Macro Consumption Problems 33-43

Transcription

1 Macro Consumption Problems rd October 6 Problem 33 This is a very simple example of questions involving what is referred to as "non-convex budget sets". In other words, there is some non-standard aspect to the constraint that the person faces when maximizing utility. Often, this will occur because of the presence of some government program which will guarantee a minimum level of income if the person qualies. When solving a problem of this sort, it will most often be the case that you will nd some trade-o between participating in the program and not participating. When you participate, you achieve some amount of total consumption that is larger, but you sacrice your ability to plan your own consumption. When you don't participate, you have a lower amount of total consumption, but can plan your consumption optimally. The key thing to nd in these problems is what is the point at which you are indierent between participating and not participating. This is done by equating the utility you receive under each choice. Now, for this problem specically. Because the interest rate and discount rate are both zero, you want smooth consumption if possible. You have no assets, and only live for two periods, and you receive income W. Consider how you would consume if there were no government program: C = C = W () T C = C + C = W () s = S W = Where T C is the total consumption, and s is the savings rate. If you participate in the program, then what is your optimal consumption plan? (3) C = W, C = C min T C = C + C = W + C min s = S W = Notice that with the program, you have higher total consumption. But that is gained at the expense of smooth consumption. In general W C min, so you have dierent consumption in the two periods under the program. There will be some point at which the utility of the extra income is totally oset by the loss of utility due to non-smooth consumption. How do we nd that point? We simply set the utility of the two scenarios equal. U(with program) = ln W + ln C min (4)

2 Setting equal U(no program) = ln W + ln W (5) ln W + ln C min = ln W + ln W Rearrange and then apply e to both sides and you get: W = 4C min What this says is that if your income is exactly 4 times as large as the minimum payment, then you are completely indierent between being in the program and not in the program. If W > 4C min, then your income is high enough that the extra consumption on the program isn't enough to make you join. You therefore save half your income, and have a saving rate of s = S W =. If W < 4C min, then the extra income is signicant to you, and you join the program, and hence have a saving rate of s = S W =. Problem 34. Part A This problem is exactly the same as problem 33. Using the results from there, W = 4C min, it follows that C min = Part B First, lets consider the problem as if there were no program. Expected life-time utility is E(U NP ) = ln(c ) + ln( C ) C max C = 3, C = 3 (6) E(U NP ) = ln( 3 ) + ln( 3 ) = ln( 4 7 ) (7) What if there is a program, and the individual chooses to participate? The expected utility in this case will be: E(U P ) = ln() + ln(c min) = ln(c min) The individual will be indierent if E(U NP ) = E(U P ), or C min = Problem 35 We need to set C min so that people maximize their utility on the downward sloping part of the line instead of choosing the consumption bundle {, C min }. But it's not as straightforward as the example in class because there is uncertainty about the government program. People will choose to take the gamble on the existence of the program if E(U P ) > E(U NP ). People will choose to stay o of the government support if E(U NP ) E(U P ). In words, they will behave as if there

3 were no program if the expected utility from the program is less than the expected utility from the consumption plan from without the program. Optimal consumption without the plan is {/, /}, (r = θ = ). Use the standard expected utility formulation to solve for optimal consumption with the 5% probability of the plan. E(U P ) = ln C + ln( C ) + ln C min (8) E(U) C = C ( C ) = (9) C = 3 () So the optimal consumption bundle with the 5% chance of the program is { 3, C min}. Note that C P = 3 regardless of the size of C min. Why is that? We need to set C min so that people choose their rst period consumption as if there were no plan. Therefore set C min such that E(U NP ) E(U P ) ln + ln ln 3 + ln( 3 ) + ln C min ln 4 ln 3 + ln 3 + ln C min ( 3 8 ) 3 C min C min 7 64 So with C min 7 64 people will consume {, } which is as if there were no program! 4 Problem 36 The previous problems helped us build intuition about the optimal consumption behavior in the presence of a government program. We have shown that a person with relatively high assets would ignore the program, while a relatively poor person would consume all his assets and switch to welfare. Think about the problem at hand - initially, the person will nance his own consumption. As time passes, his assets will be decreasing, so at some point he will switch to welfare. Moreover, at the time of the switch he will have no assets left. There is no uncertainty in this problem, which means that the individual can plan for his entire future. The trade-o is the following: if one decides to switch to welfare sooner this means that one can aord higher consumption for a shorter period of time, and a longer period consuming C min. On the other hand, decreasing marginal utility would suggest that a person would like to smooth consumption. These two tensions would determine the timing of the switch, T. Life-time utility would be: T max V = e θt ln C(t)dt + e θt ln C min dt T,C(t) T s.t. T C(t)dt = A ()

