the number comes out to be . Of course it can very well be that this is because people have liquidity constraints So Shea runs the test on people that have liquid assets and could thus borrow from themselves . These people can not have a liquidity constraint . Yet he still the same result . Then he splits people into two groups those that are facing declining incomes and those for which income is growing . Those facing

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE a future fall in income should reduce their consumption and save , so you should not an effect of liquidity constraints . Yet , it seems that , again , changes in current income help predict changes in consumption . This type of exercises has been replicated in many other . and Noel ( 2019 ) for example , that household consumption falls 13 percent when households receiving ment reach the ( anticipated ) end of their . Food stamp recipients and Social security also show monthly patterns of consumption that are related to the payment cycle . function The value function is a useful tool to estimate optimal paths . VVe review two approaches to solve for these paths guess and replace and value function iteration . While important , the quadratic case is a very special case that allows a simple of the consumption path . Can we solve for more general ?

Here is where the value function approach comes in handy There are several ways of using the value function to approximate the mal path . If the problem is , one can work the problem backwards from the last period . But this is not very useful in problems with no terminal time , which is our typical . One way to approach the problem is to simply guess the value function . This can be done in simple cases , but is not typically available , particularly because no problem should rely on having a genius at hand that can out the solution beforehand . An alternative is to do an iteration process that the tion through a recursive estimation . This is easier , and may actually deliver a solution in some cases . However , this approach can also be implemented by a recursive estimation using devices . So that you get a sense of how these methods work , we will solve a very simple problem through the guess and replace solution , and then through the value function iteration method . It is a bit tedious but will allow you to get a feel of the methodology involved . A guess and replace example Imagine we take the special case of ( log ( and guess ( spoiler we already know it will work ! the form ( that is , with a form equal to utility and with constants a and to be determined , If this is the value function , then consumption has to maximise ) Remember that 17 , 17 , as in ( 122 ) where we just assumed to be zero to lighten up notation . Now take the derivative of ( relative to , We leave this computation to you but it is easy and you should that this gives (

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE 175 Are we done ?

No , because we need to the value of . To do this we are going to use our guess ( using ( This means writing 17 la ( i ( What we have done is write the value function on the left and replacing optimal consumption from ( and la . from the budget constraint ( using optimal consumption again ) The expectation goes because all variables are now dated at Now the log makes things simple . factor out the logs on the right hand side and pile up all the coefficients of 112 on the side . If the value function is right , these should be equal to a the of la . in the value function on the left . After you clear this out , you will get the deceptively simple equivalence , which is an equation that you can use to solve for . Trivial algebra gives that a which , in ( gives our solution . This , by now is an expected result . You consume a fraction of your current wealth . The log tion cancels the effect of returns on consumption and thus the solution ) Iteration Now , lets get the intuition for the solution by the value function . Let imagine we have no idea what the value function could be , so we are going the make the arbitrary assumption that it is zero . Let us track the iteration by a subindex on the value function , So , with this assumption . So our iteration implies that ( log ( subject to the budget constraint in ( The solution to this problem is trivial . As assets have no value going forward , so our log ( Now lets iterate to the second stage by using . This means log ( log ( log ( log ( Again , maximise this value function relative to , This is not complicated and you should get that ( The more tricky part is that we will use this to compute our equation . Replace in ( to get Notice that the log will simplify things a lot so this will end up looking like log ( log ( log ( log (

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE The important part is the one that multiplies , the other is a constant which we see quickly becomes unwieldy . To the solution lets try this one more time , Our last iteration uses our to compute ( we omit the constant term ) log ( log ( I 10 Use again the budget constraint and maximise respect to You should be able to that , an ( 1224 ) We leave , for the less fainthearted , the task of replacing this in ( to compute the version of . Fortunately , we do not need to do this . You can see a clear pattern in the solutions for . If you iterate and iterate to , the denominator will add up to This implies that the solution is 17 , Not surprisingly , the same as in ( 1218 ) faced with uncertainty consumers will be more precautionary , tilting the consumption upwards throughout their lifetimes . The Caballero model provides a simple that that slope and shows how it increases with volatility . Let ask ourselves how savings and consumption react when uncertainty increases . Our intuition that an increase in uncertainty should tilt the balance towards more savings , a phenomenon dubbed precautionary savings . To illustrate how this works we go back to our equation ( Assume again that before ! to simplify matters , Thus , the condition reduces to ( we ve seen this ( 1225 ) Now assume , in addition to the typical and , that , This last condition is new and says that marginal utility is convex . This seems to be a very realistic assumption , It means that the marginal utility of consumption grows very fast as consumption approaches very low levels . Roughly speaking , people with convex marginal utility will be very concerned with very low levels of consumption . Figure shows how marginal utility behaves if this condition is met . Notice that for a quadratic utility ( 1227 ) But the graph shows clearly that if marginal utility is convex then ( and that the stronger the convexity , the larger the difference . The bigger ( is , the bigger , needs to be to keep the expected future utility equal to ( the marginal utility of consumption today . Imagine , for example that you expect one of your consumption possibilities for next period

