Macro Consumption Problems 12-24

Transcription

1 Macro Consumption Problems 2-24 Still missing 4, 9, and 2 28th September 26 Problem 2 Because A and B have the same present discounted value (PDV) of lifetime consumption, they must also have the same PDV of lifetime wages. In a world with fully functioning credit markets, since A and B have identical preferences, they attempt to solve identical maximization problems. As a result, they would have identical optimal consumption paths In this ideal setting we would have that either Ct > Ct for all periods, or that Ct < Ct always. Additionally, there is the trivial case that Ct = Ct across all periods. This, of course, should be the same for both A and B. However, in the face of potential liquidity constraints we observe that A's consumption path is upward sloping, while B's is downward sloping. A stylized representation is given in the gure on the following pages. How can liquidity constraints explain this divergence? Suppose that the optimal consumption path in a world without liquidity constraints were upward sloping. If this were the case, liquidity constraints could not explain B's behavior. In the early periods B could simply reduce consumption and save the extra income, shifting consumption forward in time. The same argument holds for all possible consumption paths that are below B's in the early periods. As a result, in a world without liquidity constraints Ct < Ct must hold along the optimal consumption path. B is on or close to this ecient benchmark. This must be because B's wages are relatively high at the beginning of his life. A's deviation from the benchmark can be rationalized using liquidity constraints. It must be the case that A's wages are relatively low at the beginning of her life (and so relatively higher later in life). The credit market prevents A from shifting consumption from later to earlier periods by borrowing against her eventually higher earnings. As a result, she cannot come close to the optimal upward-sloping consumption path. So, B is following (or closer to following) the plan of ecient resource allocation even in an economy that allows borrowing, and A must substantially deviate from this because of binding liquidity constraints. As a result, it must be the case the B enjoys higher lifetime utility. 2 Problem 3 In general, from the FOCs of optimal control problem with a CARA utility function we can derive: Integrating both sides we get: ċ = (r θ) () α c t = (r θ)t + A (2) α

2 And from the initial condition we get c = A c t = α (r θ)t + c (3) the time path of consumption. In our particular case α =, r = and θ =., so: c t =.t + c (4) which basically says that I would like consumption to be decreasing with time, that is I would like to borrow on my future income, but I cannot because I have a liquidity constraint. Therefore I will have to consume my income. But this is not all. We have to look at the discontinuity points. At t = 2 we cannot have positive assets, because I would like consumption to be declining with time, nor negative assets because of the liquidity constraint A t t, so I must have that A 2 =. So I must consume my income at time period 2, so c 2 = 2! Now at t = suppose I continue consuming my income at the instant just before and at the instant just after t =. Then at the instant just before t = my consumption is c t = 2 and at the instant just after t = it is c t = and this is not optimal. To see why let's use the perturbation argument. Your marginal utility at any time period is: U (c t ) = e αct. Comparing the marginal utility just before and just after t = we see that e 2 < e so it cannot be optimal, in fact I would like to transfer consumption (and therefore income) from just before to just after. You will want to do so until the area of the amount of income you save is equal to the area of the amount you will consume later. If you are not consuming your income you would like your consumption to follow the path we determined before c t =.t + c. Putting these two facts together you should start saving at t = 5, with your consumption falling by. up to t = 5, when you start consuming your income again. 3 Problem 5 3. Part A From the CRRA utility function, with no time discounting the desired by the continuous time FOC: path of consumption is given ċ i c i = r i σ i Under autarky, however, since there is no saving, we must have that each economy consumes all of its available resources in each period: C,t = Y t C 2,t = Y t Noting that income for each country is exogenously given by Y t = e gt : C,t = Y t ln C,t = ln Y t

3 d dt ln C,t = d dt ln Y t Ċ C = g Combining the growth rate of consumption with the FOC, we have that: r = gσ Similar analysis shows that the interest rate for country 2 is: r 2 = gσ 2 And the fact that σ > σ 2 means that r > r Part B When the world is opened up, the two nations can trade by giving or receiving some of their income each period. The key result of this trade is that only one interest rate can prevail in the world. This interest rate r will necessarily be: r 2 < r < r This means that no matter where r lies in this range, we will have: Ċ C = r σ < r σ = gσ σ = g Ċ 2 = r > r 2 = gσ 2 = g C 2 σ 2 σ 2 σ 2 Total income is still growing at the rate g. So therefore country 2's consumption is growing faster than total income, and therefore eventually C 2 > Y + Y 2, which is obviously unsustainable. In order to slow down the growth of country 2's consumption, it is necessary to decrease the interest rate. In fact, as you can probably guess, there is only one interest rate at which the growth of country 2's consumption is possibly sustainable. And at this interest rate: r = r 2 Ċ 2 C 2 = g This is potentially sustainable. What happens to country when r = r 2? Ċ C = r σ = r 2 σ = gσ 2 σ < g

