Economics 456. International Macroeconomics and Finance: Section 5. Geoffrey Dunbar. UBC, Winter March 14, 2013

Transcription

1 Economics 456 International Macroeconomics and Finance: Section 5 Geoffrey Dunbar UBC, Winter 2013 March 14, 2013 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

2 To model the consumption smoothing benefits of international borrowing and lending, we need to write down a model with time-varying income. We shall consider two types of models, one endowment economy and one production economy. Recall that the difference between an endowment economy and a production economy is that a production economy allows investment into the production process. The reason that we want to consider both types is that an endowment economy does not permit agents to smooth consumption via capital investment whereas a production economy does. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

3 Our first model is a simple variant of what we have studied before. We will imagine a representative consumer in the Home country who lives for two periods, current and future. We will imagine that she has preferences over consumption in both periods and that her preferences are such that she likes to consume both. We could think of her preferences as we often do: U = ln(c) + β ln(c ) The consumer has income in both periods and this income is assumed to be exogenously specified. Hence, she earns Y in the current period and Y in the future period. There is no uncertainty over her income. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

4 There is a government which spends G and G in the current and future periods, respectively. The government finances its spending through taxes, T and T, in the current and future periods. Thus, the consumer has a (standard) budget constraint: C + C 1 + r = Y T + Y T 1 + r The government s budget constraint is also typical: G + G 1 + r = T + T 1 + r Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

5 We know from our earlier two-period model that there is a solution to the consumer s problem of maximizing utility subject to her constraint. We will call her optimal consumption sequence C and C in the current and future periods, respectively. Recall from our earlier model that the solution to the problem: max U subject to: C + C 1 + r = Y T + Y T 1 + r gives a marginal rate of substitution between C and C of C = βc(1 + r). This, in turn implied optimal current consumption, C : C = 1 [ Y T ] Y T β 1 + r Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

6 Thus, we can write private savings of the consumer in the current period as: S p = Y T C since intuitively her savings must be the residual difference between her net income, Y T, and her optimal choice of consumption, C, given the real interest rate. Hence, we could write optimal first period savings, S p = Y T C as: S p = β(y T ) 1 + β 1 [Y T 1 + β 1 + r ] Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

7 We can similarly write the government s savings in the first period as: S g = T G since intuitively the government s savings must be the residual difference between its income, T, and its spending G. S g is also government borrowing, B. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

8 In a closed economy model the clearing condition for borrowing and lending must be: S p + S g = S p + B = 0 This will give us a market clearing real interest rate r. To see why, note that this expression is one equation in one unknown, r, since we have an expression for S p and B is an exogenously specified level of government borrowing. β(y T ) 1 + β 1 [Y T 1 + β 1 + r ] + B = 0 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

9 However, this is not the clearing condition in a SOE model because the country can borrow or lend internationally. Hence, government borrowing need not equal private savings. In our two-period SOE model, the current account can be written as: CA = S I = S p + S g 0 = Y C G The difference here is that the world interest rate, r, is exogenously specified since Home is a SOE and hence a price-taker (also recall the implications of the Fisher Effect). The real interest rate is the price of todays consumption in terms of consumption tomorrow. Thus, the current account, CA, will move around depending on the world risk-free real interest rate. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

10 Where S g = T G then this implies that the current account can be written: CA = βy + T 1 + β 1 [Y T ] G 1 + β 1 + r Thus, we are now in a position to examine the factors that lead to shifts in the current account in the context of our model. (Our model is representative enough that it captures the appropriate dynamics qualitatively although clearly other utility specifications could lead to different quantitative results.) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

11 current income Y : An increase in current period income increases consumption both today and tomorrow since both C and C are normal goods. Thus, the consumer saves more and the current account increases. In our example, the current account increases by the amount CA Y = β 1+β. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

12 current government spending, G: Two effects. First, the increase in government spending directly causes the current account to fall (because of the G). The second effect is that the increase in government spending implies an increase in the present value of lifetime taxes so either T or T must increase. Since Ricardian equivalence holds, the consumer will be indifferent as to in which period taxes increase but she will lower consumption in both periods. For example, imagine T increases to pay for the government spending. Then, G = T so the change in the current account is CA G = 1 1+β 1 = β 1+β. Thus, the current account falls when G increases. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

