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�� production function is Cobb-Douglas. Then the expression for steady state R∗ is: R∗ = αAk∗α−1. Plug in the steady state expression for k∗: R∗ = αA ( sA δ ) α−1 1−α. (5.46) (5.47) The exponent here is −1, which means we can ﬂip numerator and denominator. In other words, the A cancel out, leaving: R∗ = αδ s. (5.48) As mentioned above, we can think about there being two eﬀects of an increase in A on the variables of the model. There is the direct eﬀect, which is what happens holding kt ﬁxed. Then there is an indirect eﬀect that comes about because higher A triggers more capital accumulation. This indirect eﬀect is qualitatively the same as what happens when s changes. What diﬀers across the two cases is that the increase in A causes an immediate eﬀect on the variables in the model. Figure 5.11: Dynamic Path of Output Growth As we did for the case of an increase in s, we can think about what happens to the growth rate of output following a permanent increase in A. This is shown in Figure 5.11. Qualitatively, it looks similar to Figure 5.8, but the subtle diﬀerence is that output growth jumps up immediately in period t, whereas in the case of an increase in s there is no increase 96 𝑔𝑡𝑦 𝑡𝑖𝑚𝑒 𝑡 0 in output growth until period t + 1. In either case, the extra growth eventually dissipates, with output growth ending back up at zero. At this point, it is perhaps useful to pause for a moment and foreshadow some of what we will do in the next chapter. We sat out to study economic growth, but then wrote down a model in which the economy naturally converges to a steady state in which there is no growth. Is the Solow model therefore ill-suited to study sustained growth over long periods of time, of the type documented in Chapter 4? We think not. As we will show in Chapter 6, the model can be tweaked in such a way that there is steady state growth. But the
basic model presented in this chapter gives us the insight of where that steady state growth must come from. Sustained growth cannot come from capital accumulation per se. As shown above, an increase in s triggers temporarily high growth because of capital accumulation, but this dissipates and eventually growth settles back down to zero. Even if an economy repeatedly increased its saving rate, it would eventually run out of room to do so (as s is bound from above by 1), so even repeated increases in s cannot plausibly generate sustained growth over long periods of time. What about changes in A? It is true that a one time change in A only generates a temporary burst of output growth which is magniﬁed due to capital accumulation as the economy transitions to a new steady state. But unlike changes in s, there is no logical limit on productivity repeatedly increasing over time. This means that continual productivity improvements could plausibly generate sustained growth in output per capita over time. 5.5 The Golden Rule As discussed in reference to Figure 5.7, there is an ambiguous eﬀect of an increase in the saving rate on the steady state level of consumption per worker. Increasing the saving rate always results in an increase in k∗, and hence an increase in y∗. In other words, a higher saving rate always results in a bigger “size of the pie.” But increasing the saving rate means that households are consuming a smaller fraction of the pie. Which of these eﬀects dominates is unclear. We can see these diﬀerent eﬀects at work in the expression for the steady state consumption per worker: c∗ = (1 − s)Af (k∗). (5.49) A higher s increases f (k∗) (since a higher s increases k∗), but reduces 1 − s. We can see that if s = 0, then c∗ = 0. This is because if s = 0, then k∗ = 0, so there is nothing at all available to consume. Conversely, if s = 1, then we can also see that c∗ = 0. While f (k∗) may be big if s = 1, there is nothing left for households to consume. We can therefore intuit that c∗ must be increasing in s when s is near 0, and decreasing in s when s is near 1. A 97 hypothetical plot of c∗ and against s is shown
below: Figure 5.12: s and c∗: The Golden Rule We can characterize the golden rule mathematical via the following condition: Af ′(k∗) = δ. (5.50) The derivation for (5.50) is given below. What this says, in words, is that the saving rate must be such that the marginal product of capital equals the depreciation rate on capital. This expression only implicitly deﬁnes s in that k∗ is a function of s; put diﬀerently, the Golden Rule s (denoted by sgr), must be such that k∗ is such that (5.50) holds. Mathematical Diversion We can derive an expression that must hold at the Golden Rule using the total derivative (also some times called implicit diﬀerentiation). The steady state capital stock is implicitly deﬁned by: sAf (k∗) = δk∗. (5.51) Totally diﬀerentiate this expression about the steady state, allowing s to vary: sAf ′(k∗)dk∗ + Af (k∗)ds = δdk∗. Solve for dk∗: [sAf ′(k∗) − δ]]dk∗ = −Af (k∗)ds. (5.52) (5.53) 98 𝑠𝑠 𝑐𝑐∗ 𝑠𝑠𝑔𝑔𝑔𝑔 Steady state consumption is implicitly deﬁned by: c∗ = Af (k∗) − sAf (k∗). (5.54) Totally diﬀerentiate this expression: dc∗ = Af ′(k∗)dk∗ − sAf ′(k∗)dk∗ − Af (k∗)ds. (5.55) Re-arranging terms: dc∗ = [Af ′(k∗) − sAf ′(k∗)] dk∗ − Af (k∗)ds. (5.56) From (5.53), we know that −Af (k∗)ds = [sAf ′(k∗) − δ] dk∗. Plug this
into (5.56) and simplify: dc∗ = [Af ′(k∗) − δ] dk∗. Divide both sides of (5.57) by ds: dc∗ ds = [Af ′(k∗) − δ] dk∗ ds. (5.57) (5.58) For s to maximize c∗, it must be the case that dc∗ ds be the case if: = 0. Since dk∗ ds > 0, this can only Af ′(k∗) = δ. (5.59) Figure 5.13 below graphically gives a sense of why (5.50) must hold. It plots yt = Af (kt), it = sAf (kt), and δkt against kt. For a given kt, the vertical distance between yt and it is ct, consumption. At the steady state, we must have sAf (k∗) = δk∗; in other words, the steady state is where the plot of it crosses the plot of δkt. Steady state consumption is given by the vertical distance between the plot of yt and the plot of it at this k∗. The Golden rule saving rate is the s that maximizes this vertical distance. Graphically, this must be where the plot of yt = Af (kt) is tangent to the plot of δkt (where sAf (kt) crosses in the steady state). To be tangent, the slopes must equal at that point, so we must have Af ′(kt) = δ at the Golden rule. In other words, at the Golden Rule, the marginal product of capital equals the depreciation rate on capital. 99 Figure 5.13: The Golden Rule Saving Rate What is the intuition for this implicit condition characterizing the Golden rule saving rate? Suppose that, for a given s, that Af ′(k∗) > δ. This means that raising the steady state capital stock (by increasing s) raises output by more than it raises steady state investment (the change in output is the marginal product of capital, Af ′(k∗), and the change in steady state investment is δ). This means that consumption increases, so this s cannot be the s which maximizes steady state consumption. In contrast, if s is such that Af �
�(k∗) < δ, then the increase in output from increasing the steady state capital stock is smaller than the increase in steady state investment, so consumption declines. Hence that s cannot be the s which maximizes steady state consumption. Only if Af ′(k∗) = δ is s consistent with steady state consumption being as big as possible. Let us now think about the dynamic eﬀects of an increase in s, depending on whether the saving rate is initially above or below the Golden Rule. The important insight here is that the Golden Rule only refers to what the eﬀect of s is on steady state consumption. An increase in s always results in an immediate reduction in ct in the short run – a larger fraction of an unchanged level of income is being saved. After the initial short run decline, ct starts to increase as the capital stock increases and hence income increases. Whether the economy ends up with more or less consumption in the long run depends on where s was initially relative to the Golden Rule. If initially s < sgr, then a small increase in s results in a long run increase in consumption in the new steady state. If s > sgr, then an increase in s 100 𝑦𝑦𝑡𝑡,𝑖𝑖𝑡𝑡,𝛿𝛿𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡 𝛿𝛿𝑘𝑘𝑡𝑡 𝑖𝑖𝑡𝑡=𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡) 𝑘𝑘∗ 𝑦𝑦𝑡𝑡=𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡) 𝑦𝑦∗ 𝑖𝑖∗ 𝑐𝑐∗ results in a long run decrease in consumption in the new steady state. These features can be seen in Figure 5.14, which plots out hypothetical time paths of consumption. Prior to t, the economy sits in a steady state. Then, in period t, there is an increase in
s. Qualitatively, the time path of ct looks the same whether we are initially above or below the Golden Rule. What diﬀers is whether ct ends up higher or lower than where it began. Figure 5.14: Eﬀects of ↑ s Above and Below the Golden Rule We can use these ﬁgures to think about whether it is desirable to increase s or not. One is tempted to say that if s < sgr, then it is a good thing to increase s. This is because a higher s results in more consumption in the long run, which presumably makes people better oﬀ. This is not necessarily the case. The reason is that the higher long run consumption is only achieved through lower consumption in the short run – in other words, there is some short run pain in exchange for long run gain. Whether people in the economy would prefer to endure this short run pain for the long run gain is unclear; it depends on how impatient they are. If people are very impatient, the short run pain might outweigh the long run gain. Moreover, if the dynamics are suﬃciently slow, those who sacriﬁced by consuming less in the present may be dead by the time the average level of consumption increases. In that case, people would decide to save more only if they received utility from knowing future generations would be better oﬀ. Without saying something more speciﬁc about how people discount the future (i.e. how impatient they are) or how they value the well-being of future generations, it is not possible to draw normative conclusions about whether or not the saving rate should increase. What about the case where s > sgr. Here, we can make a more deﬁnitive statement. In particular, households would be unambiguously better oﬀ by reducing s. A reduction in 101 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑐𝑐∗ 𝑐𝑐∗ 𝑐𝑐0∗ 𝑐𝑐1∗ 𝑐𝑐1∗ 𝑐