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE Figure Precautionary savings ( to be zero . If marginal utility at zero is co then ( will also be co , and therefore you want to increase future consumption as much as possible to bring this expected marginal utility down as much as possible . In the extreme you may choose not to consume anything today ! This means that you keep some extra assets , a buffer stock , to get you through the possibility of really lean times . This is what is called precautionary savings Precautionary savings represents a departure from the permanent income hypothesis , in that it will lead individuals to save more than would be predicted by the latter , because of uncertainty The Caballero model Caballero ( 1990 ) provides a nice example that allows for a simple solution . Consider the case of a constant absolute risk aversion function . 1229 ) Assuming that the interest rate is equal to the discount rate for , this problem has a traditional equation of the form , Caballero proposes a solution of the form , were is related to the shock to income , the source of uncertainty in the model Replacing in the equation gives ,

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE which , taking logs , to er , log , 02 If is distributed ( then we can use the fact that to the value ofF ( as the value is constant , we can do away with the subscript ) in ( 1233 ) 91 log eH , or , simply , 903 ( This is a very simple expression . It says that even when the interest rate equals the discount rate the consumption is upward sloping , The higher the variance , the higher the slope . The precautionary savings hypothesis is also useful to capture other stylised facts families tend to show an consumption path while the uncertainties of their labour life get sorted out . Eventually , they reach a point were consumption and they accumulate assets , and Parker ( 2002 ) describe these dynamics . Roughly the pattern that emerges is that families have an increasing consumption pattern until sometime in the early , after which consumption starts to . New frontiers in consumption theory Consumption shows deviations from the optimal framework . One such deviation is explained by early bias , a tendency to give a stronger weight to present consumption . This leads to time inconsistency in consumption plans . Consumption restrictions , such as requesting a stay period before consumption , may solve the problem . Though our analysis of consumption has taken us quite far , many consumption decisions can not be suitably explained with the above framework as it has been found that consumers tend to develop biases that move their decisions away from what the model prescribes . For example , if a family receives an extra amount of money , they will probably allocate it to spending on a wide range of goods and maybe save at least some of this extra amount . Yet , if the family receives the same amount of extra money on a discount on food purchases , it is found that they typically increase their food consumption more ( we could even say much more ) than if they would have received cash . Likewise , many agents run up debts on their credit cards when they could pull money from their retirement accounts at a much lower cost . One way of understanding this behaviour is through the concept of mental accounting , a term coined by Richard Thaler , who won the Nobel Prize in economics in 2017 . In Thaler view , consumers mentally construct baskets of goods or categories . They make decisions based on these categories not as if they were connected by a unique budget constraint , but as if they entailed totally independent decisions . A similar anomaly occurs regarding defaults or reference points which we mentioned at the end of our Social Security chapter . Imagine organising the task of allocating yellow and red mugs to a

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE group of people . If you ask people what colour they would like their mugs to be , you will probably get a uniform distribution across colours , say 50 choose yellow and 50 choose red . Now allocate the mugs randomly and let people exchange their mugs . When you do this , the number of exchanges is surprisingly low . The received mug has become a reference point which delivers utility . This type of reference point explains why agents tend to stick to their defaults . Brigitte has shown that when a savings contribution was imposed as default ( but not compulsory ) six months later 86 of the workers remained within the plan , relative to 49 if no plan had been included as default , and 65 stuck to the contribution only choosing that contribution when such percentage was not . In fact , shows that the effect of defaults is much stronger than providing economic incentives for savings , and much cheaper ! One of the biases that has received attention is what is called present bias . Present bias is the tendency to put more weight to present consumption relative to future consumption . Let discuss the basics of this idea . We follow ( et al . 2006 ) in assuming a model with three periods . In period zero the consumer can buy ( but not consume ) an amount of a certain good . In period one , the consumer can buy more of this good ( and now consume it , Total consumption is . In period , the consumer spends whatever was left on other goods . The budget constraint can be written as ( where and are taxes over and , and is a lump sum transfer . Income is assumed equal to . 5010 5111 , where the bars indicate the average values for each variable . As the economy is large , these variables are unchanged by the individual decision to consume . Introducing taxes and lump sum transfers is not necessary , but will become useful below to discuss policy . Summing up , the structure is Period buys co at after tax price of ( Period buys an additional amount , at an after tax price of ( Consumes co Period buys and consumes good at price with the remaining resources ( 10 ) 11 ) Time inconsistency in consumer behaviour The key assumption is that the consumer has a discount factor with sequence , We assume to capture the fact that the consumer discounts more in the short run than the long run . As we will see , this will produce preferences that are not consistent over time . In addition , we will assume the good provides immediate satisfaction but a delayed cost ( a good example would be smoking or gambling ) Let assume that the utility of consuming is ( a A ) log (