13 taxes, T : Since Ricardian equivalence holds in our model, then there is no effect on current consumption of a change in the tax schedule (unless there are also changes to government spending). Hence, any change in T must be offset by a change in T. Thus, there should be no effect on the current account. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

14 real interest rate, r: This will depend on whether the consumer is a borrower or a lender. The change in the real interest rate decreases the present value of her lifetime wealth. It also makes it more costly to borrow from the future. Hence, if she is a borrower, she will borrow less and reduce her current consumption. This would cause an increase in the current account. If she was a lender, there are both income and substitution effects. She gets more for lending but has lower lifetime wealth. Hence, current consumption could rise or fall. Typically, it is thought that the income effects are not too large and hence she will decrease her current consumption and thus increase the current account. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

15 future government spending, G : If G increases then the government must increase the present value of taxes to pay for the increased spending. Since Ricardian equivalence holds, we will assume for simplicity that taxes in the future period increase to pay for the increased spending. Then the effect on the current account today is: CA 1 T = (1+β)(1+r). Thus we would expect the current account to increase. Intuitively, the increase in the present-value of taxes causes current consumption to fall which leads to a fall in the current account. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

16 permanent government spending change: Finally, if there is a permanent change to government spending then there are no intertemporal smoothing opportunities for the consumer. Hence, she decreases her consumption by the amount of the increase in government spending. In this case, there would be no change to the current account. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

17 Note that in this simple model the role of the government and household budget constraints. The budget constraints impose the condition that all borrowing is repaid. This arises here because we have a simple two-period example. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

18 Let s extend this basic model a little bit. One empirical observation that we have made in this course is that some countries default on their obligations. Here, our budget constraints have ruled this out. One way we might relax this environment to permit some idea of default is to embed this model into an overlapping generations model. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

19 Let s assume that in every period there are two generations alive, the Young and the Old. We will maintain the same preferences as before so that for a Young agent: U = ln(c t ) + β ln(c t) The consumer has income in both periods and this income is assumed to be exogenously specified. Hence, she earns Y t in the current period and Y t in the future period. There is no uncertainty over her income. The initial Old have income Y 0. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

20 The optimal consumption paths for the Young agents are still: and implicit savings: Ct = 1 [ Yt T t + Y t T t+1 ] 1 + β 1 + r S p,t = β(y t T t ) 1 + β 1 [Y t T t β 1 + r Since individual agents are still finitely-lived, it is still sensible to claim that they must repay their debts given no income uncertainty. Note as well that for simplicity, I have assumed that lump-sum taxes for the Young and Old are the same in a given period. ] Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

21 The difference the OLG environment makes becomes clear when we consider the government budget constraint. The government s budget constraint must now be satisfied only over the infinite horizon: G t (1 + r) (t 1) = t=1 T t (1 + r) (t 1) This constraint permits many possible sequences. For instance, the government could simply set G t = T t in every period. Or the government could spend everything in the first period: G 1 = t=1 T t (1 + r) (t 1) t=1 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

22 What this OLG environment hopefully makes clear is the importance of describing how G is set. Suppose, for example, we allow all currently alive generations to vote over the level of G. Does this matter? Well, yes. If they could vote here they would set G = 0 to avoid paying taxes because government spending provides no utility benefits. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

23 So let s suppose something a little bit different. Let s imagine that a fraction 0 < φ < 1 of government spending is diverted to private incomes. Thus, the budget constraint for the individual agent becomes: C t + C t 1 + r = Y t T t + Y t T t r + φg t + φ G t r So what would an individual agent vote the government to do? An individual agent would vote for the highest G t possible and to defer taxation until t + 2. Indeed, all future generations have the same incentive to vote to defer taxation if confronted with a demand for repayment. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

24 So in this world, no debts will be repaid until some point at the infinite horizon which may never come. This is the optimal plan for the borrower here. So what would cause this plan to be infeasible? There are a few possibilities. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

25 1 Current foreign lenders decide to demand repayment and no other foreign lenders decide to replace them. 2 A rise in the world real interest rate which lowers the present-value of expected future surplusses. 3 A change in the perception of risk that increases the Home countries international risk premium. 4 A change in the expected future income of the Home country. E.g. a fall in population growth. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