4 So country 's consumption is still growing, but at a rate lower than total income growth. Can we sustain this situation? We need to check whether the growth in total consumption C T = C +C 2 is less than or equal to the growth in total income. Ċ T C T = Ċ + Ċ2 C + C 2 = gσ 2 σ C + gc 2 = g[ C + C 2 So the question becomes what happens to C C 2 σ 2 σ C + C 2 ] = g[ C + C 2 over time. σ 2 C σ ( d C dt C 2 ) Ċ C = Ċ2 = gσ 2 g = g( σ 2 ) < C C C 2 σ σ 2 C 2 + C C 2 + ] As this growth rate is less than zero, the ratio C C 2 will eventually go to zero. If this ratio goes to zero, then: Ċ T C T g Which means that this pattern of consumption is stable over time. You can also pull the following results from the above analysis: Ċ C = gσ 2 σ Ċ 2 C 2 = g C C 2 C > C 2 The last result simply states that the initial consumption in country must be higher than in country 2. This follows from the fact that both countries still have to fulll their lifetime budget constraints. Country must have relatively high initial consumption in order to compensate for having relatively low consumption later in life. Country 2 has relatively low initial consumption to counteract their relatively high consumption later in life. The fact that C > C 2 implies that the interest rate must initially be r > gσ 2+gσ 2 and then asymptotes down to r = gσ 2. 4 Problem 6 4. Solving through intuition only If β >, then this means that purple people prefer consumption more in the second period than in the rst. This means they will want to save their income into the second period, but since no storage exists, the only way they can store is to get green people to give them some of their income in the second period. Green people, however, like consumption in both periods equally well, so purple people will have to make it worth their while to consume more in the rst period and less in the second. In order to induce green people to consume more in the rst period, they must oer them more consumption today than what they will give up tomorrow, or a negative interest rate. If β <, then purple people want to consume in the rst period more than the second. Therefore, in order to induce green people to give up some of their rst period consumption, they must promise them more consumption tomorrow that they are giving up today, or a postive interest rate.

5 4.2 Grinding through the math The FOC for green people is U (c G ) U (c G 2 ) = + r + θ c G 2 c G = + r Plugging this into the budget constraint we get The FOC for purple people is c G 2 = ( + r)c G c G + + r + r cg = w + + r w 2 c G = 2 + r 2 + 2r U (c P ) U (c P 2 ) = + r + θ c P 2 βc P = + r Plugging this into the budget constraint we get c P 2 = β( + r)c P c P + β + r + r cp = w + + r w 2 c P = 2 + r ( + β)( + r) In order for the market to clear, we must have c P + cg = 2 2 = 4 = 4 = Notice that r > when 2 2β > β <. 2 + r ( + β)( + r) r 2 + 2r 4 + 2r ( + β)( + r) r + r 4 + 2r + ( + β)(2 + r) ( + β)( + r) 4( + β)( + r) = 4 + 2r + ( + β)(2 + r) ( + β) [4( + r) (2 + r)] = 4 + 2r ( + β) (2 + 3r) = 4 + 2r 2 + 3r + 2β + 3βr = 4 + 2r r(3β + ) = 2 2β r = 2 2β 3β +

6 5 Problem 7 In either country, we use the rst order condition U (c t ) U (c t+ ) = + r + θ e αct e αc t+ = + r + θ c t+ c t = α ln ( + r + θ Solving this dierence equation we have c t = c + ( ) + r α ln t + θ We now plug in α = and use an approximation for x = ln( + x) c t = c + (r θ)t Let c A t be consumption in country A, and c B t be consumption in country B. Then we have c t = c + (r.)t c 2 t = c 2 + (r +.)t At any point in time t, we must have that c t + c 2 t = 2 2 = c + c 2 + 2rt Notice that the only way for this to hold at all t is that r = (since c and c 2 are constants). Another way to see it is that c + c2 = 2, so 2 = 2 + 2rt = 2rt t =,,..., r = However, notice that there are loans taking place even though the interest rate is. Solving the general intial condition problem, we have ) c t = (c θt) = c θ 2 c = + 5θ = Therefore, we have that the intial condition for each country is c = +.5 =.5 c 2 =.5 =.5