𝑐0∗ ↑𝑠𝑠,𝑠𝑠<𝑠𝑠𝑔𝑔𝑔𝑔 ↑𝑠𝑠,𝑠𝑠>𝑠𝑠𝑔𝑔𝑔𝑔 s would result in more consumption immediately, and higher consumption (relative to the status quo) at every subsequent date, including the new steady state. Regardless of how impatient people are, a reduction in s gives people more consumption at every date, and hence clearly makes them better oﬀ. We say that an economy with s > sgr is “dynamically ineﬃcient.” It is ineﬃcient in that consumption is being “left on the table” because the economy is saving too much; it is dynamic because consumption is being left on the table both in the present and in the future. 5.6 Summary • The production function combines capital and labor into output. It is assumed that both inputs are necessary for production, that the marginal products of both inputs are positive but diminishing, and there the production function exhibits constant returns to scale. • Capital is a factor of production which must itself be produced, which helps produce output, and which is not completely used up in the production process. Investment is expenditure on new physical capital which becomes productive in the future. • The Solow model assumes that the household (or households) obey simple rules of thumb which are not necessarily derived from optimizing behavior. The households consumes a constant fraction of its income (and therefore saves/invests a constant fraction of its income) and supplies labor inelastically. • Given these assumptions, the Solow model can be summarized by one central equation that characterizes the evolution of the capital stock per worker. • Starting with any initial level of capital per worker greater than zero, the capital stock converges to a unique steady state. • An increase in the saving rate or the productivity level results in temporarily higher, but not permanently higher, output growth. • The golden rule is the saving rate that maximizes long run consumption per capita. If the saving rate is less than the golden rule saving rate, consumption must be lower in the short term in order to be higher in the long run. If the saving rate is higher than the golden rule, consumption can increase at every point in time. Key Terms 102 • Capital • Constant returns to scale •
Saving rate • Inada conditions • Steady state • Golden rule Questions for Review 1. We have assumed that the production function simultaneously has constant returns to scale and diminishing marginal products. What do each of these terms mean? Is it a contradiction for a production function to feature constant returns to scale and diminishing marginal products? Why or why not? 2. What are the Inada conditions? Explain how the Inada conditions, along with the assumption of a diminishing marginal product of capital, ensure that a steady state capital stock exists. 3. Graph the central equation of the Solow model. Argue that a steady state exists and that the economy will converge to this point from any initial starting capital stock. 4. What is the Golden Rule saving rate? Is it diﬀerent than the saving rate which maximizes present consumption? 5. What would be the saving rate which would maximize steady state output? Would the household like that saving rate? Why or why not? 6. In words, explain how one can say that a household is deﬁnitely better oﬀ from reducing the save rate if it is initially above the Golden Rule, but cannot say whether or not a household is better or worse if it increases the saving rate from below the Golden Rule. 7. Critically evaluate the following statement: “Because a higher level of A does not lead to permanently high growth rates, higher levels of A are not preferred to lower levels of A.” Exercises 103 1. Suppose that the production function is the following: Yt = A [αK ν−1 ν t + (1 − α)N t ν−1 ν ν ν−1 ]. It is assumed that the parameter ν ≥ 0 and 0 < α < 1. (a) Prove that this production function features constant returns to scale. (b) Compute the ﬁrst partial derivatives with respect to Kt and Nt. Argue that these are positive. (c) Compute the own second partial derivatives with respect to Kt and Nt. Show that these are both negative. (d) As ν → 1, how do the ﬁrst and second partial derivatives for this production function compare with the Cobb-Douglas production discussed in the text? 2. Suppose that you have a standard Solow model. The central equation governing the dynamics of the level of capital is given by (5.18). In terms of capital per
worker, the central equation is given by (5.19). The production function has the normal properties. (a) Suppose that the economy initially sits in a steady state in terms of the capital stock per worker, kt. Suppose that, at time t, the number of workers doubles (say, due to an inﬂux of immigrants). The number of workers is expected to remain forever thereafter constant at this new higher level, i.e. Nt+1 = Nt. Graphically analyze how this will impact the steady state capital stock per worker and the dynamics starting from an initial capital stock. (b) Draw diagrams plotting out how capital, output, and the real wage ought to respond dynamically to the permanent increase in the workforce. 3. Suppose that we have a Solow model with one twist. The twist is that there is a government. Each period, the government consumes a fraction of output, sG. Hence, the aggregate resource constraint is: Yt = Ct + It + Gt. Where Gt = sGYt. Deﬁne private output as Y p = Yt − Gt. Suppose that t investment is a constant fraction, s, of private output (consumption is then 1 − s times private output). Otherwise the model is the same as in the text. 104 (a) Re-derive the central equation of the Solow model under this setup. (b) Suppose that the economy initially sits in a steady state. Suppose that there is an increase in sG that is expected to last forever. Graphically analyze how this will aﬀect the steady state value of the capital stock per worker. Plot out a graph showing how the capital stock per worker will be aﬀected in a dynamic sense. 4. Suppose that we have a standard Solow model with a Cobb-Douglas produc- tion function. The central equation of the model is as follows: kt+1 = sAkα t + (1 − δ)kt. Consumption per worker is given by: ct = (1 − s)Akα t. (a) Solve for an expression for the steady state capital stock per worker. In doing so, assume that the level of productivity is ﬁxed at some value A. (b) Use your answer on the previous part to solve for an expression for steady state consumption per worker. (c) Use calculus to derive an expression for the s which maximizes steady
state consumption per worker. 5. Excel Problem. Suppose that you have a standard Solow model with a Cobb-Douglas production function. The central equation of the model can be written: kt+1 = sAkα t + (1 − δ)kt. Output per worker is given by: yt = Akα t. Consumption per worker is given by: ct = (1 − s)yt. (a) Suppose that A is constant at 1. Solve for an expression for the steady state capital per worker, steady state output per worker, and steady state consumption per worker. 105 (b) Suppose that α = 1/3 and δ = 0.1. Create an Excel sheet with a grid of values of s ranging from 0.01 to 0.5, with a gap of 0.01 between entries (i.e. you should have a column of values 0.01, 0.02, 0.03, and so on). For each value of s, numerically solve for the steady state values of capital, output, and consumption per worker. Produce a graph plotting these values against the diﬀerent values of s. Comment on how the steady state values of capital, output, and consumption per worker vary with s. (c) Approximately, what is the value of s which results in the highest steady state consumption per worker? Does this answer coincide with your analytical result on the previous question? 6. Excel Problem. Suppose that you have a standard Solow model with a Cobb-Douglas production function. The central equation of the model can be written: kt+1 = sAkα t + (1 − δ)kt. (a) Analytically solve for an expression for the steady state capital stock per worker. (b) Suppose that A = 1 and is ﬁxed across time. Suppose that s = 0.1 and δ = 0.10. Suppose that α = 1/3. Create an Excel ﬁle. Using your answer from the previous part, numerically solve for k∗ using these parameter values. (c) Create a column in your Excel sheet corresponding to periods. Let these periods run from period 1 to period 100. Suppose that the capital stock per worker equals its steady state in period 1. Use the central equation of the Solow model to compute the capital stock in period 2, given this capital stock in period 1. Then iterate again, computing the capital
stock in period 3. Continue on up until period 9. What is true about the capital stock in periods 1 through 9 when the capital stock starts in the steady state in period 1? (d) Suppose that in period 10 the saving rate increases to 0.2 and is expected to forever remain there. What will happen to the capital stock in period 10? (e) Compute the capital stock in period 11, given the capital stock in period 10 and the new, higher saving rate. Then iterate, going to period 11, and then period 12. Fill your formula down all the way to period 100. Produce a plot of the capital stock from periods 1 to 100. 106 (f) About how many periods does it take the capital stock to get halfway to its new, higher steady state value when s increases from 0.1 to 0.2? 107 Chapter 6 The Augmented Solow Model We developed the basic Solow model in Chapter 5. The model is intended to study long run growth, but has the implication the economy converges to a steady state in which it does not grow. How, then, can the model be used to understand growth? In this chapter, we augment the basic Solow model to include exogenous growth in both productivity and the population. Doing so requires transforming the variables of the model, but ultimately we arrive at a similar conclusion – the model converges to a steady state in which the transformed variables of the model are constant. As we will see, the transformed variables being constant means that several of the actual variables will nevertheless be growing. This growth comes from the assumed exogenous growth in productivity and population. The model makes predictions about the long run behavior of these variables which is qualitatively consistent with the stylized time series facts we documented in Chapter 4. In the augmented model, we conclude that the only way for the economy to grow over long periods of time is from growth in productivity and population. For per capita variables to grow, productivity must grow. Increasing the saving rate does not result in sustained growth. In a sense, this is a bit of a negative result from the model, since the model takes productivity growth to be exogenous (i.e. external to the model). But this result does pinpoint where sustained growth must come from – it must come from productivity. What exactly is productivity? How can we make it grow faster? Will productivity growth continue forever into the future? We address these questions in this chapter. 6.1 Introducing Productivity and Population Growth The production
function is qualitatively identical to what was assumed in Chapter 5, given by equation (5.1). What is diﬀerent is how we deﬁne labor input. In particular, suppose that the production function is given by: Here, Yt is output, A is a measure of productivity which is assumed to be constant going Yt = AF (Kt, ZtNt). (6.1) 108 forward in time, and Kt is capital. Nt is still labor input. The new variable is Zt. We refer to Zt as “labor augmenting productivity.” The product of this variable and labor input is what we will call “eﬃciency units of labor.” Concretely, consider an economy with N1,t = 10 and Z1,t = 1. Then there are 10 eﬃciency units of labor. Consider another economy with the same labor input, N2,t = 10, but suppose Z2,t = 2. Then this economy has 20 eﬃciency units of labor. Even though the economies have the same amount of labor, it is as if the second economy has double the labor input. An equivalent way to think about this is that the second economy could produce the same amount (assuming equal capital stocks and equal values of At) with half of the actual labor input. At a fundamental level, A and Zt are both measures of productivity and are both taken to be exogenous to the model. The higher are either of these variables, the bigger will be Yt for given amounts of Kt and Nt. We refer to A as “neutral” productivity because it makes both capital and labor more productive. Zt is labor augmenting productivity because it only (directly) makes labor more productive. We will make another distinction between the two, which is not necessary but which simpliﬁes our analysis below. In particular, we will use Zt to control growth rates of productivity in the long run, while A will impact the level of productivity. For this reason, as Zt will be evolving going forward in time while A will not, we need to keep a time subscript on Zt but can dispense with it (as we did in the previous chapter) for A. This ought to become clearer in the analysis below. The function, F (⋅), has the same properties laid out in Chapter 5