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE where A and I are fixed . The utility from is assumed linear , as it represents all other goods . For simplicity , we assume . Expected utility as seen in period zero is ( Notice that the delayed consumption penalty disappears when seen from afar . In period , the utility function is ( A a ) log ( Ia log ( Notice that relative utility between the good and is not the same when seen at time and when seen at time . At period zero , the other goods were not penalised relative to , but from the perspective of period the of consumption are stronger because satisfaction is immediate relative to the delayed cost and relative to the utility of other goods to which the present bias applies . This will lead to time inconsistency Imagine consumption is determined at time zero and for now 10 11 . This would give the optimal consumption . Maximising ( 1239 ) subject to ( can easily be shown to give . Notice that this implies A and A . Thus , expected utility as of period is ( A log ( A ) A ) This will be our benchmark . The free equilibrium Keeping , now imagine that consumption is chosen in period . This is obtained ing ( subject to ( This gives ( ar ( the case . From the perspective of period , the marginal utility of is now smaller than the utility of consuming . Thus , this free equilibrium is not optimal at least from the perspective of period zero . Notice that now which can easily be shown to be higher than the value obtained in Optimal regulation Las Vegas or taxation ?

Are there policies that may restore the equilibrium from the perspective of period zero utility ?

One option is an early decision rule that allows the purchase of only during period zero . This is like having an tax in period . A application of this policy is , for example , to move gambling activities far away from living areas ( Las Vegas ) This way , the consumer decides

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE on the consumption without the urgency of the instant satisfaction . This case trivially replicates the case above and need not be repeated here . This outcome can also be replicated with optimal taxation . To see how , let consider a tax policy of the form 10 . In order to solve this problem , let revisit the of ( 1240 ) subject to ( The solution for gives ( To obtain the optimal I , replace ( in ( and maximise with respect to The first order condition gives ( A ( which gives the optimal tax rate ( which delivers A , replicating the optimal equilibrium . So a tax policy can do the trick . In fact , with no heterogeneity both policies are equivalent . Things are different if we allow for heterogeneity Allow now for individual differences in A . We can repeat the previous steps replacing A with A , and get the analogous conditions A ( A ( which gives the same tax rate ' With heterogeneity , consumers will move towards the first best but faced with a unique tax rate will consume amounts . Notice that ( A log ( A ) A ) for all A , which happens to the extent that A A . As this happens for a nonzero mass of consumers if heterogeneity is going to be an issue at all ( A ) A . The result is quite intuitive , as each consumer knows its own utility an early decision mechanism is superior to a tax policy because each individual knows his own utility and attains the best with the early decision mechanism . These biases have generated attention in recent years , generating important policy .

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE and typical corporate uses asset prices to explain investment or decisions , attempts to understand asset prices in a general equilibrium format , ie in a way that is consistent with the aggregate outcomes of the economy . The basic pricing equation , is remarkable expected returns are not associated with volatility but to the correlation with the stochastic discount factor . We ve come a long way in understanding consumption . Now it is time to see if what we have learnt can be used to help us understand what asset prices should be in equilibrium . To understand this relationship , we can use Lucas ( 1978 ) metaphor imagine a tree that provides the economy with a unique exogenous income source . What is this tree worth ?