26 For simplicity let s assume that lenders want the Home country to repay all its debt (note that this is not a given). Let s also assume that if a country defaults, it is permanently excluded from international capital markets. Under these assumptions, what will a lender do? It will loan an amount such that the Home country is indifferent between repayment and default. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

27 We can relatively easily describe the situation for a country without any access to international markets. In this environment, it implies: S p + S g = S p + B = 0 in every period. As noted above, we can solve for the closed-economy equilibrium r. Let s call this autarky. But with access to international capital markets, we saw that: at the world real interest rate r. CA = S I = S p + S g 0 = Y C G Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

28 Given preferences, each equilibrium provides a certain amount of utility to the agent. Agents in the Home country will default whenever the utility from autarky is at least as great as the utility in the model with international capital markets. So the question for lenders becomes: at what point is it optimal for a borrower to default. Under some assumptions, we can solve this. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

29 Let s begin with the outside option, autarky. What we need to find is the utility for a generation when it is faced with autarky. Let s return to our simple 2 period OLG example with which we have been working. For simplicity, we will abstract from government spending to illustrate the main point of incentive compatibility constraints. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

30 Let s assume that all generations have the same preferences: U = ln(c t ) + ln(c t) The consumer has income in both periods and this income is assumed to be exogenously specified. Hence, she earns Y t in the current period and Y t in the future period. There is no uncertainty over her income. We will assume (1 + r)y t < Y t. The initial Old have income Y 0. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

31 Let s assume that the population in every generation is constant. Because all agents in a generation are identical, there is no possibility of a domestic credit market. The young and the old only meet once and so cannot exchange intertemporal credit. And, because all agents in a generation are identical, all agents either want to save or borrow. For the credit market to clear, aggregate savings must equal zero which implies that all agents save nothing. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

32 Since there is no domestic credit market and savings is equal to zero, then the best that individual agents can do is to consume their endowments. This gives them lifetime utility of: U a = ln Y t + ln Y t where I have subscripted U a to indicate that this is the autarky level of utility. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

33 The optimal allocations for a generation must equate the marginal utility of each type of consumption, otherwise a generation could be made better off by re-allocating from one consumption choice to another. Thus, it must be that: By implication: along the optimal path. 1 C t = 1 C t C t = C t Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

34 If the agent has access to international capital markets then he or she can save or borrow internationally. Thus, the agents period budget constraints are: and C t + S t = Y t C t = Y t + (1 + r)s t This gives a lifetime budget constraint of: C t + C t 1 + r = Y t + Y t 1 + r Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

35 So now we can solve the agents problem: subject to: This should give us: max ln C t + ln C C t,c t t C t + C t 1 + r = Y t + Y t 1 + r C t = (1 + r)c t Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

36 Using the budget constraint we should get: Savings is therefore: C t = 1 2 (Y t + Y t 1 + r ) S t = Y t C t = Y t 2 Y t 2(1 + r) Notice how both consumption and savings depend on r. Because of our assumption on the endowments, S t < 0 and this country runs a current account deficit. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

37 Now note that because S t < 0 and this generation is only alive for one period, agents in this generation are strictly better off defaulting on the debt incurred by borrowing internationally. If they default, C t > Y t and C t = Y t. This is clearly better then repaying in which case C t < Y t. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

38 But if they do not repay then future generations cannot borrow internationally. This means that all future generations will be worse off. So how might one evaluate this tradeoff? Clearly, if the utility of all future generations was considered relevant then, unless the future generations are discounted heavily, it will be optimal to enforce repayment. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

39 But perhaps the utility of all future generations is not relevant. Suppose we have a voting mechanism. Is there a point a which the young and the old would vote for autarky? The old have already consumed Y t S t. This decision is sunk. What matters for the old is the cost of repaying versus the cost of bribing the young with an amount X. ln(y t X ) > ln(y t + (1 + r)s t ) For the young the incentive compatibility constraint is: ln(y t+1 + X ) + ln(y t+1) > ln(y t+1 S t+1 ) + ln(y t+1 + (1 + r)s t+1 ) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