4 We are trying to nd an optimal path of consumption C(t), and an optimal time to run out of assets, T. The budget constraint only holds until time T because after that the government is paying for our consumption and we don't have to worry about where they get the money. Finding the Optimal Consumption Path Let's start by holding T constant, and maximizing over C(t). Notice that the second term of V does not involve C(t) so we can ignore it for this maximization. In essence, we have: T max e θt ln C(t)dt C(t) s.t. T C(t)dt = A We already know the FOC for a continuous time maximization problem. Ċ C = (r θ) σ But we know that σ = and that r =, so we have: is: Ċ C = θ Notice that this is a simple dierential equation of the type that we've seen before. It's solution C(t) = C e θt We have now described the optimal path of consumption over time, but haven't constrained it with any budget yet. Plug the above equation into the budget constraint to do this: T C e θt dt = A T C e θt dt = A C [ e θt θ ]T = A C = θa e θt We now have an equation for initial consumption in terms of T. From this we can also map out consumption into the future by using the result of our FOC: θa C(t) = [ e θt ]e θt Finding the Optimal time T Now we have to gure exactly what this optimal time T is. Notice that there will be two eects at work. As T gets smaller, I have to spend my assets faster so I run out at T. This means consumption and therefore utility is higher in the early periods. But it also means that I spend more time having at consumption of C min. Flat consumption is not optimal for me (θ > r), and C min will be lower than what I could have had consuming my assets a little longer, so this will make my overall utility lower. As T gets larger, the opposite eect takes place. I have more time with

5 nicely falling consumption as I would prefer, but I have to have lower consumption at each point from to T to not use up my assets. Also, note that there will not be a continuous ow of consumption at time T. It is not the case that I will consume C(T ) = C min. The reason is that I can improve my situation by doing the following. I can reduce T by some small amount, dt. I will now only consume C min in that small amount of time. So I've lost consumption there. But, I can take all the money I would have spent in dt, and use it earlier in my life. So I'll lose utility in dt, but I'll gain utility back by spending the amount C(T ) in earlier periods. How much does this change in T cost me in utility? e θt ln C(T ) = e θt ln C e θt () What do I gain by this change in T? I get utility from consuming C min, and I get the bonus consumption of C(T ), which I can spend at the rst instant of my life. The amount is so small (remember, this is all for an innitely small value dt ), that I can consume it right away without messing up my consumption plan. So I'll get C(T ) times the marginal utility of consumption in C. Mathematically, I have the following two gains: From consuming C min From consuming more in period e θt ln C min (3) U (C )C(T ) = ( C )C e θt = e θt (4) So if our benets must equal our costs, we combine equation A with equations 3 and 4: This reduces to e θt ln C e θt = e θt ln C min + e θt ln C e θt C min = C e θt C min = e C = ec min e θt It basically xes the initial amount of consumption that should occur when we have the optimal value of T. But we also know that there is an initial amount of consumption that should occur to full our budget constraint when we have the optimal value of T. That is: C = θa e θt Set these two conditions on initial consumption equal and solve for T and we get that: ec min e θt = θa e θt (5) T = θ ln[ θa ec min + ] (6)