Optimal consumption theory can be used to think about this question , except that we turn the analysis upside down . cally , we would have the price of an asset and have the consumer choose how much to hold of it . But in the economy the amount held and the returns of those assets are given because they are what the economy produces . So here we will use the to derive what price makes those exogenous ings optimal . By looking at the at a given equilibrium point as an asset pricing equation allows us to go from actual consumption levels to asset pricing . Lets see an example . Start with the first order condition for an asset that pays a random return ' Vi . Remember that ( xy ) so , applying this equation to ( we have that ( This is a remarkable equation . It says that you really don care about the variance of the return of the asset , but about the covariance of this asset with marginal utility . The variance may be very large , but , if it is not correlated with marginal utility , the consumer will only care about expected values . The more positive the correlation between marginal utility and return means a higher side , and , therefore , a higher value ( more utility ) Notice that a positive correlation between marginal utility and return means that the return is high when your future consumption is low . Returns , in short , are better if they are negatively correlated with your income and if they are , volatility is welcomed ! As simple as it is , this equation has a lot to say , for example , as to whether you should own your house , or whether you should own stocks of the company you work for . Take the example of your house . The return on the house are capital gains and the rental value of your house . Imagine the booms . Most likely , prices of property and the corresponding rental value goes up . In these cases your marginal utility is going down ( since the boom means your income is going up ) so the lation between returns and marginal utility is negative . This means that you should expect housing to deliver a very high return ( because its hedging properties are not that good ) Well , that right on the dot . Remember our mention to Kaplan et al . 2014 ) in Chapter , who show that housing has an amazingly high return . There may be other things that play a role in the opposite direction , as home ownership provides a unique sense of security and belonging , which our discussion of precautionary

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE 183 savings indicates can be very valuable . Buying stocks of the you work in is a certain no go , so , to rationalise it , you need to appeal to asymmetric information , or to some cognitive bias that makes you think that knowing more about this asset makes you underestimate the risks . In fact , optimal icy would indicate you should buy the stock of your competitor . So far , we have been thinking of interest rates as given and consumption as the variable to be determined . However , if all individuals are , equilibrium returns to assets will have to satisfy the same conditions . This means we can think of equation ( as one of equilibrium returns . To make things simple , assume once again that ( Then ( 1253 ) becomes ( i ) which can also be written as ' Notice that for a asset , for which ( we will have ( Before proceeding , you may want to ponder on an interesting result , Notice that in the denominator you have the expected marginal utility of future consumption . This produces two results . If tion growth is high , the interest rate is higher ( if , is big , its marginal utility is low ) But at the same time , notice that if the volatility of consumption is big , then the denominator is bigger ( remember our discussion of precautionary savings ) To see this , imagine that under some scenarios , falls so much that the marginal utility becomes very large . In higher volatility economies , the rate will be lower ! So , using ( in ( we obtain ( This equation states that the of an asset is determined in equilibrium by its covariance with aggregate consumption . VVe show that the basic pricing equation can be written as ) rind . Risk depend on the assets covariance with market returns with a coefficient called that can be computed by running a regression between individual and market returns . This is the capital asset pricing model (

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE Consider now an asset called the market that negatively with marginal utility of tion ( as the market represents the wealth of the economy , consumption moves with it , and therefore in the opposite direction as the marginal utility of consumption ) That is , Applying ( 1258 ) to this asset , we have var ( I Consider now an individual asset with return . Applying the same logic we have that , Um ! CHIN Combining both equations ( just replace , from ( into ( we have that , Var ( You may have seen something very similar to this equation it is the , used to mine the equilibrium return of an asset , The formula says that the asset will only get a for the portion of its variance that is not , An asset can have a very large return , but if the tion of the return with the market is zero , the asset will pay the rate in spite of all that ity ! Another way of saying this is that all idiosyncratic ( by holding a large portfolio ) risk is not paid for . This is the reason you should typically not want to hold an individual asset it will carry a lot of volatility you are not for . The popularity of the model also hinges on how easy it is to compute the slope of the risk it is just the regression obtained from running the return of the asset ( relative to the risk free ) and the market return . The value of that coefficient is called . This version , derived from the optimal behaviour of a consumer under uncertainty , is often referred to as ( Equity premium puzzle The for equities is given by ( But this does not hold in the data unless risk aversion is unreasonably high . This is the equity premium puzzle . Our asset pricing model can help us think about some asset pricing puzzles that have long left economists and finance practitioners scratching their heads , One such puzzle is the equity premium puzzle . The puzzle , in the , refers to the fact that equities have exhibited a large ( around on average ) relative to bonds , and this has remained relatively constant for about 100 years . As equities are riskier than bonds a is to be expected . But does make sense ?