40 The question then is there an X which would make both agents vote for default? Clearly the answer is yes. Notice that for the old if: X > (1 + r)s t then the old are better off bribing. For the young, receiving a bribe X > S t makes them better off than they would be by borrowing in the first period and they are definitely better off in the second period because they don t have to repay. Thus, loaning against the lifetime budget constraint would yield a voting outcome of default! Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

41 So now we must ask why this can happen. The reason is that the OLG structure means that the costs of default are not borne by any currently alive generation. The costs of default are all borne by the future generations who return to autarky and utility U a. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

42 What can lenders do? Is there an amount they can lend and still hope to be repaid? Yes, though the amount will depend on our assumptions of what triggers a default. Let s assume that we only need one generation to prefer no-default to prevent default from occurring. Our focus is on the young, since the old always prefer default if X > (1 + r)s t. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

43 For the young, we must investigate the incentive compatibility constraint: ln(y t+1 + X ) + ln(y t+1) > ln(y t+1 S t+1 ) + ln(y t+1 + (1 + r)s t+1 ) We know that: S t+1 = Y t+1 2 Y t+1 2(1 + r) which we shall also use. Rearranging the incentive compatibility constraint gives: ln(y t+1 + X ) ln(y t+1 S t+1 ) > ln(y t+1 + (1 + r)s t+1 ) ln(y t+1) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

44 Substituting in the savings function gives: ln(y t+1 + X ) ln 1 2 (Y t+1 + Y t r ) > ln 1 2 (Y t+1 + (1 + r)y t+1 ) ln Y t+1 If we rearrange a little more, we find: ln 2(Y t+1 + X ) Y t+1 + Y t+1 1+r > ln Y t+1 + (1 + r)y t+1 2Y t+1 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

45 Next, note that we can dispense with the logarithms since the only way this is true is if: 2(Y t+1 + X ) Y t+1 + Y t+1 1+r > Y t+1 + (1 + r)y t+1 2Y t+1 Thus, solving a little bit more we should get: X > (Y t+1 + (1 + r)y t+1) 2 4(1 + r)y t+1 Y t+1 As long as this condition is satisfied then the young generation will prefer default. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

46 If we assume that the young vote for no-default when indifferent then the minimum bribe X is: X = (Y t+1 + (1 + r)y t+1) 2 4(1 + r)y t+1 Y t+1 We can investigate how this threshold is different from the optimal savings when repayment is certain. Recall that: S t+1 = Y t+1 2 Y t+1 2(1 + r) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

47 A little manipulation of the minimum bribe equation will yield: X = (1 + r)s 2 t+1 Y t+1 This is the threshold for default for the Young. Notice that the bribe required depends on the desired savings, the real interest rate and the income when old. So, how can the lender prevent the old from wanting and being able to bribe the young by this amount? They can t. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

48 Let s assume that endowments are identical across generations such that Y t = Y = 1 and Y t = Y = 4. We will also assume r = 0. Then (if my arithmetic is correct): X = S = 1.5 Thus, a bribe that is 37.5 % of the amount he or she could receive from international lenders would be sufficient to induce default in this example. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

49 This means that the bribe is easily affordable by the Old if they can borrow S!!! So a lender won t want to let the old borrow this amount. How do lenders prevent default from occurring? Let s recall the incentive compatibility condition for the young. ln(y t+1 + X ) + ln(y t+1) > ln(y t+1 S t+1 ) + ln(y t+1 + (1 + r)s t+1 ) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

50 Now, let s ask the question: what is the maximum bribe the old pay would pay? The answer is (1 + r)s t (recall S t < 0). If they pay more, then they are worse off. So let s use this in the young s incentive compatibility constraint for default. ln(y t+1 (1 + r)s t ) + ln(y t+1) > ln(y t+1 S t+1 ) + ln(y t+1 + (1 + r)s t+1 ) Notice that this is a difference equation in S t. Thus, given S t it tells us how much S t+1 can be before the young default. Thus, the optimal borrowing constraint will be in terms of debt level growth!!! Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