6 Putting it together to nd the Optimal Value for Initial consumption First note that if we use our optimal value of T from above and manipulate it a little So we have: e θt = θa ec min + = ec min (7) ec min + θa And nally that: e θt = θa ec min + θa C = = θa e θt θa θa ec min +θa = ec min + θa Finding the Optimal time T with Math You can use Leibnitz's rule for dierentiating under an integral. b(z) z a(z) And take the derivative: f(x, z)dx = f(b(z), z) z b(z) f(a(z), z) z a(z) + b(z) T [ T e θt θa ln[ e θt ]e θt dt + Use Leibnitz's rule separately for each term in the brackets. T T T a(z) e θt ln C min dt] = [ f(x, z)]dx z e θt θa ln[ e θt ]e θt dt = e θt θa ln[ e θt e θt ] e θt (8) T Combining these together you get: Rearrange this to see that we get T e θt ln C min dt = e θt ln C min (9) e θt θa ln[ e θt e θt ] e θt e θt ln C min = e θt ln C e θt = e θt ln C min + e θt Which is exactly the condition we derived above using intuition. identical from this point on. The results are obviously

7 5 Problem 37 Let us rst determine the path of consumption and ignore the fact that we have tow dierent interest rates. The Lagrangian of the problem is then: FOCs: L = + t= e c(t) + ( + θ) t λ[ t= c(t) + ( + r) t A() t= ( + r) t ] () L c(t) = ec(t) ( + θ) t λ ( + r) t = () L c(t + ) = ec(t+) ( + θ) t+ λ = () ( + r) t+ Dividing these two equations we get: e c(t) ( + r) = ec(t+) ( + θ) Taking logs and using the approximation: c(t + ) c(t) = r θ c(t + ) = c(t) + r θ, which is the time path of consumption. If the individual needs to borrow ( c(t) > A(t) + ) then c(t + ) = c(t) =. If he saves ( c(t) < A(t) + ) then c(t + ) = c(t) θ. Now what you need to notice is that as long as you still have some assets (from the beginning), you will be consuming more than your income in THAT time period (interest on your assets and labor income of ), that is you will be eating away your assets, up until the point where you have no more assets, after which you will only consume your labor income at every time period. What we need to determine is time period t when you have consumed all your initial assets A(), and you start consuming in each time period your income. So from your budget constraint: t t= c(t) = A() + t t= (3) We only care about the budget constraint up to time period t because afterwards I will consume my income that time period. So: t c() t (t + ) θ = A() + t (4) This would be the usual way you proceed. But in this case you have two unknowns t and C(). So, instead of substituting recursively backwards, lets substitute consumption forward, up to t. So: t c(t ) + t (t + ) θ = A() + t In this case we know what c(t ) =, because at this time period you start consuming your labor income, as you have exhausted your initial assets. So now we can solve for t. So:.5(t + t) = t + t 4 = So t = 9.5. Finally you can plug everything into 4 to get: 9.5c() = c() =.5

8 6 Problem 38 If the man had no probability of dying, he would like to smooth consumption equally between the two periods by consuming. Alternatively, if the man had a positive probability of dying, but faced the same ln(c ) utility in the second period no matter what his rst period consumption was, he would want to front load consumption. The trick with this problem is that if he front loads consumption, then he will face a negative hit to his utility of. We can solve this problem by comparing the utility of the corner solution c = c = with the utility of taking the hit and front loading consumption. There will be some ρ for which front-loading will be better, and some ρ for which consuming equally will be better. Specically, the agent's utility will be if he consumes c = c =. If he doesn't consume this, then he will choose the solution in which he front loads consumption. This is the solution to the problem max ln(c ) + ρ ln(c ) s.t c + c = From the FOC's we get c = ρ ρc = c c Plugging this into the budget constraint we have ()c = c = c = ρ If the agent dies after the rst period, then his utility is ( ) ln. If he dies in the second period, then his utility is ( ln His expected utility of the gamble is [ ρ ln ( ) + ln ) + ln ( ) ρ ( ) ] ( ) ρ + ( ρ) ln If he chooses to consume c = c =, then he gets utility with surety. Therefore, he will decide to front load consumption if [ ρ ln ( ) ( ) ] ( ) ρ + ln + ( ρ) ln > What we have stated is sucent. However, for those who are interested, we can solve further for ρ and then get a numerical solution. ( ) [ ( ) ] ρ ln + ρ ln > ( ) [ ( )] ln > ρ ln ln() ln() > ρ Solving this numerically, we have that the individual will decide to front load consumption and consume c = +ρ if ρ >.4597