: it is increasing in both arguments (ﬁrst partial derivatives positive), concave in both arguments (second partial derivatives negative), with a positive cross-partial derivative, and constant returns to scale. The rest of the setup of the model is identical to what we had in the previous chapter. In particular, we have: Kt+1 = It + (1 − δ)Kt It = sYt. (6.2) (6.3) The real wage and rental rate on capital are still equal to the marginal products of capital and labor. For the rental rate, this is: Rt = AFK(Kt, ZtNt). (6.4) For the real wage, we have to be somewhat careful – we will have wt equal the marginal product of labor, but it is important to note that the marginal product of labor is the partial derivative of the production function with respect to actual labor, Nt, not eﬃciency units of 109 labor, ZtNt. This means that the real wage can be written: wt = AZtFN (Kt, ZtNt). (6.5) Why does the Zt show up outside of FN (⋅) in (6.5)? FN (⋅) here denotes the partial derivative of F (⋅) with respect to the argument ZtNt; the derivative of ZtNt with respect to Nt is Zt. Hence, we are using the chain rule to derive (6.5). We will make two assumptions on how Nt and Zt evolve over time. Like we did in Chapter 5, we will assume that labor is supplied inelastically (meaning it doesn’t depend on the wage or anything else in the model). Unlike Chapter 5, however, we will allow Nt to grow over time to account for population growth. In particular, let’s assume: Nt = (1 + n)Nt−1, n ≥ 0. (6.6) In other words, we allow Nt to grow over time, where n ≥ 0 is the growth rate between two periods. If we iterate back to period 0, and normalize the initial level N0 = 1, then we get: Nt = (1 + n)t. (6.7) Equation (6.7) embeds what we had in the previous chapter as a special
case. In particular, if n = 0, then Nt = 1 at all times. What we are assuming is that time begins in period t = 0 with a representative household which supplies N0 = 1 unit of labor. Over time, the size of this household grows at rate n, but each member of the household continues to supply 1 unit of labor inelastically each period. We will also allow Zt to change over time. In particular, assume: Zt = (1 + z)Zt−1, z ≥ 0. (6.8) Here, z ≥ 0 is the growth rate of Zt across periods. As with labor input, normalize the period 0 level to Z0 = 1 and iterate backwards, meaning we can write (6.8) as: Zt = (1 + z)t. (6.9) Again, the setup we had in Chapter 5 is a special case of this. When z = 0, then Zt = 1 at all times and could be omitted from the analysis. As we will see below, z > 0 is going to be the factor which allows the model to account for growth in output per worker in the long run. 110 In summary, the equations characterizing the augmented Solow model can be written: Kt+1 = It + (1 − δ)Kt It = sYt Yt = AF (Kt, ZtNt) Yt = Ct + It Rt = AFK(Kt, ZtNt) Nt = (1 + n)t Zt = (1 + z)t wt = AZtFN (Kt, ZtNt). (6.10) (6.11) (6.12) (6.13) (6.14) (6.15) (6.16) (6.17) The behavior of the capital stock is what drives everything else. As in Chapter 5, we can combine equations to focus on the capital accumulation: Kt+1 = sAF (Kt, ZtNt) + (1 − δ)Kt. (6.18) (6.18) describes how the capital stock evolves, given an exogenous initial capital stock, Kt, the exogenous levels of Nt and Zt (which evolve according to (6.15) and (6.16)), the exogenous value of A, and the value of
the parameters s and δ. Once we know how Kt evolves across time, we can ﬁgure out what everything else is. As in Chapter 5, for the analysis to follow it is helpful to re-write the equations in transformed variables. In Chapter 5, we re-wrote the equations in terms of per worker variables, with xt = Xt/Nt denoting a per worker version of some variable Xt. Let’s now re-write the equations in terms of per eﬃciency units of labor. In particular, for some variable Xt, deﬁne ̂xt = Xt. Transforming the variables in this way is useful because variables relative ZtNt to per eﬃciency units of labor will converge to a steady state, while per worker / per capita variables and level variables will not. Let’s start with the capital accumulation equation. Begin by dividing both sides of (6.18) by ZtNt: Kt+1 ZtNt = sAF (Kt, ZtNt) ZtNt + (1 − δ) Kt ZtNt. (6.19) Because we continue to assume that F (⋅) has constant returns to scale, we know that ) = F (̂kt, 1). Deﬁne f (̂kt) = F (̂kt, 1). Hence, (6.19) can be written: = F ( Kt ZtNt, ZtNt ZtNt F (Kt,ZtNt) ZtNt Kt+1 ZtNt = sAf (̂kt) + (1 − δ)̂kt. 111 (6.20) To get the left hand side of (6.20) in terms of ̂kt+1, we need to multiply and divide by Zt+1Nt+1 as follows: Kt+1 Zt+1Nt+1 Zt+1Nt+1 ZtNt = sAf (̂kt) + (1 − δ)̂kt. (6.21) From (6.6) and (6.8), we know that Zt+1/Zt = (1 + z) and Nt+1/Nt = (1 + n
). Hence, we can write the capital accumulation equation in terms of per eﬃciency units of capital as: ̂kt+1 = 1 (1 + z)(1 + n) [sAf (̂kt) + (1 − δ)̂kt]. (6.22) The other equations of the model can be re-written in terms of eﬃciency units as follows: ̂it = ŝyt ̂yt = Af (̂kt) ̂yt = ̂ct +̂it Rt = Af ′(̂kt) wt = Zt [Af (̂kt) − Af ′(̂kt)̂kt]. (6.23) (6.24) (6.25) (6.26) (6.27) Mathematical Diversion How does one derive equations (6.23)–(6.27)? Here, we will go step by step. Start with (6.11). Divide both sides by ZtNt: = s It ZtNt ⇒ ̂it = ŝyt. Yt ZtNt Similarly, divide both sides of (6.12) by ZtNt: Yt ZtNt = AF (Kt, ZtNt) ZtNt ̂yt = AF ( Kt ZtNt ̂yt = AF (̂kt, 1) ̂yt = Af (̂kt)., ZtNt ZtNt Next, divide both sides of (6.13) by ZtNt: 112 (6.28) (6.29) (6.30) (6.31) (6.32) ) = Ct Yt ZtNt ZtNt ⇒ ̂yt = ̂ct +̂it. + It ZtNt (6.33) For the rental rate on capital, note that, because F (⋅) is constant returns to scale, the partial derivatives are homogeneous of degree 0. This means: Rt = AFK(Kt, ZtNt) Rt = AFK ( Kt ZtNt Rt = Af ′(̂kt)., ZtNt ZtNt ) (6.34
) (6.35) (6.36) For the wage, because of Euler’s theorem for homogeneous functions, we know that: F (Kt, ZtNt) = FK(Kt, ZtNt)Kt + FN (Kt, ZtNt)ZtNt. (6.37) Divide both sides by ZtNt: F (Kt, ZtNt) ZtNt = FK(Kt, ZtNt)̂kt + FN (Kt, ZtNt). (6.38) Since F (⋅) is homogeneous of degree 1, this can be written: f (̂kt) − f ′(̂kt) = FN (Kt, ZtNt). (6.39) Since wt = ZtAFN (Kt, ZtNt). This means that: ZtAFN (Kt, ZtNt) = Zt [Af (̂kt) − Af ′(̂kt)̂kt]. (6.40) 6.2 Graphical Analysis of the Augmented Model We can proceed with a graphical analysis of the augmented Solow model in a way similar to what we did in Chapter 5. Diﬀerently than that model, we graphically analyze the capital stock per eﬃciency units of labor, rather than capital per unit of labor. We wish to plot (6.22). We plot ̂kt+1 against ̂kt. The plot starts in the origin. It increases at a decreasing rate. Qualitatively, the plot looks exactly the same as in the previous chapter 113 (see Figure 5.1). The only slight diﬀerence is that the right hand side is scaled by which is less than or equal to 1. 1 (1+z)(1+n), Figure 6.1: Plot of Central Equation of Augmented Solow Model As in the previous chapter, we plot a 45 degree line, showing all parts where ̂kt+1 = ̂kt. Via exactly the same arguments as in the basic Solow model, the plot of ̂kt+1 against ̂kt must cross this 45 degree line exactly once (other than at the origin). We call this point the steady state capital stock per e�
��ciency unit of labor, ̂k∗. Moreover, via exactly the same arguments as before, the economy naturally converges to this point from any initial starting point. While we are not per se interested in per eﬃciency unit of labor variables, knowing that the economy converges to a steady state in these variables facilitates analyzing the behavior of per worker and level variables. 6.3 The Steady State of the Augmented Model Graphically, we see that the economy converges to a steady state capital stock per eﬃciency unit of labor. Once we know what is happening to ̂kt, everything else can be ﬁgured out from equations (6.23)–(6.27). It is important to note that we are not really particularly interested in the behavior of the “hat” variables (the per eﬃciency units of labor variables). Writing the model in terms of these variables is just a convenient thing to do, because the model converges to a steady state in these variables. In this section, we pose the question: what happens to per worker 114 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1=𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1=1(1+𝑧𝑧)(1+𝑛𝑛)�𝑠𝑠𝐴𝐴𝑓𝑓�𝑘𝑘�𝑡𝑡�+(1−𝛿𝛿)𝑘𝑘�𝑡𝑡� 𝑘𝑘�∗ 𝑘𝑘�∗ and actual variables once the economy has converged to the steady state in the per eﬃciency variables? Note that being at ̂k∗ means that ̂kt+1 = ̂kt. Recall the deﬁnitions of these variables: ̂kt+1 = Kt+1 Zt+1Nt+1 and ̂kt = Kt ZtNt. Equate these and simplify:
Kt+1 Zt+1Nt+1 Kt+1 Kt Kt+1 Kt = Kt ZtNt = Zt+1Nt+1 ZtNt = (1 + z)(1 + n). (6.41) (6.42) (6.43) Here, Kt+1 Kt is the gross growth rate of the capital stock, i.e. 1 + gK. This tells us that, in the steady state in the eﬃciency units of labor variables, capital grows at the product of the growth rates of Zt and Nt: 1 + gK = (1 + z)(1 + n) ⇒ gK ≈ z + n. (6.44) The approximation makes use of the fact that zn ≈ 0. We can also re-arrange (6.44) to look at the growth rate of the capital stock per worker: Kt+1 Nt+1 kt+1 kt = Zt+1 Zt Kt Nt = 1 + z ⇒ gk = z. (6.45) (6.46) In other words, the capital stock per worker grows at the growth rate of Zt, z, in steady state. The same expressions hold true for output, consumption, and investment: Yt+1 Yt Ct+1 Ct It+1 It = (1 + z)(1 + n) ⇒ gY ≈ z + n = (1 + z)(1 + n) ⇒ gC ≈ z + n = (1 + z)(1 + n) ⇒ gI ≈ z + n. (6.47) (6.48) (6.49) 115 This also applies to the per worker versions of these variables: yt+1 yt ct+1 ct it+1 it = (1 + z) ⇒ gy = z = (1 + z) ⇒ gc = z = (1 + z) ⇒ gi = z. (6.50) (6.51) (6.52) In other words, the economy naturally will converge to a steady state in the per eﬃciency units of variables. In this steady state, output and capital per worker will grow at constant rates, equal to z. These are the same rates, so the capital-output ratio will