51 Rearranging this constraint and using our logarithms are irrelevant trick, we get: (Y t+1 + (1 + r)s t+1)(y t+1 S t+1 ) Y t+1 = Y t+1 (1 + r)s t This is a quadratic equation (so it is a bit messy) but it tells us how much the young can borrow as a function of the borrowing of the old before they could be bribed to default!! Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

52 This is useful. But there is one more thing to consider: do the young want to borrow the S t+1 that solves this problem? Only if it is less than or equal to their desired savings level when there is no repayment risk!! This let s us rule out some roots of the quadratic equation. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

53 An example. Suppose Y t = Y = 1 and Y t = Y = 4 for all t. Also assume that S t = 0.1 and r = 0. Then: (4 + (1)S t+1 )(1 S t+1 ) = 1 (1)( 0.1) = This is a bit ugly, but we can rearrange to find: S 2 t+1 + 3S t = 0 And using the quadratic equation we should find S t+1 = If the young borrow at least this amount then they will choose not to default. (There is also another root but the young will never borrow this much so it is irrelevant.) Then we ask, do the young wish to borrow this amount? Recall that optimal savings is: S t+1 = Y t+1 2 So they do want to borrow this amount! Y t+1 2(1 + r) = 1.5 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

54 This lending sequence could keep going with debts growing each period. But there is a point at which lenders cannot lend an amount such that the borrowers will repay. At which point they cease lending and there is a sudden stop! Suppose Y t = Y = 1 and Y t = Y = 4 for all t. Also assume that S t = 0.5 and r = 0. Then: (4 + (1)S t+1 )(1 S t+1 ) = 1 ( 0.5)1 = I find S t+1 = 1. But for the next generation, there is no real root that solves this equation. I.e. there is no level of borrowing that would prevent the Young from defaulting (and which they would want to borrow)! So lenders will not lend them anything! This is a sudden stop and a borrowing crisis! Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

55 So far we have seen how the long-run borrowing constraint and the incentive compatibility constraints affect the current account. In particular, when default is possible then lenders increase borrowing limits over time to give the young an incentive to borrow and repay. But this sequence has a limit and default crises can occur. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

56 So why then do we not see more examples of default? I.e. our simple model suggests that default is inevitable!! The answer is that it is unusual for incomes to be static over time. We observe countries running deficits for some periods and surpluses for others. Thus, unlike the simple model above, countries are sometimes borrowers and sometimes lenders. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

57 What might make a country switch from being a borrower to being a lender? One answer is income risk. The current account can be a risk-sharing mechanism between a country and the rest of the world. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

58 Let s go back to our basic example. We will imagine a representative consumer in the Home country who lives for two periods, current and future. We will imagine that she has preferences over consumption in both periods and that her preferences are such that she likes to consume both. We could think of her preferences as we often do: U = u(c) + u(c ) The consumer has income in both periods and this income is assumed to be exogenously specified. Now, however, let s assume that the consumer faces income risk. Specifically, the consumer s income can be either Y h with probability p or Y l with probability 1 p. We will assume that the consumer knows her current period income. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

59 Thus, the consumer is in one of two possible states. She either has income Y h in which case she faces period budget constraints: and with probability p and with probability 1 p: C + S = Y h C = Y h + (1 + r)s C = Y l + (1 + r)s If her income is Y l then her first period budget constraint is: C + S = Y l The second period budget constraints are not affected. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

60 So how do the probabilities affect the consumer s problem assuming she is in state Y h? Well, with probability p, she faces the lifetime budget constraint: C + C 1 + r = Y h + Y h 1 + r With probability 1 p, she faces the lifetime budget constraint: C + C 1 + r = Y h + Y l 1 + r Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

61 Turning to her utility, with probability p she receives: u(c) + u(c (Y h )) where I have written C (Y h ) to indicate the level of consumption with second period income Y h. With probability 1 p, she receives: u(c) + u(c (Y l )) (where I have written C (Y l ) to indicate the level of consumption with second period income Y l. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

62 Thus, her expected utility is: pu(c) + pu(c (Y h )) + (1 p)u(c) + (1 p)u(c (Y l )) This reduces to: E[U] = u(c) + pu(c (Y h )) + (1 p)u(c (Y l )) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