9 7 Problem 39 So what the problem says is that when you take your consumption decision at a certain date you take into consideration your consumption in the period before. So you should be able to derive the Lagrangian for this problem as: This only complicates the FOCs a bit: L = ln(c()) + ln(c() γc()) λ[c() + c() w] (5) L c() = c() γ c() γc() λ = (6) L c() = c() γc() λ = (7) L = c() + c() w = (8) λ Solving the second FOC for λ and plugging it into the rst FOC you get: c() = + γ c() γc() Which we can use in the budget constraint to get: 8 Problem 4 8. Part A c() γc() = ( + γ)c() c() = ( + γ)c() c() = c() = In this case, the agent maximizes the problem w + γ ( + γ)w + γ max U = ln(c ) + β ln(c ) + β ln(c 3 ) s.t. c + c + c 3 = A The FOC's of this problem are This implies = λ c β = β = λ c c 3 Plugging this into the budget constraint we have c = β c = β c 3 βc = c = c 3 c + βc + βc = A ( + β) c = A c = A + β c = c 3 = βa + β

10 8. Part B For the rst period, the agent solves the same problem as in part A, and therefore will consume c = A + β However, once the agent gets to the next period, he will consider whether he wants to reoptimize. His assets are now A = A c = A A ( + β = ) A = β + β + β A, so he will solve the problem max U = ln(c ) + β ln(c 3 ) s.t. c + c 3 = The rst order conditions of this problem are: c = λ β c 3 = λ c = β c 3 Plugging this into the budget constraint we have: β c 3 + c 3 = ( ) β + c 3 = c 3 = c = β + β A β + β A β ( + β) ( + β) A β ( + β) ( + β) A β + β A. So now we have that c > c > c 3, whereas in part A we had that c > c = c 3. Thus we see that the uncommitted agent wishes a certain plan of action in the rst period, but then breaks his plan and consumes more than was optimal under his st period plan when he gets to the next period because the current period is so important. 9 Problem 4 9. Part A The decision problem of the woman looks like this: max {ln(c ) + {c t} 8 ln(c s )} s.t. 8 A = c s (9) s=t+ s=

11 The rst order conditions imply that c = c =... = c 8 (3) c = c (3) Using the budget constraint, we can solve for the optimal consumption path: c = A 5, c = c =... = c 8 = A 9. Part B Lets examine the consumption decision at date. The woman will choose to consume c = A 5 and she will plan to consume A thereafter. She will start the second period of her life (t = ) with assets A = A c = 4 5 A. The consumption problem at t = will look like this: max {ln(c ) + {c t} Again, the rst order conditions will imply that 8 ln(c s )} s.t. 8 A = c s (3) s=t+ s= And using the BC c =... = c 8 (33) c = c (34) c = 9 A = 8 45 A c =... = c 8 = 9 A = 4 45 A A = A c = 7 9 A = 8 45 A The decision at t = would imply that c = A 4 = 7 45 A c 3 =... = c 8 = A 8 = 7 9 A Continuing on for each time period, we nd that consumption in each period is Problem 4 c t = 9 t 45 for t =,,..., 8. Before we start on the four cases, we should notice the following: The marginal utility of consuming c t c at any point in time is αe θt. The marginal utility of consuming c t > c at any point in time is βe θt