If an asset earns more than another , it means that the asset value will be 80 higher at the end of 10 years , 220 more at the end years , 1740 higher at the end years , and higher at the end of 100 CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE Figure Equity premium puzzle , from and ( 1999 ) I We I ' I Hungary ' Norway I Finland Ir a I Germany tra Netherlands at I France I Italy I Brazil New Zealand I I Belgium I Portugal , I Pakistan I I Spam India ' Egypt I Philippines I Poland I Columbia I Argentina I Peru I Greece I I I I I I I I 20 40 60 80 100 Years of Existence since Inception years ! You get the point there is no possible risk aversion that can deliver these differences as an equilibrium spread . Figure , taken from and ( 1999 ) shows that the equity premium puzzle is a common occurrence , but does not appear in all countries In fact the seems to be the country where its result is most extreme . To have more , we need to move away from a quadratic utility function and use a more general utility function instead Now our looks like , which can be written as . I Take a second order expansion of the term within the square brackets on the at ( notice that in this case the usual Ar becomes , and Ag becomes )

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE At ( but keeping the deviations ) this to ( With this result , and using ( 1252 ) we can approximate ( 1264 ) as 50 ) where we can drop the quadratic terms ( and ( as these may be exceedingly small . This again to ( 95 ( em ( is ( var ( For a risk free asset , for which ( the equation becomes ( var ( which again shows the result that the higher the growth rate , the higher the risk free rate , and that the bigger the volatility of consumption the lower the risk free rate ! Using ( in ( 1268 ) we obtain ( This is the risk for an asset i . We will see that this equation is incompatible with the observed spread on equities ( per year ) To see this , notice that from the data we know that 63 , and , 140 . This implies that , Now we can plug this into ( to get that the following relation has to hold , 00024 ( and this in turn implies 25 , which is considered too high and incompatible with standard measures of risk aversion ( that are closer to ) and Prescott ( 1985 ) brought this issue up and kicked off a large of literature on potential explanations of the equity premium . In recent years the premium seemed , if anything , to have increased even further . But be careful , the increase in the may just the convergence of the prices to their equilibrium without the . So we can really say how it plays out from here on . What next ?

Perfect or not , the idea of consumption smoothing has become pervasive in modern . Many of you may have been taught with an undergraduate textbook using a consumption function a bY , with a marginal propensity to consume from income equal to 17 . Modern , both in the version with and without uncertainty , basically states that this equation does not make much sense , Consumption is not a function of current income , but of wealth , The distinction is important because it affects how we think of the response of consumption to shocks or taxes , A permanent tax increase will imply a one to one reduction in consumption with no effect on aggregate spending , while transitory taxes have a more muted effect on consumption . These

CONSUMPTION UNDER UNCERTAINTY AND MACRO FINANCE differences are indistinguishable in the traditional setup but essential when thinking about policy The theory of consumption has a great tradition . The permanent income hypothesis was stated by Milton who thought understanding consumption was essential to modern . Most of his thinking on this issue is in his 1957 book A Theory of the Consumption Function ( 1957 ) though this text , today , would be only of historical est . The life cycle hypothesis was presented by and ( 1954 ) again , a historical reference . Perhaps a better starting point for those interested in consumption and savings is Angus ( 1992 ) Understanding Consumption . For those interested in exploring value function you can start easy be reviewing Chiang ( 1992 ) Elements of Dynamic Optimization , before diving into and Sargent ( 2018 ) Recursive Theory . Miranda and ( 2004 ) Applied Computational and Finance is another useful reference , Eventually , you may want to check out Sargent and ( 2014 ) Quantitative Economics , which is graciously available online at . There are also several computer programs available for solving dynamic programming models . The toolbox ( a toolbox accompanying Miranda and ( 2004 ) textbook ) and the website by Sargent and with Python and scripts . If interested in , the obvious reference is Asset Pricing ( 2009 ) of which there have been several editions . Sargent and provide two nice chapters on asset pricing theory and asset pricing empirics that would be a wonderful next step to the issues discussed in this chapter . If you want a historical reference , the original and Prescott ( 1985 ) article is still worth reading . Notes Later on in the chapter we will allow the return also to be stochastic . If such is the case it should be inside the square brackets . We will come back to this shortly You can check easily that this has positive marginal utility ( or can be so ) and negative second derivative of utility relative to consumption . It might also be because policy , such as federal tax of mortgage interest ( and not of rental payments ) encourages excessive home ownership . We can also appeal to irrationality people may convince themselves that the company they work for is a can miss investment . And how often have you heard that paying rent is a waste of money ?

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