63 How do we determine what the consumer chooses for consumption and savings? We maximize her expected utility. To do this we can simply write expected utility as: E[U] = u(y h S) + pu(y h + S(1 + r)) + (1 p)u(y l + S(1 + r)) Note I am assuming that the consumer initially has high income. Taking the first order conditions with respect to S then yields the optimal savings response given the income uncertainty. E[U] S = u (Y h S)+p(1+r)u (Y h +S(1+r))+(1 p)(1+r)u (Y l +S(1+r)) Solving this for S gives the savings decision in the face of income uncertainty. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

64 Let s consider an example. Suppose: Then: u(c) = αc 2 E[U] = 2α(Y h S)+p2(1+r)α(Y h +S(1+r))+(1 p)2(1+r)α(y l +S(1+r S If we simplify this expression we will find: S = 2αY 2pα(1 + r)y h 2(1 p)α(1 + r)y l 2α(1 + (1 + r) 2 ) Note that: S p = (1 + r)(y l Y h ) 1 + (1 + r) 2 < 0 So as the risk of low income falls, savings falls too. Thus, as risk falls the current account falls. The converse is also true. As risk of low income increases, then savings increases and the current account rises. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

65 What are some examples of countries that face risk of lower future incomes? We might expect that the oil producing nations are enjoying current high incomes but possibly lower future incomes due to the finite nature of their resource. If our little example is true and these countries are planning for the future then we might expect to see these countries running current account surplusses and high net wealth. Indeed we do. Abu Dhabi ($654 Billion), Norway ($ 471 Billion) and Saudia Arabia ($ 415 Billion) are among the countries with the largest net external wealth positions. Only China is larger than these 3. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

66 Perhaps interestingly, both Norway and Abu Dhabi have about 5 million inhabitants. Thus, in per-capita terms Abu Dhabi has roughly $125,000 per person. Norway has roughly $100,000 per person. These are, in per capita terms, the largest net external wealth positions worldwide. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

67 The last issue with respect to the current account that we will examine is the role of investment. Recall the simple national income accounting statement: CA = S I Thusfar, we have assumed that I = 0. We will now dispense with that assumption. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

68 Clearly, the motivation for saving and the motivation for investment can be quite different. Savings is the act of shifting consumption across periods. Investment is the act of increasing the means of future production. We shall see that they are related but nevertheless quite distinct. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

69 How important is investment to the current account? Famously, two economists Feldstein and Horioka found evidence that suggested S = I in most economies. Since national wealth and the expected returns to investment are thought to be quite different across countries this would be a puzzle. In fact, some people still refer to apparent finding as the Feldstein-Horioka puzzle. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

70 A different economist, Nobel laureate Robert Lucas asked the question: Why doesn t capital flow from rich countries to poor countries? In a very basic model in which production technologies are equalized across countries, one should expect to see the same capital-labour ratios. We don t. In fact, using these basic models suggest that the returns to investment in India during the 1990 s might have been 58 times the returns in the US over the same period. This too is perhaps puzzling. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

71 Lucas suggested that the answer to this question was largely due to productivity differences across countries. Productivity is a measure of how much output a specific amount of inputs can produce in a given period of time. Lucas found that productivity in the US was much greater than in India and that this accounted for the majority of the difference in rates of return. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

72 How might we model productivity? Consider the canonical model of macro output: Y = zf (K, L) z is referred to as total factor productivity because it measures the effect of both capital, K, and labour, L, on output, Y. The function F (K, L) describes how the production process happens. One often used function is called Cobb-Douglas production: F (K, L) = K α L 1 α Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

73 Thus, total output is given by: Y = zk α L 1 α α has a special interpretation. If we take a logarithmic transformation and differentiate, we will find: dy Y = dz z + αdk K + (1 α)dl L If we hold productivity and labour fixed, then the elasticity of output with respect to capital is: dy K dk Y = α The converse is true for labour and 1 α. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

74 This production function also exhibits constant returns to scale (CRS). To see this, suppose we increase both input factors by µ. Then: zf (µk, µl) = z(µk) α (µl) 1 α = zµ α+1 α K α L 1 α = µzk α L 1 α = µy In words, changing both inputs by the same proportionate amount simply changes final output by the same amount. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