7 6 Problem 8 From the rst order conditions for both countries, we have U (c t ) U (c t+ ) = + r + θ c t+ c t We also must have that at any point in time, = + r + θ Notice c t + c 2 t = Y h + Y l c t+ + c 2 t+ = + r + θ (c t + c 2 t ) = Y h + Y l + r + θ (Y h + Y l ) = Y h + Y l + r + θ = r = θ Therefore, both countries will want consumption to be at. It suces to nd the level of consumption for each country. Country has the following budget constraint: c c ( + r) t = +r Y h ( + r) 2t + = Y h ( +r This can be simplied and solved for c. Country 2 has the following budget constraint: c 2 c 2 ( + r) t = +r This can be simplied and solved for c 2. Y l ( + r) 2t + r ) 2 + Y l ( ) 2 +r Y h ( + r) 2t + = Y h + r ( +r ) 2 + Y l Y l ( + r) 2t ( +r ) 2

8 7 Problem 2 7. Part A Notice that the probability of death works as an additional time discount rate. Therefore, the rst order condition for CRRA is ċ c = (r θ ρ). σ Since r = θ =, and σ =, we have which implies The budget constraint is because θ =. Plugging in for c t yeilds 7.2 Part B A = Now the woman faces the following problem: ċ c = ρ, c t = c e ρt. A = c t dt c e ρt dt = c ρ c = ρa. max U(ĉ t )e θ dt + U( c t )e (θ+ρ)t dt subject to A = ĉ t e rt dt + c t e rt dt. In other words, we will have one function ĉ t which will hold from time to, and another function c t which will hold after time. This must be the case since we have a dierent Euler equation governing optimality before time, and another thereafter. From time to we have ĉ ĉ = σ (r θ) = ĉ t = ĉ. From time to we have c t c t = σ (r θ ρ) = ρ c t = c e ρt. Notice that since there is no new information being gained at time, there cannot be a jump in consumption. Therefore, we must have that c = ĉ, or ĉ = c e ρ. The budget constraint is A = A = ĉ dt + ĉ t dt + c t dt ĉ e ρ e ρt dt

9 A = ĉ + ĉe ρ ρ = ĉ ρ + e ρ ρ ĉ = ρa ρ + e ρ < ρa. So consumption is less in the rst period now that the woman has some time before she has a chance of dying, unlike in the last problem where we was faced with the possibility of death her whole life. 7.3 Part C: This problem is hard, and might be incorrect The woman now tries to solve max U(ĉ t )e θ dt + [ ] U( c t )e (θ+ρ)t + U( c 2 t )e (θ+ρ 2)t dt 2 subject to A = ĉ t e rt dt + c t e rt dt Just as before, this problem will be governed by three Euler equations: ĉ = for t (, ) ĉ c c = ρ for t (, ) with probability 2 c 2 c 2 = ρ 2 for t (, ) with probability 2 Notice that the rst Euler equation implies that the agent will consume a constant ĉ between time and, and then will follow the path c t = c e ρ e ρ t or c t = c 2 e ρ 2 e ρ 2t after time, with a 5% chance each. At period, we will have that A = A ĉ. Since the remaining consumption must now equal A, we must have that c e ρ e ρ t = c 2 e ρ 2 e ρ 2t Since ρ ρ 2, then c c 2. This implies that the two potential paths from time to will start and dierent points, and will both asymptotically decline to. Also, the curve which starts higher at time will decline faster and ultimately cross the other line. Notice that this implies that there will be a jump in consumption at time. How can this happen if the agent was optimizing? Because new information is being discovered, and the agent re-optimizes based on the new information. In this case, the new information she receives is the probability of death for the rest of her life. If the probability of death is higher, then she will wish to front-load consumption, and have her consumption decline faster. If the probability of death is lower, then she will wish to spread out consumption, and have her consumption decline slower. Consider the agent's new problem at time. She has some stock of assets A. She wishes to solve max U(c t )e ρ(t ) dt subject to A = c t dt We have already solved the Euler equation for this problem c t = c e ρ(t ) for t (, )

10 Plugging this into the budget constraint we have A = c e ρ(t ) dt ( A = c ) ρ e ρ(t ) t= c = ρa t= = c ρ Therefore, we must have the following initial conditions for the two time paths after t = : c = ρ A c 2 = ρ 2 A Although we have a jump in consumption at t =, if the agent is optimizing, there will never be a jump in expectation (we haven't dealt with this yet in the class, and won't until asset pricing). Therefore, we can set the expected marginal utilities equal before the jump and after the jump: 8 Problem 22 The utility function for this man is E ( U (ĉ ) ) = E ( U (c ) ) = ĉ 2 c + 2 c 2 = ĉ 2 + ρ A 2 ρ 2 A = ĉ 2 ρ (A ĉ ) + 2 ( ρ + ρ2 ) ĉ A ĉ = 2 ( ) ρ + ρ 2 A = + 2ρ ρ 2 ( ρ + ρ 2 + 2ρ ρ 2 A = 2ρ ρ 2 ĉ = ρ 2 (A ĉ ) ĉ ) ĉ 2ρ ρ 2 ρ + ρ 2 + 2ρ ρ 2 A U = ln(c ) + ln(c 2 ) ln(c 5 ) + 2 ln(c 5) ln(c ) From the rst order conditions, we get that c = c 2 =... = c 5 c 5 = c 52 =... = c c = 2c 5

