Advanced Macroeconomics An Easy Guide Consumption

CHAPTER 11 Consumption The usefulness of the tools that we have studied so far in the book goes well beyond the issue of nomic growth and capital accumulation that has kept us busy so far . In fact , those tools enable us to think about all kinds of dynamic issues , in and beyond . As we will see , the same arise again and again how do individuals trade off today tomorrow ?

It depends on the rate of return , on impatience , and on the willingness to shift consumption over time , all things that are captured by our old friend , the equation ! What are the constraints that the market imposes on individual behaviour ?

You won be able to borrow if you are doing something unsustainable . Well , its the game condition ! How should we deal with shocks , foreseen and unforeseen ?

This leads to lending and borrowing and , at an aggregate level , the current account ! All of these issues are raised when we deal with some of the most important ( and inherently dynamic ) issues in the of consumption and investment , policy , and monetary policy To these issues we will now turn . We start by looking at one of the most important , namely tion . In order to understand consumption , we will go back to the basics of individual and the choice of how much to save and consume . Our investigation into the of consumption will proceed in two steps . First , we will analyse the consumers choice in a context of full certainty We will be careful with the algebra , so readers who feel comfortable with the solutions can skip the detail , while others may the careful procedure useful . Then , in the next chapter , we will add the realistic trait of uncertainty ( however simply it is modelled ) In the process , we will also see some important connections between and . Consumption without uncertainty The main result ofthe consumption theory without uncertainty is that smoothing . ple try to achieve as smooth a consumption as possible , by choosing a consumption level that is consistent with their resources and saving and borrowing along the way to smooth the volatility in income paths . Let start with the case where there is one representative consumer living in a closed economy , and no population growth . All quantities ( in letters ) are . The typical provides one unit of labour . Their problem is how much to save and how much to consume How to cite this book chapter , and , A . 2021 . Advanced An Easy Guide . 11 . Consumption , London Press . DOI License .

CONSUMPTION over their lifetime of length This ( unlike in the analysis of choice that we pursued in the context of the Neoclassical Growth Model ) will be partial equilibrium analysis we take the interest rate and the wage rate as exogenous . The consumer problem This will be formally very similar to what we have encountered before . The utility function is ( where denotes consumption and ( is the rate oftime preference . Assume ( and that conditions are . The resource constraint is 15 , where , is the real wage and , is the stock of bonds the agent owns . Let us assume that the real interest rate is equal to The agent is also constrained by the game ( or solvency condition ( Solution to consumer problem The consumer ( subject to ( and ( for given and 170 . The current value for the problem can be written as ( CI . Note that is a control variable ( jumpy ) la is the state variable ( sticky ) and A is the costate variable ( the multiplier associated with the budget constraint , also jumpy ) The costate has an intuitive interpretation the marginal value as of time an additional unit of the state ( assets 17 , in this case ) The conditions are ( I , 116 ) This last expression is the transversality condition ( Solving for the time and level of consumption Take ( and differentiate both sides with respect to time ( a ,

CONSUMPTION 163 Divide this by ( and rearrange ( it . it Now , as we ve seen before , as the elasticity of substitution in consumption . Then , becomes ) Finally using ( 116 ) in ( we obtain ( what a surprise ! 51 Equation ( says that consumption is constant since we assume It follows then that ) so that consumption is constant . We now need to solve for the level of consumption . Using ( in ( we get ' a ) which is a differential equation in 17 , whose solution is , for any time , lace . where time is any moment between and Evaluating this at ( the terminal period ) we obtain the stock of bonds at the end of the agents life ' hoe . Dividing both sides by and rearranging , we have , be . Notice that using ( 115 ) and ( the can be written as ( Since clearly ( this would require oo ) for the to hold it must be the case that . Applying this to ( and rearranging we have ( be . The of this equation is the net present value ( of consumption as of time , and the the of resources as oftime .

CONSUMPTION The permanent income hypothesis Dividing ( through by ( and multiplying through by we have rho The of this expression can be thought of as the permanent income of the agent as of time . That is what they consume . What is savings in this case ?