75 CRS means we can use a simple trick to transformation the production function into per-capita production. Define µ = 1/L. Then by the above algebra: zf (µk, µl) = zf ( K, 1) = zf (k) L where k = K/L. Using the Cobb-Douglas assumption above, this yields: Y /L = zk α (L/L) 1 α = zk α = y where y = Y /L. We now have a simple representation of per-capita production. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

76 Why does this matter? Because we can write the marginal return to investment as: MP K = Y K = αzkα 1 = α y k Thus, the marginal return to capital depends on productivity, the elasticity of output with respect to capital and per-capita capital stock. We usually assume that investment returns to capital investment equal the marginal product of capital. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

77 So now we can begin to answer the question: why does investment flow across countries? The answer will be to seek the highest marginal product. This implies that savings and investment are necessarily very correlated. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

78 One simple exercise is to examine how much of additional dollar of savings will be invested in the domestic country. This question can be written as: I = β S Here β is the fraction of an additional dollar of domestic savings that becomes domestic investment. Feenstra and Taylor (p ) report estimates of β for three types fo countries. They find β = 0.26 for the EU countries, β = 0.39 fr developed countries and β = 0.67 for emerging markets. This suggests that financial market integration may have a role to play. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

79 The disconnect between savings and investment raises an interesting question. If the returns to investment are sufficiently high, why not fund the investment domestically? For instance, Norway financed much of the North Sea oil exploration via international investment. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

80 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

81 The reason is that it is costly to save in terms of foregone consumption. Think back to our basic intertemporal model. The cost of one unit of additional savings is the marginal utility of consumption. Thus, a relatively poor country that has a high marginal product of capital may run a current account deficit because being poor makes the marginal utility cost of savings too high to finance the investment itself, but the high investment returns are atttractive to richer investors. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

82 Let s return to the basic marginal product equation and let s assume that there are two countries, i and j. The ratio of the marginal products of capital is: MPK i MP j = αi y i k i K α j y j k j We usually assume that countries have the same production processes so that α i = α j. In which case: MP i K MP j K = y i k j y j k i Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

83 If international capital markets are complete then we would expect MPk i = MPj K. If we examine data from the US (j) and Mexico (i), we find that: y i y j = 0.43 and k j k i = 3 This suggests that the relative marginal productivity is roughly 1.3. So they are close. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

84 What is the reason? Productivity differences. Lucas noted their importance for explaining why capital doesn t flow from the US to India. And the same is true in our example for Mexico and the US. y k = zkα 1 So another way to write the ratio of the marginal products is: MP i K MP j K = zi z j (ki k j )α 1 The term z i /z j is the relative productivity difference. For most countries, relative to the US this is less than one. (See Table 6-5 in the textbook). Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

85 So one commonly accepted conclusion to explain why we don t observe more capital investment into poor countries is productivity differences. Where do these differences come from? Mainly, education, regulation and transportation (infrastructure). A second type of difference observed by Caselli and Feyrer (QJE 2007) is that the prices of factors differ across countries. Capital, in particular, is more expensive in poor countries and so investors need a higher payoff to compensate for the higher price. Once these corrections are made, marginal products seem reasonably equalized across countries. One conclusion from these studies is that giving aid money for capital investment will not matter much for output growth in developing countries. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

86 One final factor that matters is current account risk. Why? Because investors are concerned about repayment risk. As we have seen, repayment risk is both empirically and theoretically a concern. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

87 Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

88 Risk premiums can shift capital flows from high marginal product regions to lower marginal product regions. To see why, consider that the risk-free real interest rate is usually equal to the marginal product of capital. Thus, r i = MPK i = y i αi k i = r where the last equality stems from the Fisher effect and also our finding that the marginal products across countries are basically the same. If country i has a real risk premium of, say, 6.4 percent, then it requires a real return that is 6.4 per cent higher than r to compensate investors for risk! So in this instance, r i = r implies that the expected return in i is lower than the world return by 6.4 per cent. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

89 There are a number of concerns that we should bear in mind using the production approach to the current account. First, the production approach to the current account suggest that capital inflows should happen quickly as investment flows into a country to equalize the marginal products. For instance, suppose country i initially has a capital stock k0 i. If the world real interest rate is r and assuming no risk premia, then it s optimal leve of capital should be: ˆk i = α i y i Thus the current account inflow should be CA = (ˆk i k i 0 ). r Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