12 . Part A: w = c and αe θt > β Let us propose a feasible consumption path, and then see if we can consume it. Since her wage is w, she could consume that in each period. Now consider a possible deviation from this path. Since this agent is impatient, she will generally want to front-load consumption, so I am going to see whether she gains a higher utility from taking an ɛ amount of consumption away from the terminal period T and adding it to her initial consumption. Since she was already consuming c in the rst period, now she consumes c + ɛ > c. This implies that her marginal utility gain of from this change is β, while her marginal utility loss in is αe θt. Since αe θt > β, she is losing more utility from this change than she is gaining, and therefore she will not deviate from her initial consumption path. A similar argument will eliminate not only switch consumption between the rst and last period, but between any two periods. Therefore, the optimal path here is to consume her wage in each period and have at consumption where c t = c.. Part B: w > c and αe θt > β Now the PDV of total lifetime consumption is more than enough to consume c is each period. By the argument given in part A, she will use this income to consume at least c in each period. Now she must decide what to do with the extra income. She will consume it in the period in which she gains the highest marginal utility, unless she has diminishing marginal utility, in which case she will smooth it out over time. Since her marginal utility at any one point in time is constant, we will consume all the extra income at the moment in time where the marginal utility is the greatest. Since all the extra consumption at any point in time will give her a marginal utility of βe θt (remember that we are saying she already consumes at least c in each time period), the time at which the marginal utility is maximized is t =. Therefore, her consumption path will be to consume c t = c at each moment, and consume the rest at c..3 Part C: w < c and αe θt > β Now the woman does not have enough income to consume at least c at each point in her life. We know from the argument in A that she will never want to consume more than c in any period, because then the marginal utility of consumption will fall from αe θt to βe θt. However, since the marginal utility of consumption is constant for any point in time, she would like to consume up to c at those points in time where her marginal utility is highest. Since MU = αe θt, she nds that front-loading consumption gives her a higher marginal utility. So she will consume c up until the point where she runs out of income, at which point she will consume..4 Part D: w = c and αe θt < β Now the problem is very tricky. The present lifetime wages of the individual is the same as in part A, but now we don't have a condition which tells us that consumption above c in the rst period is less preferred to consumption under c in the last period. Therefore, there will be some t (call it s) such that αe θs = β. After this time s, the marginal utility of consuming less that c in the last period is less than the marginal utility of consuming over c in the rst period. Therefore, the agent is going to take all the consumption that otherwise would have occured after time s and consume

13 it in the rst period. Therefore, the optimal consumption is c if t (, s] c t = if t [s, T ] everything else if t = This is not necessary, but for those who are interested, the value of s is αe θs = β β α = e θs ( β ln = θs s = α) ( ) β θ ln α Her total income is T c, and therefore, her consumption in the rst instant is the area [ T ( )] β θ ln c α Problem 43 First, we have to notice that this is not our usual smooth consumption function - this particular utility function is called quasi-linear. To understand the characteristics of the preferences described by this utility function, lets consider the following one period -good problem (for simplicity, I have assumed that the prices of consumption and donations are both equal to ; the results, therefore, apply to this special case; you can set up the problem with dierent prices, and examine the more general problem): max{g + ln(c)} s.t. I = c + g (35) c,g You can solve this problem using the Kuhn-Tucker conditions, but I will discuss it graphically. The rst thing to notice is that the indierence curves associated with this utility function are horizontal displacements of each other. To see that, graph several indierence curves in the space of g and c (c = e u e g ): Now, draw in several budget constraints. You should nd that the expansion path is vertical, and then horizontal (after the income exceeds ). Why is that? As long as the income is smaller that, the marginal utility from spending an extra dollar on c is higher than the marginal utility of g, which is for any level of g. Once the marginal utility of c becomes equal to one, i.e. c =, the further consumption of c will bring the marginal utility below. Consequently, the man will spend all remaining money on g. The optimal consumption would look like this: c = { [c]ci, I, I > } (36) g = { [c]c, I I, I > } (37) Let's go back to the original problem. The marginal utility from consumption at t = equals c(), and the marginal utility from charity donations equals. Think about the marginal utility from consumption and charity at some future date t = T. The marginal utility of consumption will be c(t ) e.5t, and the marginal utility of charity will be e.5t. Comparing the marginal utilities of charity donations we can conclude that if if the man decided to make any donations at all, he

14 will do them in the very rst period when the marginal utility from them is highest. Now, compare the marginal utilities from consumption in period and T. If the man is choosing consumption optimally, these marginal utilities should be equal (remember that the interest rate is ). What does this imply for the path of consumption: c() = e.5t c(t ) (38) c(t ) = c()e.5t (39) Finally, we have to nd out how much c() is. To do that we have to compare the marginal utilities from spending money on consumption and charity. Not surprisingly, c() =. To tie the loose ends, we have to determine the size of the donation at date by using the life-time budget constraint: = A() = [c(s) + g(s)]ds = g() + c()e.5s ds = g() + e.5s ds g() = 98, c() =, c(t) = c()e.5t