11 His budget constraint is Plugging all this in, we have c t = 5 2c 5 + 5c 5 = 5c 5 = c 5 = c 52 =... = c = 2 3 c = c 2 =... = c 5 = Problem 23 This is another expected utility problem. The agent does not know whether he will live to see the second or third periods. Therefore, he wishes to maximize his expected utility. I claim the correct problem is Why is this the case? E(U) = U(c ) + max U(c ) + 2 U(c 2) + 4 U(c 3) subject to A = c + c 2 + c 3. E(U) = EU(c ) + EU(c 2 ) + EU(c 3 ) ( 2 U(c 2) + 2 ) + ( 2 2 U(c 2) ) E(U) = U(c ) + 2 U(c 2) + 4 U(c 3) From here, the problem is standard. ( Just by ) looking at the objective function, we can realize that the answer will be (c, c 2, c 3 ) = 4 7 A, 2 7 A, 7 A. Why? Because if we want one share of the asset in the last period, then we will want two shares in the second to last period because we value the second period twice as much as the third. Similarly, we would want twice as much in the rst period as we would the second, which means four times as much as we would like in the third. Summing the shares, we have seven shares, one in the third period, two in the second, and four in the rst. (Warning! This logic only works for log utility!) Analytically, we have as our rst order conditions: which implies L c = c λ = L = λ = c 2 2c 2 L = λ =, c 3 4c 3 c 2 = 2 c and c 3 = 4 c Plugging this into the budget constraint, we have c + 2 c + 4 c = 7 4 c = A c = 4 7 A, c 2 = 2 7 A, c 3 = 7 A.

12 Problem 24. Part A First: a bequest is going to be the assets of the individual that are left when she dies. This problem is similar to the example in the class notes, except for two things. First, we have a specic form for the probability of dying, and second, we are optimizing over an innite amount of potential lifespan. In order to say anything about how this person acts over time, we'll have to consider optimizing her utility subject to some budget constraint. In this kind of problem, despite the known probability of dying, the optimization acts as if the person might live forever. The key point is that the person has to create an optimal consumption plan for every potential period of life, just in case they aren't dead yet. You can get the intuition for part A without all of the math, but you need the math for part B, so I'm going to set all of the math up in part A to conrm our intuition. Here's the intuition: if r > p then the person gets a higher interest rate than her probability of dying so she'll want to save more now to take advantage of the interest rate and will have consumption rising over time. An increase in σ will cause her to have atter consumption, so consumption rises more slowly causing her to spend more now and leave less in the bequest. It's easiest to see in a picture. If r < p then her probability of dying is higher than the interest rate, so she'll have falling consumption over time. An increase in σ will again atten her consumption so it will be falling more slowly causing her to spend less now and leave a larger bequest. It's easiest to see in a picture. Ok, math time: Remember that our consumer has to have a complete plan as if she were going to live forever, so the relevant budget constraint is one which takes in to account consumption in every possible period. In this specic case: C t ( + r) t = A This simply says that the present discounted value of consumption in every possible period of life is equal to the present discounted value of initial assets. So we can now optimize the following Lagrangian: [ ] L = ( p) t C σ t σ + λ C t A ( + r) t Take derivatives with respect to C t and C t+. dl = ( p) t Ct σ dc t λ ( + r) t = (5) dl = ( p) t+ Ct+ σ dc λ = (6) t+ ( + r) t+ You can solve these two conditions together to get that: Note the following: C t+ C t = [( + r)( p)] σ ( + r)( p) ( + r p)