be constant in the steady state. These results are consistent with the stylized facts presented in Chapter 4. What will happen to factor prices in the steady state? Recall that Rt = Af ′(̂kt). Since ̂kt → ̂k∗, and A does not grow in steady state, this means that there exists a steady state rental rate: R∗ = Af ′(̂k∗). (6.53) This will be constant across time. In other words, Rt+1/Rt = 1, so the rental rate is constant in the steady state. This is consistent with the stylized fact that the return on capital is constant over long stretches of time. What about the real wage? Evaluate (6.27) once ̂kt → ̂k∗: wt = Zt [Af (̂k∗) − Af ′(̂k∗)̂k∗] (6.54) The term inside the brackets in (6.54) does not vary over time, but the Zt does. Taking this expression led forward one period, and dividing it by the period t expression, we get: wt+1 wt = Zt+1 Zt = 1 + z ⇒ gw = z. (6.55) In other words, once the economy has converged to a steady state in the per eﬃciency units of labor variables, the real wage will grow at a constant rate, equal to the growth rate of Zt, z. This is the same growth rate as output per worker in the steady state. This is also consistent with the stylized facts presented in Chapter 4. Finally, since wt and yt = Yt/Nt both grow at rate z, labor’s share of income is constant, also consistent with the time series stylized facts. In other words, the augmented Solow model converges to a steady state in per eﬃciency units of labor variables. Since the economy converges to this steady state from any initial 116 starting point, it is reasonable to conclude that this steady state represents where the economy sits on average over long periods of time. At this steady state, per worker variables and factor prices behave exactly as they do in the stylized facts presented in Chapter 4. To the extent to which a model is judged by the quality of the predictions it makes
, the augmented Solow model is a good model. 6.4 Experiments: Changes in s and A Let us consider the same experiments considered in Chapter 5 in the augmented model: permanent, surprise increases in s or A. Let us ﬁrst start with an increase in s. Suppose that the economy initially sits in a steady state associated with s0. Then, in period t, the saving rate increases to s1 > s0 and is expected to remain forever at the higher rate. Qualitatively, this has exactly the same eﬀects as it does in the basic model. This can be seen in Figure 6.2: Figure 6.2: Increase in s The steady state capital stock per eﬃciency unit of labor increases. The capital stock per eﬃciency unit of labor starts obeying the dynamics governed by the blue curve and approaches the new steady state. Given the dynamics of ̂kt, we can infer the dynamic responses of the other variables. These dynamic responses are shown in Figure 6.3 below. With the exception of the behavior of wt, these look exactly as they did after an increase in s in the basic model. The path of wt looks diﬀerent, because, as shown above in (6.27), the 117 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1=𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1=1(1+𝑧𝑧)(1+𝑛𝑛)�𝑠𝑠0𝐴𝐴𝑓𝑓�𝑘𝑘�𝑡𝑡�+(1−𝛿𝛿)𝑘𝑘�𝑡𝑡� 𝑘𝑘�0∗ 𝑘𝑘�𝑡𝑡+1=1(1+𝑧𝑧)(1+𝑛𝑛)�𝑠𝑠1𝐴𝐴𝑓𝑓�𝑘𝑘�𝑡𝑡�
+(1−𝛿𝛿)𝑘𝑘�𝑡𝑡� 𝑠𝑠1>𝑠𝑠0 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡+2 𝑘𝑘�𝑡𝑡=𝑘𝑘�0∗ 𝑘𝑘�1∗ 𝑘𝑘�1∗ wage depends on Zt, and so inherits growth from Zt in the steady state. After the increase in s, the wage grows faster for a time as capital per eﬃciency unit of labor is accumulated. This means that the path of wt is forever on a higher level trajectory, but eventually the growth rate of wt settles back to where it would have been in the absence of the change in s. Figure 6.3: Dynamic Responses to Increase in s What we are really interested in is not the behavior of the per eﬃciency units of labor 118 𝑘𝑘�𝑡𝑡 𝑦𝑦�𝑡𝑡 𝑐𝑐̂𝑡𝑡 𝚤𝚤̂𝑡𝑡 ln𝑤𝑤𝑡𝑡 𝑅𝑅𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡
𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑘𝑘�0∗ 𝑦𝑦�0∗ 𝑐𝑐̂0∗ 𝚤𝚤̂0∗ 𝑅𝑅0∗ 𝑘𝑘�1∗ 𝑦𝑦�1∗ 𝑐𝑐̂1∗ 𝚤𝚤̂1∗ 𝑅𝑅1∗ variables, but rather the per worker variables. Once we know what is going on with the per eﬃciency unit variables, it is straightforward to recover what happens to the per worker variables, since xt = ̂xtZt, for some variable Xt. Figure 6.4: Dynamic Responses to Increase in s, Per Worker Variables The paths of the per worker variables are shown in Figure 6.4. These look similar to what is shown in Figure 6.3, but these variables grow in the steady state. So, prior to the increase 119 ln𝑘𝑘𝑡𝑡 𝑅𝑅𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑅𝑅0∗ 𝑅𝑅1∗ ln𝑦𝑦𝑡𝑡 ln�
��𝑐𝑡𝑡 ln𝑡𝑡𝑡𝑡 ln𝑤𝑤𝑡𝑡 in s, yt, kt, ct, and it would all be growing at rate z. Then, after the saving rate increase, these variables grow faster for a while. This puts them on a forever higher level trajectory, but eventually the faster growth coming from more capital accumulation dissipates, and these variables grow at the same rate they would have in the absence of the increase in s. The dynamic path of the growth rate of output per worker after the increase in s can be seen in Figure 6.5 below. This looks very similar to Figure 5.8 from the previous chapter, with the exception that output growth starts and ends at z ≥ 0, instead of 0. In other words, increasing the saving rate can temporarily boost growth, but not permanently. Figure 6.5: Dynamic Path of Output Per Worker Growth Next, consider a one time level increase in A from A0,t to A1,t. In terms of the main diagram, this has eﬀects very similar to those of an increase in the saving rate, as can be seen in Figure 6.6: 120 𝑔𝑔𝑡𝑡𝑦𝑦 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡+1 𝑧𝑧 Figure 6.6: Increase in A Given the inferred dynamic path of ̂kt from Figure 6.6, the paths of the other variables can be backed out. These are shown below: 121 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1=𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡+1=1(1+𝑧𝑧)(1+𝑛𝑛)�𝑠𝑠𝐴𝐴0𝑓𝑓�𝑘𝑘�𝑡𝑡�+(1−�
�𝛿)𝑘𝑘�𝑡𝑡� 𝑘𝑘�0∗ 𝑘𝑘�𝑡𝑡+1=1(1+𝑧𝑧)(1+𝑛𝑛)�𝑠𝑠𝐴𝐴1𝑓𝑓�𝑘𝑘�𝑡𝑡�+(1−𝛿𝛿)𝑘𝑘�𝑡𝑡� 𝐴𝐴1,𝑡𝑡>𝐴𝐴0,𝑡𝑡 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡+1 𝑘𝑘�𝑡𝑡+2 𝑘𝑘�𝑡𝑡=𝑘𝑘�0∗ 𝑘𝑘�1∗ 𝑘𝑘�1∗ Figure 6.7: Dynamic Responses to Increase in A As in the case of the increase in s, we can transform these into paths of the per worker variables by multiplying by Zt. These paths are shown in Figure 6.8: 122 𝑘𝑘�𝑡𝑡 𝑦𝑦�𝑡𝑡 𝑐𝑐̂𝑡𝑡 𝚤𝚤̂𝑡𝑡 ln𝑤𝑤𝑡𝑡 𝑅𝑅𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑘𝑘�0∗ 𝑦𝑦�0∗ 𝑐𝑐̂0∗ 𝚤𝚤̂0∗ 𝑅𝑅0∗ 𝑘𝑘�1∗ 𝑦𝑦�1∗ 𝑐𝑐̂1∗ 𝚤𝚤̂1∗ 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 Figure 6.8: Dynamic Responses to Increase in A, Per Worker Variables Example The preceding analysis is all qualitative. It is possible do similar exercises quantitatively, using a program like Excel. To do things quantitatively, we need to make a functional form assumption on the production function. Let us assume that it is Cobb-Douglas: 123 ln𝑘𝑘𝑡𝑡 𝑅𝑅𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑅𝑅0∗ ln𝑦𝑦𝑡𝑡 ln𝑐𝑐𝑡𝑡 ln𝑤𝑤𝑡𝑡
ln𝑡𝑡𝑡𝑡 Yt = AK α t (ZtNt)1−α. (6.56) The accumulation equation for the capital stock per eﬃciency unit of labor is: ̂kt+1 = 1 (1 + z)(1 + n) [sÂkα t + (1 − δ)̂kt]. (6.57) In terms of the per eﬃciency unit variables, the variables of the model can be written solely in terms of the capital stock per eﬃciency units of labor: t t ̂yt = Âkα ̂ct = (1 − s)Âkα ̂it = sÂkα Rt = αÂkα−1 wt = Zt(1 − α)Âkα t. t t (6.58) (6.59) (6.60) (6.61) (6.62) One can solve for the steady state capital stock per eﬃciency unit of labor by setting ̂kt+1 = ̂kt = ̂k∗ and solving (6.57): ̂k∗ = [ sA (1 + z)(1 + n) − (1 − δ) ] 1 1−α. (6.63) To proceed quantitatively, we need to assign values to the parameters. Let’s assume that s = 0.2, A = 1, δ = 0.1, and α = 0.33. Furthermore, assume that z = 0.02 and n = 0.01. This means that Zt grows at a rate of 2 percent per year and Nt grows at a rate of 1 percent per year, while the capital stock depreciates at a rate of 10 percent per year. With these parameters, the steady state capital stock per eﬃciency unit of labor is 1.897. Assume that time begins in period t = 0. From (6.15)–(6.16), this means that the initial values of Zt and Nt are both 1. Assume that the capital stock per eﬃciency unit of labor begins in steady state. Once we know ̂kt in the �
�rst period, as well as the initial level of Zt, one can use (6.58)–(6.62) to determine values of ̂yt, ̂ct, ̂it, Rt, and wt. We can also determine levels of the per worker variables, kt, yt, ct, and it, by multiplying the per eﬃciency unit of labor variables by the level of Zt. Given an initial value of ̂kt in period t = 0, we can determine the value in the next period using Equation (6.57). Once we know ̂kt+1, we can then determine the per 124 eﬃciency unit of labor and per worker versions of the remaining variables. We can then continue to iterate this procedure moving forward in time (by simply ﬁlling formulas in an Excel worksheet). Since we assume that we begin in the steady state, the per eﬃciency unit variables will remain in that steady state until something changes. Consider the following experiment. From periods t = 0 through t = 8, the economy sits in the steady state. Then, in period t = 9, the saving rate increases from 0.2 to 0.3, and is forever expected to remain at this higher level. Using the new value of the saving rate and the existing capital stock per eﬃciency unit of labor in period t = 9, we can determine values of all the other variables in that period. Then we can use (6.57) to determine the period 10 value of the capital stock per eﬃciency unit of labor, and then use this to compute values of all the other variables. Figure 6.9: Dynamic Responses to Increase in s, Quantitative Exercise 125 11.522.533.54051015202530354045505560657075khat 11.11.21.31.41.51.6051015202530354045505560657075yhat 0.80.850.90.9511.051.1051015202530354045505560657075chat 0.20.250.30.350.40.450.5051015202530354045505560657075ihat -0.4-0.200.20.40.60.811
.21.41.6051015202530354045505560657075log(w) 0.10.120.140.160.180.20.220.24051015202530354045505560657075R Figure 6.9 plots the dynamic paths of the per eﬃciency unit variables, as well as the real wage and rental rate on capital, from periods t = 0 to t = 75. The saving rate changes in period 9. These plots look similar to what is shown in Figure 6.3. We plot the natural log of the wage, since it grows as Zt grows and plotting in the log makes the picture easier to interpret. Figure 6.10 plots the log levels of the per worker variables from the same experiment. This ﬁgure looks similar to Figure 6.4. The higher saving rate causes variables to grow faster for a while. They end up on a permanently higher level trajectory, but the slope of the plots is eventually the same as it would have been had the saving rate remained constant. Figure 6.10: Dynamic Responses to Increase in s, Quantitative Exercise, Per Worker Variables Figure 6.11 plots the growth rate of output per worker (the log ﬁrst diﬀerence of output per worker). Prior to period 9, the growth rate is constant at 2 percent. Then, starting in period 10 (the period after the increase in s), the growth rate jumps up. It remains higher than before for several periods, but eventually comes back to where it began. This illustrates the key point that saving more can 126 0.511.522.53051015202530354045505560657075log(k) 0.20.40.60.811.21.41.61.82051015202530354045505560657075log(y) -0.200.20.40.60.811.21.41.61.8051015202530354045505560657075log(C) -2-1.5-1-0.500.51051015202530354045505560657075log(i) temporarily boost growth, but not for a long period of time. Over long periods of time, the growth rate of output per worker is driven by the growth rate of labor augmenting technology, z.