90 Second, we have assumed that the labor share of income is identical across countries. There is some reason to believe that this is roughly true (Gollin, JPE, 2002). But there is likely to be some difference across countries. Lowering α lowers the marginal product of capital. Finally, as I mentioned above, this model assumes that investment goods can be acquired at the same price around the world. This is often not true both for reasons of taxation (exise taxes) and also networks and transportation costs. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

91 The Balassa-Samuelson Hypothesis Relative productivities also matter for exchange rates. This point was first noticed by Balassa and Samuelson. To see the effect of relative productivities for the real exchange rate, we shall use a simple traded goods/non-traded goods model. We shall assume that there are two countries, H and F. Both countries have two goods, traded goods T and non-traded goods N. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

92 The Balassa-Samuelson Hypothesis We shall assume that the price of traded goods is the same in both countries. Thus, letting p be the price in country F, the price of traded goods must satisfy: p T = Ep T = 1 where E is the exchange rate and the = 1 is just a normalization. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

93 The Balassa-Samuelson Hypothesis We will also assume that productivity in traded goods determines wages. Let s define A to be the productivity in country H and A to be the productivity in country F. If we assume that firms use only labour to produce and that the labour market is competitive then: and w = A Ew = A Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

94 The Balassa-Samuelson Hypothesis For non-traded goods we make a crucial assumption: productivities are equal everywhere. For simplicity we shall assume the productivity in the non-traded good sector is equal to 1. If wages are equalized across sectors and assuming the price of the non-traded good equals the wage then: and p N = A Ep N = A Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

95 The Balassa-Samuelson Hypothesis Now what happens if there is a change in productivity. First, let s consider the case where A changes by A. Assume that aggregate prices are a weighted average of traded and non-traded goods prices where η is the weighted on non-traded goods. Then: p p = (1 η) p T p T + η p N p N But since the traded goods price is always = 1 by assumption then: p p = η A A Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

96 The Balassa-Samuelson Hypothesis By the same logic, the same is true for the Foreign country. Thus: which reduces to: Ep Ep = (1 η) Ep T Ep T + η Ep N Ep N Ep Ep = η A A Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

97 The Balassa-Samuelson Hypothesis Now we need to consider the real exchange rate. The real exchange is given by: q = Ep p Combining the expressions for Ep and p yields: q q = η( A A A A ) Thus, relative productivities in the traded goods sector drive relative prices through their effects on wages in and prices in the non-traded goods sector. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

98 The Balassa-Samuelson Hypothesis The Balassa-Samuelson hypothesis has some implications for exchange rates that are worth noting. The most important observation is that it implies that PPP does not hold! There is no reason why q = 1 in the Balassa-Samuelson model. Let s consider the real exchange rate equation again. In growth rates this implies: E E = q q q = Ep p + ( p p Substituting for the real exchange rate, we get: p p ) ( ) E E = η( A A A A ) + ( p p p p ) Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

99 The Balassa-Samuelson Hypothesis The Balassa-Samuelson hypothesis has empirical support (See figure 11-3 in the textbook). It can also be used to evaluate whether an exchange rate is over-valued or undervalued. In 2000, the real exchange rate between the US and China was q = Taking into account productivities suggested that the equilibrium level should be ˆq = Thus the yuan was undervalued by 38 per cent (0.088/0.231) according to equation ( ). Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91

100 The Balassa-Samuelson Hypothesis The final point to note about productivities and exchange rates is that the relative productivities may themselves be a function of the nominal exchange rates. This is sort of a reverse Balassa-Samuelson hypothesis. The thesis here is that when a country must import productive capital, low values of its currency make that capital expensive. This in turn lowers productive capital investment and thus lowers productivity. Some commentators argued that Canada suffered from this reverse productivity effect during the late 1990 s. The weak exchange rate made productive capital investment expensive while at the same time the weak currency made exporting easier. Some feel this may, in part, explain why Canada s productivity has tended to lag that of the US. Geoffrey Dunbar (UBC, Winter 2013) Economics 456 March 14, / 91