1121 ) Hence , savings is high when a ) are high relative to their permanent level , and ) current wage income is high relative to its permanent level . Conversely , when current income is less than permanent income , saving can be negative . Thus , the individual uses saving and borrowing to smooth the path of consumption . Where have we seen that before ?

This is the key idea of ( 1957 ) Before then , economists used to think of a rule of thumb in which consumption would be a linear function of current disposable income . But if you think about it , from introspection , is this really the case ?

It turns out that the data also belied that vision , and ( 1957 ) gave an explanation for that . The case of constant labour income Note also that if , the expression for consumption becomes rho rho , 1122 ( Moreover , if oo , this becomes 13 rho , which has a clear interpretation rho is what the agent can consume on a permanent ( constant ) basis forever . What is the path of over time ?

Continue considering the case in which is constant , but remains . In that case the equation for bonds ( becomes 17 , Using ( in here we get ( la , be 170 . 1125 ) Notice that ( CONSUMPTION Figure with constant income Bonds be I 50 Savings , so that decline , and at an accelerating rate , until they are exhausted at time Figure 111 shows this path . The effects of labour income Suppose now that wages have the following path ) Then , we can use ( to out what constant consumption is rho My If ( 1129 ) rho (

CONSUMPTION For , saving is given by ( 1131 ) 170 ( What are along this path ?

In this case the equation for bonds ( becomes , for ( 1132 ) Notice , in ( et ( 1134 ) so that are increasing at an increasing rate for I if 170 is small . Plugging this into ( we obtain ( Mi Mi so that , yet again savings is high when current wage income is above permanent wage income . Simplifying , this last expression becomes ( 1135 ) A Notice ?

re ( 5125 ?

1137 ) so that , if 170 is small , rise , and at an accelerating rate , until time shows this path . This is consumption smoothing since the current wage is higher than the future wage , the agent accumulates assets .

CONSUMPTION Figure Saving when income is high Bonds A Sa The hypothesis The most notable application of the model with labour income is that of consumption over the life cycle . Assume be , and also that income follows , so that now the works for the periods of his life , and is retired for the ing periods Then , consumption is simply given by ( 11129 ) with be , so that consumption per instant is less than the wage . Let us now out what are during working years ( Looking at ( and using ( we can see that

CONSUMPTION II ' I I I . I . II IE I IT ! I cu I um ( 1139 ) By the same token , savings during the working years ( can be obtained simply by differentiating this expression with respect to time , we ( 1140 ) so that 5151 ' en . What happens after the time of retirement To calculate , notice that for I , 1125 ) gives ( so that , 15 ! I ( 71 ' 73 ( so that savings decrease over time . Figure 113 shows this path The agent accumulates assets until retirement time I , then depletes them between time and time of death This is the basic of the of and ( 1954 )

CONSUMPTION Figure The hypothesis Bonds A Sa Vin Of course , the hypothesis is quite intuitive . One way or the other we all plan for retirement ( or trust the government will ) et al . 2006 ) show that 80 of the households over age 50 had accumulated at least as much wealth as a model prescribes , and the the wealth of the remaining 20 is relatively small , thus providing support for the model . On the other hand , many studies have also found that consumption falls at retirement . For example , et al . 2001 ) show that there is a drop in consumption at retirement and that it is larger with families with a lower replacement rates from Social Security and pension . This prediction is at odds with the cycle hypothesis . Notes Do you remember our discussion of the Ramsey model , and the implications of or ?

What explains the curvature ?

In other words , why is it that the individual accumulates at a faster rate as she approaches retirement , and then at a faster rate as she approaches death ?

The intuition is that , because of her asset accumulation , the individuals interest income increases as she approaches retirement for a constant level of consumption , that means she saves more and accumulates faster . the of this argument happens close to the death threshold , as interest income gets lower and intensifies as a result .

CONSUMPTION References , Skinner , 2001 ) What accounts for the variation in retirement wealth among households ?

American Economic Review , 91 ( 1957 ) The permanent income hypothesis A theory ( 37 ) Princeton University Press . 1954 ) Utility analysis and the consumption function An tion of data . Franco , 2006 ) Are Americans saving for retirement ?

Economy , 114 (