Figure 6.11: Dynamic Responses to Increase in s, Quantitative Exercise Per Worker Output Growth 6.5 The Golden Rule The Golden Rule saving rate is deﬁned in a similar way to Chapter 5. It is the s which maximizes ̂c∗ (i.e. the saving rate which maximizes the steady state value of consumption per eﬃciency unit of labor). We can think about the Golden Rule graphically in a way similar to what we did in Chapter 5. But before doing so, we must ask ourselve, what must be true about steady state investment per eﬃciency unit of labor. Write the capital accumulation equation in terms of investment as follows: (1 + z)(1 + n)̂kt+1 = ̂it + (1 − δ)̂kt. In the steady state, ̂kt+1 = ̂kt. This means: ̂it = [(1 + z)(1 + n) − (1 − δ)] ̂kt. (6.64) (6.65) Note that (1 + z)(1 + n) − (1 − δ) ≈ z + n + δ (since zn ≈ 0). This implies that in steady state: ̂it = (z + n + δ)̂kt. (6.66) (6.66) is “break-even” investment per eﬃciency unit of labor – i.e. the amount of investment per eﬃciency unit of labor necessary to keep the capital stock per eﬃciency unit of labor from declining. What is slightly diﬀerent from the previous chapter is that break-even 127 00.0050.010.0150.020.0250.030.0350.040.045051015202530354045505560657075gy investment depends not just on the depreciation rate but also the growth rates of labor augmenting productivity and population. Put slightly diﬀerently, in the augmented model capital per eﬃciency unit of labor will naturally decline over time due to (i) depreciation of physical capital, δ; (ii) more labor input through population growth, n; (iii) more productive labor input through productivity growth, z. Break-even investment needs
to cover all three of these factors to keep the capital stock per eﬃciency unit of labor constant. In a graph with ̂kt on the horizontal axis, let us plot ̂yt, ̂it, and (z +n+δ)̂kt (i.e. break-even investment) against ̂kt: Figure 6.12: The Golden Rule Saving Rate ̂ct is given by the vertical distance between the plots of ̂yt and ̂it, given a value of ̂kt. The steady state occurs where the plot of ̂it cross the line (z + n + δ)̂kt. Hence, the steady state level of consumption per eﬃciency unit of labor is given by the vertical distance between the plot of ̂yt and the plot of ̂it at the value ̂k∗. This vertical distance is maximized when the slope of the ̂yt plot is equal to the slope of the (z + n + δ)̂kt plot, which means: Af ′(̂k∗) = z + n + δ. (6.67) In words, the Golden rule s is the s which generates a ̂k∗ where the marginal product of capital equals z + n + δ. If z = n = 0, then this is the same condition we saw in the basic model. 128 𝑦𝑦�𝑡𝑡,𝚤𝚤̂𝑡𝑡,(𝑧𝑧+𝑛𝑛+𝛿𝛿)𝑘𝑘�𝑡𝑡 𝑘𝑘�𝑡𝑡 (𝑧𝑧+𝑛𝑛+𝛿𝛿)𝑘𝑘�𝑡𝑡 𝚤𝚤̂𝑡𝑡=𝑠𝑠𝐴𝐴𝑓𝑓�𝑘𝑘�𝑡𝑡� 𝑘𝑘�∗ 𝑦𝑦�𝑡𝑡=𝐴𝐴𝑓
𝑓�𝑘𝑘�𝑡𝑡� 𝑦𝑦�∗ 𝚤𝚤̂∗ 𝑐𝑐̂∗ 6.6 Will Economic Growth Continue Indeﬁnitely? In the augmented Solow model, we can generate sustained growth in output per capita by simply assuming that labor augmenting productivity grows at a constant rate. In some sense, this is an unsatisfying result, as the model takes progress in labor augmenting productivity as given and does not seek to explain where it comes from. In this section we pose the provocative question: will economic growth continue into the indeﬁnite future? The model we have been working with cannot say anything about this, since the long run rate of growth is taken to be exogenous. But some historical perspective might help shed some light on this important question. Delong (1998) provides estimates of world real GDP from the beginning of recorded history to the present. From the the year 1 AD to 1600, worldwide real GDP grew by about 300 percent. While this may sound like a lot, considering compound it is an extremely slow rate of growth – it translates into average annual growth about 0.001, or 0.1 percent per year. In contrast, from 1600 to 2000, world GDP grew by an of about 0.015, or 1.5 percent – about 15 times faster than prior to 1600. Growth over the 20th century has been even higher, at about 3.6 percent per year. In other words, continuous economic growth is really only a modern phenomenon. For most of recorded human history, there was essentially no growth. Only since the beginning of the Industrial Revolution has the world as a whole witnessed continuous economic growth. While economic growth seems to have accelerated in the last several hundred years, there are some indications that growth is slowing. Since the early 1970s, measured productivity growth in the US has slowed down compared to earlier decades. The recent Great Recession has also seemed to be associated with a continual slowdown in growth. Is economic growth slowing down? Was the last half millenniem an anomaly? Economist Robert Gordon thinks so, at least in part. In Gordon (2016), he argues the period 1870-1970 was a “special century” that witnessed many new inventions and vast improvements in quality of life (e.g. the average life expectancy in the US increased by thirty years). He argues that this period
in particular was an anomaly. In essence, his thesis is that we have exhausted most life-changing ideas, and that we cannot depend on continuous large improvements in standards of living going forward. Not all economists, of course, agree with Gordon’s thesis. It is easy to say conclude that the improvements of the 20th century were historical anomalies after the fact. It is diﬃcult to predict what the future may hold. People in the 18th likely could not conceive of the breakthroughs of the 20th century (like aviation, computing, and telecommunications). Likewise, it is diﬃcult for us in the early 21th century to envision what will happen in the decades to come. Only time will tell. 129 6.7 Summary • The augmented Solow model is almost identical to the Solow model of the previous chapter except now there is sustained growth in the population and labor augmenting productivity. • The eﬀective number of workers equals labor augmenting productivity multiplied by the number of workers. Consequently, the eﬀective number of workers can increase when either the population grows or labor augmenting productivity grows. • A stable steady-state solution exists in per eﬀective worker variables. At this steady state, output, capital, consumption, and investment all grow at a rate equal to the sum of the growth rates in population and labor augmenting productivity. Per worker (or per capita) variables grow at the growth rate of labor augmenting productivity. The return on capital is constant, and the real wage grows at the rate of growth of labor augmenting productivity. These productions of the augmented Solow model are consistent with the stylized facts. Key Terms • Labor augmenting productivity Questions for Review 1. Explain, in words, what is meant by labor augmenting productivity. 2. Draw the main diagram of the Solow model with both labor augmenting productivity growth and population growth. Argue that there exists a steady state capital stock per eﬃciency unit of labor. 3. Graphically show the golden rule saving rate and explain what, if anything, a country that is below it should do. Exercises 1. Suppose that you have a standard Solow model with a Cobb-Douglas production function and both labor augmenting productivity growth and population growth. The central equation of the model is: ̂kt+1 = 1 (1 + z)(1 + n) [sÂkα t + (1 − δ
)̂kt]. 130 (a) Suppose that the economy initially sits in a steady state. Suppose that at time t there is a surprise increase in z that is expected to last forever. Use the main diagram to show how this will impact the steady state capital stock per eﬃciency unit of labor. (b) Plot out a diagram showing how the capital stock per eﬃciency unit of labor ought to react dynamically to the surprise increase in z. (c) Plot out diagrams showing how consumption and output per eﬃciency unit of labor will react in a dynamic sense to the surprise increase in z. (d) Do you think agents in the model are better oﬀ or worse oﬀ with a higher z? How does your answer square with what happens to the steady state values of capital, output, and consumption per eﬃciency unit of labor? How can you reconcile these ﬁndings with one another? 2. Suppose that you have a standard Solow model with a Cobb-Douglas production function and both labor augmenting productivity growth and population growth. The central equation of the model is: ̂kt+1 = 1 (1 + z)(1 + n) [sÂkα t + (1 − δ)̂kt] Consumption per eﬃciency unit of labor is: ̂ct = (1 − s)Âkα t. (a) Derive an expression for the steady state capital stock per eﬃciency unit of labor. (b) Use your answer from the previous part to derive an expression for the steady state value of consumption per eﬀective worker. (c) Use calculus to derive an expression for the value of s which maximizes steady state consumption per worker. Does the expression for this s depend at all on the values of z or n? 3. [Excel Problem] Suppose that you have the standard Solow model with both labor augmenting productivity growth and population growth. The production function is Cobb-Douglas. The central equation of the Solow model, expressed in per eﬃciency units of labor, is given by: ̂kt+1 = 1 (1 + z)(1 + n) [sÂkα t + (1 − δ)̂kt]. 131 The other variables of the model are governed by
Equations (6.23)–(6.27). (a) Create an Excel ﬁle. Suppose that the level of productivity is ﬁxed at A = 1. Suppose that s = 0.2 and δ = 0.1. Suppose that α = 1/3. Let z = 0.02 and n = 0.01. Solve for a numeric value of the steady state capital stock per eﬃciency unit of labor. (b) Suppose that the capital stock per worker initially sits in period 1 in steady state. Create a column of periods, ranging from period 1 to period 100. Use the central equation of the model to get the value of ̂k in period 2, given that ̂k is equal to its steady state in period 1. Continue to iterate on this, ﬁnding values of ̂k in successive periods up through period 9. What is true about the capital stock per eﬃciency unit of labor in periods 2 through 9? (c) In period 10, suppose that there is an increase in the population growth rate, from n = 0.01 to n = 0.02. Note that the capital stock per eﬃciency unit of labor in period 10 depends on variables from period 9 (i.e. the old, smaller value of n), though it will depend on the new value of n in period 11 and on. Use this new value of n, the existing value of the capital stock per eﬃciency unit of labor you found for period 9, and the central equation of the model to compute values of the capital stock per eﬃciency unit of labor in periods 10 through 100. Produce a plot showing the path of the capital stock per eﬃciency unit of labor from period 1 to period 100. (d) Assume that the initial levels of N and Z in period 1 are both 1. This means that subsequent levels of Z and N are governed by Equations (6.7) and (6.9). Create columns in your Excel sheet to measure the levels of N and Z in periods 1 through 100. (e) Use these levels of Z and N, and the series for ̂k you created above, to create a series of the capital stock per work, i.e. kt = ̂ktZt. Take the natural log of the