13 And rewrite the FOC as: C t+ C t = [ + r p] σ We can make more sense out of the FOC by seeing that they will form a pattern to consumption: C = [ + r p] σ C (7) C 2 = [ + r p] σ C = C t = [ + r p] σ Ct =. ( [ + r p] σ ) 2 C (8) ( [ + r p] σ ) t C (9) This equation for C t can be used in the budget constraint to solve out for C C t ( + r) t = A () [ + r p] σ t C ( + r) t = A () [ + r p] σ t C ( + r) t = A (2) A C = (3) [+r p] σ t (+r) t [ ] C = A + r ( + r p) σ < A (4) + r So, where are we now? We know initial period consumption. We know the FOC, and so the pattern of consumption from time until innity. We are trying to nd out how the size of the persons bequest will change with their value of σ. So we need an equation for their bequest. The bequest at any given time is simply the assets at the end of that time. So we need to solve for an equation for the assets at any point in time. Let's start with what we know is the path of assets over the person life: A = A (5) A = ( + r)(a C ) (6) A 2 = ( + r)(a C ) = ( + r) [( + r)(a C ) C ] (7). A t = ( + r)(a t C t ) = ( + r) t A ( + r) t C ( + r)c t (8)

14 So assets at time t are just total possible assets if you had consumed nothing (+r) t A, minus the series of amounts consumed over time, weighted by the interest rate. We can group the consumption terms together to obtain the following: t A t = ( + r) t A ( + r) t C s ( + r) s This equation describes assets in period t, no matter whether the person is optimizing consumption or not. It is simply an accounting identity. But we do know the person will optimize, and we know a form for the amount consumed C s at each period. Plug in from our FOC: t A t = ( + r) t A ( + r) t s= A t = ( + r) t t A C s= s= [ + r p] σ s C ( + r) s (9) [ + r p] σ s ( + r) s (2) Now we have to evaluate the summation term. This is a nite sum, but it can be broken into two parts: t s= [ + r p] σ s ( + r) s = s= [ + r p] σ s ( + r) s s=t [ + r p] σ s ( + r) s You may not see the logic behind this split, but consider this similar example: Evaluating the summation now: t s= [ + r p] σ s [ ( + r) s = = ( ) (5 + ) + r + r ( + r p) σ ] ( + r p) σ t ( + r) t + r + r ( + r p) σ So our equation for assets at t is now: [ ] A t = ( + r) t + r A C ( + r p) σ t + r + r ( + r p) σ ( + r) t + r ( + r p) σ This can be solved further by rearranging terms and plugging in for C. ( ) A t = ( + r) t + r A C ( + r p) σ t ( + r p) σ + + r ( + r) t (2) A t = (+r) t A A [ ] + r ( + r p) σ + r A t = ( + r) t A ( ) + r ( + r p) σ t + r ( + r p) σ ( + r) t ( + r p) σ t ( + r) t (22) (23)

15 A t = ( + r) t A ( + r p) σ t ( + r) t (24) A t = A ( + r p) σ t (25) And this result is surprising. It says that assets follow exactly the same path over time that consumption does. NOTE, it does not say that C t = A t, as we showed C < A in (). Rather, once C has been chosen, both assets and consumption follow the same shaped trajectory through time. Now we are ready to analyze how bequests change with σ. The rst thing to note is that the the bequest received at any time t is simply the assets at time t. So if the person were to die in period 2, then A 2 is the size of the bequest. The expected value of the bequest is technically the following: E[Bequest] = pa + ( p)pa 2 + ( p) 2 pa 3 + We don't need to evaluate this large summation. We can see that A t is a continuous and monotonic function of A. So if A t rises, then E[Bequest] rises. If A t falls, then E[Bequest] falls. So how does A t respond to changes in σ? As the problem indicates, there are two cases. Case : Interest rate higher than chance of dying: r > p ( + r p) (26) r > p C t+ > C t (27) We see that consumption is rising over time. If we now look at what happens when σ rises: σ rising σ falling ( + r p) σ falling C rises and A t falls. In other words, as you try to smooth your consumption more, you are forced to raise you initial consumption and lower your future consumption. This implies less savings at early periods, and so results in you holding less total assets in any given period. Less total assets imply a lower expected bequest. Case 2: Interest rate lower than chance of dying: r < p ( + r p) (28) r < p C t+ < C t (29) So consumption is falling over time. We can again look at what happens when σ rises: σ rising σ falling ( + r p) σ rising C falls and A t rises. In this case, in order to smooth your consumption more, you lower your initial consumption, and raise your future consumption. This in turn leads to greater savings early on in life, and so raises the amount of assets your hold in any given period. Higher assets imply a higher expected bequest. Ultimately, the σ term is acting identically in both cases. It is forcing you to smooth your consumption over your lifetime. Depending on what your optimal path of consumption looks like to start with, you'll either have to lower or raise initial consumption. This change in consumption plans changes your lifetime path of assets, resulting in dierent bequests.

16 .2 Part B We already have an equation for assets at time t: A t = A ( + r p) σ t If we impose log utility with σ =, then we have: A t = A ( + r p) t