resulting series, and plot it across time. (f) How does the increase in the population growth rate aﬀect the dynamic path of the capital stock per worker? 132 Chapter 7 Understanding Cross-Country Income Diﬀerences In Chapter 4, we documented that there are enormous diﬀerences in GDP per capita across country (see, e.g., Table 4.1). In this Chapter, we seek to understand what can account for these large diﬀerences. Our conclusion will be that, for the most part, poor countries are poor not because they lack capital, but because they are relatively unproductive. The fact that they are unproductive means that they have relatively little capital, but this lack of capital is a symptom of their lack of productivity, not the cause of their being poor. There is an analog to our work in the previous chapters (Chapters 5 and 6). To account for the time series stylized facts, the Solow model requires sustained increases in productivity over time. To account for large disparities of standards of living in a cross-sectional sense, the model requires large diﬀerences in productivity across countries. Productivity is the key driving force in the Solow model. For this section we will be focusing on the basic Solow model from Chapter 5 without productivity or population growth. We could use the extended machinery of the augmented Solow model from 6, but this would not really alter the conclusions which follow. We will illustrate the arguments by assuming that there are just two countries, although it would be straightforward to include more than two. Suppose that we have two countries, which we will label with a 1 and a 2 subscript, i.e. country 1’s capital per capita at time t will be represented by k1,t. We assume that both countries have a Cobb-Douglas production function and that the parameter α is the same across countries. We also assume that capital depreciates at the same rate in both countries. We will potentially allow three things to diﬀer across the two countries – productivity levels (i.e. A1 ≠ A2, though we again drop subscripts and hence implicitly assume that these variables are constant going forward in time), saving rates (i.e. s1 ≠ s2), and initial endowments of capital per worker (i.e. k1,t ≠ k2,t). At any point in time, output per capita in the two countries is
: 133 y1,t = A1kα 1,t y2,t = A2kα 2,t The ratio of output per capita in the two countries is: y1,t y2,t = A1 A2 (k1,t k2,t α ) (7.1) (7.2) (7.3) From (7.3), we can see that there are really only two reasons why the countries could have diﬀerent levels of output per capita – either the productivity levels are diﬀerent or the capital stocks per capita are diﬀerent. Which one is it, and what are the policy implications? 7.1 Convergence Let us suppose, for the moment, that countries 1 and 2 are fundamentally the same, by which we mean that they have the same productivity levels and the same saving rates. This means that the only thing that could potentially diﬀer between the two countries is the initial endowment of capital stocks per worker. As you might recall from Chapter 5, the steady state capital stock per worker, and hence the steady state level of output per worker, does not depend on the initial endowment of capital. Starting from any (non-zero) initial endowment of capital, the economy converges to a steady state where the steady state is determined by productivity, the saving rate, the curvature of the production function, and the depreciation rate. If we are supposing that all of the features are the same for the two countries under consideration, it means that these countries will have the same steady state capital stocks per worker and hence identical steady state levels of output per capita. Will these two countries always have the same output per worker? Not necessarily – this will only be true in the steady state. One hypothesis for why some countries are richer than others is that those countries are initially endowed with more capital than others. Suppose that this is the case for countries 1 and 2. Let country 1 be relatively rich and country 2 relatively poor, but the countries have identical productivity levels and saving rates. Figure 7.1 below plots the main Solow diagram, with one twist. We index the countries by j = 1, 2. Since the countries have identical parameters, the main equation of the Solow model is the same for both countries, so kj,t+1 = sAf (kj,t) + (1 − δ)kj,t. Suppose that country 1 starts out in period
t with a capital stock equal to the steady state capital stock, so k1,t = k∗. Country 2 starts out with an initial capital stock substantially below that, k2,t < k∗. In this scenario, 134 country 2 is poor because it is initially endowed with little capital. Because country 2 is initially endowed with less capital than country 1, it will initially produce less output than country 1. But because country 2 starts out below its steady state capital stock, its capital will grow over time, whereas the capital stock for country 1 will be constant. This means that, if country 2 is poor relative to country 1 only because it is initially endowed with less capital than country 1, it will grow faster and will eventually catching up to country 1 (since the steady state capital stocks are the same). Figure 7.1: Country 1 Initially Endowed With More Capital than Country 2 Figure 7.2 plots in the left panel the dynamic paths of the capital stock in each country from the assumed initial starting positions – i.e. it plots kj,t+s for j = 1, 2 and s ≥ 0. Since it starts in steady state, country 1’s capital stock per worker simply remains constant across time. Country 2 starts with a capital stock below steady state, but its capital stock should grow over time, eventually catching up to country 1. In the right panel, we plot the growth rate of output per worker in each country across time, gy j,t+s. Because it starts in steady state, country 1’s growth rate will simply remain constant at zero (more generally, if there were population or productivity growth, country 1’s output growth would be constant, just not necessarily zero). In contrast, country 2 will start out with a high growth rate – this is because it is accumulating capital over time, which causes its output to grow faster than 135 𝑘𝑘𝑗𝑗,𝑡𝑡+1 𝑘𝑘𝑗𝑗,𝑡𝑡 𝑘𝑘𝑗𝑗,𝑡𝑡+1=𝑘𝑘𝑗𝑗,𝑡𝑡 𝑘𝑘𝑗𝑗,𝑡𝑡+1=𝑠𝑠𝐴𝐴𝑓�
��𝑘𝑘𝑗𝑗,𝑡𝑡�+(1−𝛿𝛿)𝑘𝑘𝑗𝑗,𝑡𝑡 𝑘𝑘1,𝑡𝑡=𝑘𝑘∗ 𝑘𝑘∗ 𝑘𝑘2,𝑡𝑡 country 1. Eventually, country 2’s growth rate should settle down to 0, in line with country 1’s growth rate. This analysis suggests that the Solow model predicts convergence if two countries have the same saving rates and same levels of productivity. In other words, if one country is relatively poor because it is initially endowed with less capital than another country, that country should grow faster than the other country, eventually catching up to it. Casual observation suggests that convergence is likely not consistent with the data – there are very large and very persistent diﬀerences in GDP per capita across countries. If countries only diﬀered in their initial endowment of capital, countries should eventually all look the same, and we don’t seem to see that. Figure 7.2: Paths of Capital and Output Growth for Countries 1 and 2 Figure 7.3 plots a scatter plot of 1950 GDP per capita (measured in real US dollars) and the cumulative gross growth rate of GDP from 1950-2010 for a handful of countries. The vertical axis measures the ratio of a country’s GDP per capita in 2010 to its GDP per capita in 1950; this ratio can be interpreted as the gross growth rate over that sixty year period. The horizontal axis is the GDP per capita level in 1950. If countries which were poor in 1950 were poor because of a lack of capital to rich countries, these countries should have experienced faster growth over the ensuing 60 years. 136 𝑠𝑠 𝑠𝑠 0 0 𝑘𝑘𝑗𝑗,𝑡𝑡+𝑠𝑠 𝑘𝑘1,𝑡𝑡=𝑘𝑘∗ 𝑘𝑘2,𝑡𝑡 𝑔𝑔1,𝑡𝑡𝑦𝑦=0 𝑔�
�𝑗𝑗,𝑡𝑡+𝑠𝑠𝑦𝑦 𝑔𝑔2,𝑡𝑡𝑦𝑦 Country 1 Country 2 Figure 7.3: Initial GDP Per Capita in 1950 and Cumulative Growth From 1950–2010 Is the evidence consistent with the data? In a sense yes, though the data do not provide very strong support for the convergence hypothesis. The convergence hypothesis makes the prediction that ensuing growth rates should be negatively correlated with initial GDP per capita. We do see some evidence of this, but it’s fairly weak. The correlation between cumulative growth over the 60 year period and initial GDP is only -0.13. There are some countries which were very poor in 1950 but experienced very rapid growth (represented by dots near the upper left corner of the graph). This pattern is loosely consistent with the convergence hypothesis. But there are many countries that were very poor in 1950 yet still experienced comparatively low growth over the ensuing sixty years (these countries are represented by dots near the origin of the graph). The countries included in the scatter plot shown in Figure 7.3 include all countries for which data are available dating back to 1950. Would the picture look diﬀerent if we were to focus on a subset of countries that are potentially more similar to one another? In Figure 7.4, we reproduce a scatter plot between cumulative growth over the last 60 years and the initial level of real GDP, but focus on countries included in the OECD, which stands for Organization for Economic Cooperation and Development. These include primarily western developed economies that trade extensively with one another. 137 024681012141602,0006,00010,00014,000Y1950Y2010/Y1950Correlation between cumulative growthand initial GDP = -0.18 Figure 7.4: Initial GDP Per Capita in 1950 and Cumulative Growth From 1950–2010 OECD Countries Relative to Figure 7.3, in Figure 7.4, we observe a much stronger negative relationship between the initial level of real GDP and subsequent growth. The correlation between initial GDP and cumulative growth over the ensuing 60 years comes out to be -0.71, which is substantially stronger than when focusing on all countries. What are we to conclude from Figures 7.3 and 7.4? While there is some evidence to support convergence, particularly for a restricted set of countries that are fairly similar, overall the convergence hypothesis is not a great
candidate for understanding some of the extremely large diﬀerences in GDP per capita which we observe in the data. 7.1.1 Conditional Convergence To the extent to which the Solow model provides a reasonably accurate description of actual economies, the evidence above suggests that convergence doesn’t seem to be a very strong feature of the data – countries have diﬀerent levels of GDP per capita and these diﬀerences seem to persist over time. This either suggests that the Solow model is fundamentally wrong on one or more levels or that it is a reasonable description of reality but something other than initial endowments of capital is the primary reason behind diﬀerences in standards of living across countries. What about a weaker proposition than absolute convergence, something that we will call conditional convergence? Conditional convergence allows for countries to have diﬀerent values of s or A, but still assumes that the economies of these countries are well approximated by the Solow model. Allowing these countries to have diﬀerent s or A means that their steady 138 24681012142,0004,0006,0008,00010,00014,000Y1950Y2010/Y1950Correlation between cumulative growthand initial GDP = -0.71 states will be diﬀerent. The model would predict that if an economy begins with less capital than its steady state, it ought to grow faster to catch up to its steady state (though that steady state might be diﬀerent than another country’s steady state). World War II provides a clean natural test of conditional convergence. Let’s focus on four countries – two of which were the primary winners of the war (the U.S. and United Kingdom) and two of which were the main losers of the war (Germany and Japan).1 Figure 7.5 plots the relative GDP per capita of these countries over time (relative to the U.S.). This is over the period 1950–2010. By construction, the plot for the U.S. is just a straight line at 1. The UK plot is fairly is ﬂat, with UK GDP about two-thirds (0.66) of U.S. GDP over most of the sample period. Figure 7.5: Real GDP Per Capita Relative to the United States The plots for Germany and Japan look quite diﬀerent. These countries both started quite poor relative to the U.S.
in 1950 (immediately after the War), but grew signiﬁcantly faster than the US over the ensuing 20–30 years. In particular, from 1950–1980, Germany went from GDP per capita about one-third the size of the U.S.’s to GDP per capita about 70 percent as big as the U.S.. Japan went from GDP per capita less than 20 percent of the U.S.’s in 1950 to GDP per capita about 75 percent of the U.S.’s in 1980. After 1980, the GDP per capita of both German and Japan has been roughly stable relative to U.S. GDP per capita. The patterns evident in Figure 7.5 are consistent with the Solow model once you allow countries to diﬀer in terms of their saving rates or their levels of productivity. One can 1We do not have good data for Russia because the collapse of the Soviet Union. 139 00.20.40.60.811.21950196019701980199020002010USGermanyUKJapan think about World War II as destroying a signiﬁcant amount of capital in both Japan and Germany (while the U.S. was unaﬀected and the UK was aﬀected, but to a lesser degree). Eﬀectively, we can think about the U.S. and the UK as being close their steady state capital stocks in 1950, whereas Germany and Japan were far below their steady state capital stocks. The Solow model would predict that Germany and Japan ought to have then grown faster relative to the U.S. and the UK for several years as they converged to their steady states. This is exactly what we observe in the data. This convergence seems to have taken roughly 30 years, but seems to have stopped since then. These ﬁndings are signiﬁcant, because they are consistent with the Solow model being an accurate description of reality, but point to countries diﬀering in fundamental ways other than just initial endowments of capital. 7.2 Can Diﬀerences in s Account for Large Per Capita Output Diﬀerences? Given that absolute convergence seems to be a poor description of the data, within the context of the Solow model it must be the case that income per capita diﬀerences across economies stem from fundamental diﬀerences in
productive capacities or saving rates. Consider the standard Solow model with a Cobb-Douglas production function. Assume that two countries have the same α and same δ, but potentially diﬀer in terms of saving rates and productivity levels (where we assume that the productivity levels in each country have settled down to constants). Under these assumptions, the steady state output per worker in country j = 1, 2 is given by: (sj δ The ratio of steady state output per capita across the two economies is then: for j = 1, 2 1 = A 1−α j y∗ j ) α 1−α y∗ 1 y∗ 2 = ( A1 A2 ) 1 1−α ( s1 s2 α 1−α ) (7.4) (7.5) Very persistent diﬀerences in output per capita across the two countries (by which we mean diﬀerent steady state levels of output) can be driven either by diﬀering productivity levels (i.e. A1 ≠ A2) or diﬀerent saving rates (i.e. s1 ≠ s2). In this section we wish to pose the following question: can diﬀerences in s alone account for large and persistent diﬀerences in output per capita? Here we will not focus on any data but will instead simply conduct what one might call a plausibility test. In particular, can plausible diﬀerences in s account for large diﬀerences in steady state output per capita? The answer turns out to be no for plausible values of α. 140 To see this concretely, suppose that the two countries in question have the same level of productivity, i.e. A1 = A2. From (7.5), their relative outputs are then: y∗ 1 y∗ 2 = ( s1 s2 ) α 1−α. (7.6) Our objective is to see how diﬀerent saving rates across countries would have to be to account for a given diﬀerence in per capita output. Let’s consider a comparison between a “middle income” country like Mexico and the U.S.. As one can see from Table 4.1, Mexican output per capita is about one-fourth the size of the U.S.. Suppose that country 1 is the U.S., and country 2 is Mexico. Then
y∗ = 4. Let’s then solve (7.6) for s2 in terms of s1, given 1 y∗ 2 this income diﬀerence. We obtain: s2 = 4 α−1 α s1. (7.7) A plausible value of α is 1/3. With this value of α, 4 α−1 α = 0.0625. What Equation (7.7) then tells us is that, to account for Mexican GDP that is one-fourth of the U.S.’s, the Mexican saving rate would have to be 0.0625 times the U.S. saving rate – i.e. the Mexican saving rate would have to be about 6 percent of the US saving rate. If the U.S. saving rate is s1 = 0.2, this would then mean that the Mexican saving rate would have to be s2 = 0.0125. This means that Mexico would essentially have to be saving nothing if the only thing that diﬀered between Mexico and the U.S. was the saving rate. This is not plausible. The results are even less plausible if one compares a very poor country to the U.S.. Take, for example, Cambodia. U.S. GDP per capita is about 20 times larger than that in Cambodia. If the U.S. saving rate were s1 = 0.2, then the Cambodian saving rate would have to be s2 = 0.0025 – i.e. essentially zero. One could argue that extremely poor countries are caught in a sort of poverty trap wherein they have not reached what one might call a subsistence level of consumption, and therefore actually do not save anything. This could be an explanation for extremely poor countries, such as those in Africa. But it is not a compelling argument for middle income countries like Mexico. Note that the assumed value of α has an important role in these plausibility tests. When α = 1/3, the exponent (α − 1)/α in (7.7) is −2. If α were instead 2/3, however, the exponent would be −1/2. With this value of α, taking the Mexican versus U.S. comparison as an example, one would only need the Mexican saving rate to be about 1/2 as big as the U.S. (as opposed to 6 percent of the U.S. saving rate when α = 1/
3). This is far more plausible. Mankiw, Romer, and Weil (1992) empirically examine the relationship between saving rates and output per capita across a large set of countries. They ﬁnd that saving rates are much more strongly correlated with GDP per capita across countries than the standard Solow 141 model with a relatively low value of α (e.g. α = 1/3) would predict. Their empirical analysis points to a value of α more on the order of α = 2/3. They argue that the basic Solow model production function is misspeciﬁed in the sense that human capital, which we discussed in Chapter 4, ought to be included. Roughly speaking, they ﬁnd that physical capital, human capital, and labor input ought to each have exponents around 1/3. Human capital ends up looking very much like physical capital in their model, and in a reduced-form sense implies a weight on physical capital in a misspeciﬁed production function on the order of 2/3. With this, the Solow model predicts a much stronger relationship between saving rates and output per capita that allows for more plausible diﬀerences in saving rates to account for large the diﬀerences in output per capita that we observe in the data. 7.3 The Role of Productivity For a conventionally speciﬁed Solow model, diﬀerences in saving rates cannot plausibly account for the very large diﬀerences in GDP per capita which we observe across countries in the data. While the inclusion of human capital in the model can help, it still cannot explain all of the observable income diﬀerences. To the extent to which we believe the Solow model, this leaves diﬀerences in productivity – i.e. diﬀerent levels of A across countries – as the best hope to account for large diﬀerences in standards of living across countries. In a sense, this result is similar to our conclusion in Chapters 5 and 6 that productivity must be the primary driver of long run growth, not saving rates. This begs the question – are there are large diﬀerences in productivity across countries? We can come up with empirical measures of A across countries by assuming a function form for the production function. In particular, suppose that the production function is Cobb-Douglas: Take natural logs of (7.8) and
re-arrange terms to yield: Yt = AK α t N 1−α t. ln A = ln Yt − α ln Kt − (1 − α) ln Nt. (7.8) (7.9) If we can observe empirical measures of Yt, Kt, and Nt across countries, and if we are willing to take a stand on a value of α, we can recover an empirical estimate of ln A. In essence, ln A is a residual – it is the part of output which cannot be explained by observable capital and labor inputs. Consequently, this measure of ln A is sometimes called the “Solow residual.” It is also called “total factor productivity” (or TFP for short). 142 The Penn World Tables provide measures of TFP for countries at a point in time. From there we can also collect data on GDP per worker or per capita. Figure 7.6 present a scatter plot of GDP per worker (measured in 2011 U.S. dollars) against TFP (measured relative to the U.S., where U.S. TFP is normalized to 1) for the year 2011. Each circle represents a TFP-GDP pair for a country. The solid line is the best-ﬁtting regression line through the circles. We observe that there is an extremely tight relationship between TFP and GDP per capita. In particular, the correlation between the two series is 0.82. By and large, rich countries (countries with high GDP per worker) have high TFP (i.e. are very productive) and poor countries have low TFP (i.e. are not very productive). Figure 7.6: Scatter Plot: TFP and GDP Per Worker in 2011 In summary, the Solow model suggests that the best explanation for large diﬀerences in standards of living is that there are large diﬀerences in productivity across countries. If some countries were poor simply because they were initially endowed without much capital, the Solow model would predict that these countries would converge to the GDP per capita of richer countries. For the most part, we do not see this in the data. For plausible values of α, the diﬀerences in saving rates which would be needed to justify the large diﬀerences in GDP per capita observed in the data would be implausible. This leaves diﬀerences
in productivity as the best candidate (within the context of the Solow model) to account for large diﬀerences in standards of living. Empirically, this seems to be consistent with the data, as is documented in Figure 7.6 – rich countries tend to be highly productive and poor countries tend to be very unproductive. 143 020,00040,00060,00080,000100,000120,000140,000160,0000.20.40.60.81.01.21.41.6TFP (relative to US = 1)GDP per worker (2011 US Dollars)Correlation between TFP and GDP = 0.82 This conclusion begs the question: what exactly is productivity (measured in the model in terms of the variable A)? The model takes this variable to be exogenous (i.e. does not seek to explain it). For many years, economists have sought to better understand what drives this productivity variable. Understanding what drives diﬀerences in productivity is important for thinking about policy. By and large, countries are not poor because they lack capital or do not save enough – they are poor because they are unproductive. This means that policies which give these countries capital or try to increase their saving rates are not likely to deliver large changes in GDP per capita. Policies to lift these countries out of poverty need to focus on making these countries more productive. Below is a partial listing (with brief descriptions) of diﬀerent factors which economists believe contribute to overall productivity: 1. Knowledge and education. A more educated workforce is likely to coincide with a more productive workforce. With more knowledge, workers can better make use of existing physical capital and can come up with new and better ways to use other inputs. As documented in Chapter 4, there is a strong positive correlation between an index of human capital (which one can think of as measuring the stock of knowledge in an economy) and real GDP per person. Related to this, Cubas, Ravikumar, and Ventura (2016) present evidence that the quality of labor in rich countries is nearly as twice as large as in most poorer countries. 2. Climate. An interesting empirical fact is this: countries located in climates closer to the Equator (think countries like Mexico, Honduras, and many African countries) tend to be poor relative to countries located further from the equator (think the U.S., northern Europe, and Australia). Hot, muggy climates make it diﬃcult
for people to focus and therefore are associated with lower productivity. These climates are also ones where disease tends to thrive, which also reduces productivity. As an interesting aside, there is suggestive evidence that the economic development in the southern states of the U.S. was fueled by the rise of air conditioning in the early and middle parts of the 20th century. 3. Geography. From microeconomics, we know that trade leads to specialization, which leads to productivity gains. How much trade a country can do is partly a function of its geography. A country with many natural waterways, for example, makes the transport of goods and services easier, and results in specialization. Think about the many waterways in the U.S. (like the Mississippi River) or the Nile River in Egypt. Geographies with very mountainous and diﬃcult to cross terrain, like Afghanistan, are not well suited for trade and the gains from specialization associated with it. 144 4. Institutions. Economists increasingly point to “institutions,” broadly deﬁned, as an important contributor to productivity. By institutions we primarily mean things like legal tradition, the rule of law, etc.. Countries with good legal systems tend to be more productive. When there are well-deﬁned and protected property rights, innovation is encouraged, as innovators will have legal claims to the fruits of their innovation. In countries with poor legal protections (think undeveloped Africa, countries like Afghanistan, etc.), there is little incentive to innovate, because an innovator cannot reap the rewards of his or her innovations. Acemoglu, Johnson, and Robinson (2001) have pointed to colonial development with European-style legal traditions in countries like the U.S. and Australia as important factors in the quality institutions of these countries now, and consequently their relatively high productivity. 5. Finance. The ﬁnancial system intermediates between savers and borrowers, and allows for the implementation of large scale projects which individuals or businesses would not be able to do on their own because of a lack of current funds. Relatively rich countries tend to have good ﬁnancial institutions, which facilitates innovation. Poorer countries do not have well-developed ﬁnancial institutions. A lot of recent research in development economics concerns the use of better ﬁnance to help fuel productivity. 6. Free trade. Countries with fewer barriers to international trade to tend to have higher productivity. International trade in goods and services has two e�
��ects. First, like trade within a domestic economy, it allows for greater specialization. Second, trade in goods and services leads to knowledge spillovers from rich to poor countries, increasing the productivity levels in poor countries. Sachs and Warner (1995) document that relatively open economies grow signiﬁcantly faster than relatively closed economies. 7. Physical infrastructure. Countries with good physical infrastructure (roads, bridges, railways, airports) tend to be more productive than countries with poor infrastructure. Good physical infrastructure facilitates the free ﬂow of goods, services, and people, resulting in productivity gains. If productivity is the key to high standards of living, policies should be designed to foster higher productivity. Climate and geography are things which are largely beyond the purview of policymakers (although one could argue that Global Warming is something which might harm future productivity and should therefore be addressed in the present). Policies which promote good legal and political institutions, free trade, good and fair ﬁnancial systems, and solid physical infrastructure are good steps governments can take to increase productivity. 145 7.4 Summary • In the steady state, income diﬀerences across countries are driven by diﬀerences in saving rates and productivity levels, but not initial levels of capital. Outside of steady state however, a country’s income and growth rate are in part determined by their initial capital stocks. • The Solow model predicts that if countries share common saving rates and productivity levels, they will converge to the same steady state level of output per worker. This is called the convergence hypothesis. Analysis of data over the last 60 years suggests that countries by and large fail to converge meaning that there must be diﬀerences in either cross country saving rates or productivity levels. • The conditional convergence hypothesis allows countries to have diﬀerent levels of productivity and saving rates but still assumes that their economies are well approximated by the Solow model. There is rather strong evidence of conditional convergence in the data. Countries which had large portions of their capital stocks destroyed in WWII subsequently grew faster than other rich countries. • For typical values of α, diﬀerences in saving rates alone cannot plausibly explain long-run diﬀerences in income across countries. However, if the production function is mis-speciﬁed by omitting intangible forms of capital like human capital, the combined values of the capital shares may be much bigger which would allow for saving rates to play a bigger role in
cross country income determination. • If diﬀerences in saving rates cannot explain cross country income diﬀerences that leaves diﬀerences in TFP as the main driver of income disparities. Indeed, a country’s GDP per worker is strongly correlated with its TFP level. • Although productivity is exogenous to the Solow model, some variables that might determine a country’s TFP include: climate, geography, education of its citizens, access to trade, the ﬁnancial system, legal institutions, and infrastructure. Questions for Review 1. Explain, in words, what is meant by the convergence hypothesis. What feature of the Solow model gives rise to the prediction of convergence? 2. Explain what is meant by conditional convergence. Can you describe an historical event where conditional convergence seems to be at work? 146 3. Try to provide some intuition for why diﬀerences in saving rates cannot plausibly account for large diﬀerences in income per capita for relatively low values of α. Hint: it has to do with how α governs the degree of diminishing returns to capital. 4. Discuss several factors which might inﬂuence a country’s level of productivity. 5. Suppose that you were a policy maker interested in increasing the standard of living in a poor African country. Suppose that an aide came to you and suggested giving every resident of that country a laptop computer. Do you think this would be a good idea? If not, propose an alternative policy to help raise the standard of living in the poor African country. 6. Do you think that any of the lessons from the Solow model about understanding large cross-country diﬀerences in income could be applied to understanding income diﬀerences within a country? If so, how? Elaborate. Exercises 1. [Excel Problem] Suppose that you have two countries, call them 1 and 2. Each is governed by the Solow model with a Cobb-Douglas production function, but each each country has potentially diﬀerent values of s and A. Assume that the value of A for each country is ﬁxed across time. The central equation of the model is: ki,t+1 = siAikα i,t + (1 − δ)ki,t, i = 1, 2. Output in each country is given by: yi,t = Aikα i,t.
(a) Solve for the steady state capital stock per worker for generic country i (i is an index equal to either 1 or 2). (b) Use this to solve for the steady state level of output per worker in country i. (c) Use your answers from previous parts to write an expression for the ratio of steady state output in country 1 to country 2 as a function of the respective saving rates, productivity levels, and common parameters of the model. (d) Suppose that each country has the same value of A, so A1 = A2. Suppose that α = 1/3, and δ = 0.1. Suppose that the saving 147 rate in country 1 is s1 = 0.2. In an Excel spreadsheet, compute diﬀerent values of the relative steady state outputs (i.e. y∗ ) 1 y∗ 2 ranging from 1 to 5, with a gap of 0.1 between entries (i.e. you should create a column with 1, 1.01, 1.02, 1.03, and so on). For each value of y∗, solve for the value of s2 necessary to be 1 y∗ 2 consistent with this. Produce a graph of this value of s2 against the values of y∗. Comment on whether it is plausible that 1 y∗ 2 diﬀerences in saving rates could account for large diﬀerences in relative GDPs. (e) Redo this exercise, but instead assume that α = 2/3. Compare the ﬁgures to one another. Comment on how a higher value of α does or does not increase the plausibility that diﬀerences in saving rates can account for large diﬀerences in output per capita. 2. Excel Problem. Suppose that you have many countries, indexed by i, who are identical in all margins except they have diﬀerent levels of A, which are assumed constant across time but which diﬀer across countries. We denote these levels of productivity by Ai. The central equation governing the dynamics of capital in a country i is given by: ki,t+1 = sAikα i,t + (1 − δ)ki,t Output in each country is given by: yi,t = Aikα i,t (a) Solve for expressions for steady state capital and output in a particular country i as functions of its Ai
and other parameters. (b) Create an Excel sheet. Create a column with diﬀerent values of A, each corresponding to a diﬀerent level of productivity in a diﬀerent country. Have these values of Ai run from 0.1 to 1, with a gap of 0.01 between entries (i.e. create a column going from 0.1, 0.11, 0.12, and so on to 1). For each level of Ai, numerically solve for steady state output. Create a scatter plot of steady state output against Ai. How does your scatter plot compare to what we presented for the data, shown in Figure 148 7.6? 149 Chapter 8 Overlapping Generations In Chapters 5 and 6 we explored the Solow model which is a model designed to think about economic performance in the long run. A disadvantage of the Solow model is that the saving decision on the part of the representative household is exogenously given and not derived from an underlying economic decision-making problem. Further, without a description of preferences, it is diﬃcult to say much about normative implications of the model as they relate to policy. Finally, because the model is populated by one representative household that lives forever, it is not possible to address issues related to intergenerational transfers (i.e. things like Social Security systems, which are in eﬀect transfers from young people to old). In this Chapter we consider what is called an Overlapping Generations (OLG) model. The OLG model was ﬁrst developed by Samuelson (1958) and Diamond (1965). Like the Solow model, time runs forever. But in the OLG model, we depart from the inﬁnitely-lived representative agent assumption. A representative agent is replaced by agents that live two periods. In the ﬁrst period agents are “young” and in the second they are “old.” At the end of each period, all old agents die, the young transition to old, and a new cohort of young agents is born. In their youth, agents optimally choose saving so as to maximize the present discounted value of lifetime utility. In other words, the saving decision of a household is explicitly endogenized in a way that it is not in the Solow model. Nevertheless, the OLG model is in many ways similar to the Solow model. But because of explicit optimization on
the part of households, as well as multiple generations alive at any one point in time, we are able to address some of the issues raised in the opening paragraph. In many respects this chapter provides a bridge from Part II to Part III. In particular, the set of issues it addresses are most closely connection to the material in Part II but it makes use of tools and analysis that are more similar to Part III. 8.1 The General Overlapping Generations Model Time will be denoted with the usual subscripts – think of t as the present, t + 1 as one period into the future, t − 1 as one period in the past, and so on. At any given time, there are two types of agents alive – young and old. Each household (which we use interchangeably 150 with “agent”) lives two periods. In the ﬁrst period a household is “young” and in the second it is “old.” At the end of a period, the old agents die and a new cohort of young agents are born. Let the number of young agents born in a given period be Nt. The total number of agents alive in any period is Nt + Nt−1, where Nt−1 denotes the number of old households alive in t (equal to the number of young households born in t − 1). We assume that the number of agents born each period evolves exogenously; Nt > Nt−1 would mean that more and more young households are born each period, so the total population would be growing. We must keep track of both time and generation. As such, we will index variables chosen by agents with a y or o subscript, for “young” or “old.” Aside from the two types of households, the economy is populated by a single representative ﬁrm. This ﬁrm is similar to the representative ﬁrm in the Solow model. It simply leases factors of production and produces output. Diﬀerently than the Solow model, decisions about capital accumulation will be derived from an underlying microeconomic optimization problem. 8.1.1 Households A household is born with no wealth. The key decision-making occurs in the ﬁrst period of life. In its youth, a household supplies one unit of labor inelastically, for which it is compensated at real wage wt. The key decision a young household must make
is how much of its income to consume and how much to save. Let st denote the saving of a young household in period t (we will use lowercase letters to denote the quantities chosen by an individual household). Saving is turned into productive capital in the next period, which can be rented to the representative ﬁrm at rental rate Rt+1. The household can then consume any capital leftover after depreciation once production in t + 1 takes place, (1 − δ)st, where 0 ≤ δ ≤ 1 is the depreciation rate on capital. Old households do not supply any labor (i.e. they are retired and live oﬀ the income from their accumulated capital). Let st denote the saving of a young household and cy,t its consumption in period t. Since it supplies one unit of labor inelastically, the budget constraint it faces in period t is: cy,t + st ≤ wt (8.1) In old age, the household has income from renting capital and can consume any remaining capital after depreciation. Hence, the constraint facing an old household in t + 1 is: From the perspective of period t, a household’s lifetime utility, U, is a weighted sum of c0,t+1 ≤ [Rt+1 + (1 − δ)] st (8.2) 151 utility ﬂows from consumption in each stage of life. Consumption is mapped into utility via some function u(⋅), which we assume is increasing (u′(⋅) > 0) and concave (u′′(⋅) < 0). Future utility ﬂows are discounted relative to current utility ﬂows by 0 < β < 1, where β is called a discount factor and measures a household’s degree of impatience. Formally, lifetime utility is: U = u(cy,t) + βu(co,t+1) (8.3) A young household’s objective is to pick st to maximize (8.3) subject to the two ﬂow budget constraints, (8.1)-(8.2). As we will do later in the book (see, e.g., Part III), we can solve a constrained optimization problem by assuming that both constraints bind with equality and substituting them into the objective function. This turns the constrained problem into an unconstrained one. Doing so we get: max st U =
u (wt − st) + βu ([Rt+1 + (1 − δ)] st) (8.4) To characterize the optimum, take the derivative of (8.4) with respect to st and equate it to zero: ∂U ∂st = 0 ⇔ u′ (wt − st) = β [Rt+1 + (1 − δ)] u′(([Rt+1 + (1 − δ)] st) (8.5) (8.5) can be written in a way that is somewhat easier to interpret by writing the arguments of the utility function in terms of consumption, or: u′(cy,t) = β [Rt+1 + (1 − δ)] u′(co,t+1) (8.6) (8.6) is very similar to the canonical consumption Euler equation which is studied in Chapter 9. The intuition for why (8.6) must hold is as follows. Suppose that a young household increases its saving by one unit. This reduces consumption during youth by 1, which lowers lifetime utility by u′(cy,t). Hence, the left hand side of (8.6) may be interpreted as the marginal utility cost of saving more. What is the beneﬁt? If the household saves one more unit in its youth, it can consume Rt+1 + (1 − δ) additional units in its old age (the rental rate from leasing the capital to the representative ﬁrm plus any capital left over after depreciation). This extra consumption is valued at βu′(co,t+1). Hence, the right hand side of (8.6) represents the marginal utility beneﬁt of saving more in youth. At an optimum, the marginal utility beneﬁt must equal the marginal utility cost. Were this not so, e.g. the marginal utility beneﬁt of saving exceeded the cost, the household should be saving more, and hence could not be optimizing. Expression (8.5) implicitly determines an optimal level of saving, st, as a function of 152 factor prices, wt and Rt+1. We denote this optimal level of saving via: st = s(wt, Rt+1) (8.7) Without specifying a functional form for u(⋅), we cannot say much speciﬁc about the (unknown)
function s(⋅). We can conclude that it is increasing in wt, i.e. sw(⋅) > 0. If wt increases, it must be the case that st increases. If st did not change, the marginal utility cost of saving (the left hand side of (8.5)) would decrease but there would be no change in the marginal utility beneﬁt. In contrast, if st decreased, the marginal utility cost of saving would decrease while the marginal utility beneﬁt of saving (the right hand side of (8.5)) would increase. Since the marginal utility beneﬁt must equal the marginal utility cost, an increase in wt must be met by an increase in st. In contrast, it is not possible to say with certainty how Rt+1 impacts optimal saving. On the one hand, a higher Rt+1 works to increase the marginal utility beneﬁt of saving by increasing the extra consumption a household may enjoy in its old age; but on the other hand, for a ﬁxed level of st a higher Rt+1 reduces the way in which this extra consumption is valued (i.e. for a given st a higher Rt+1 makes u′ ([Rt+1 + (1 − δ)] st) smaller given the concavity of u(⋅)). Hence, we cannot say with certainty how Rt+1 impacts desired saving.1 8.1.2 Firm As in the Solow model, there is a single, representative ﬁrm (or many identical ﬁrms, the total size of which may be normalized to one). Output is produced using capital and labor according to the same sort of production function we previously encountered. Formally: Yt = AtF (Kt, Nt) (8.8) Note that we are here using capital letters, whereas when describing the problem of a particular houehold we used lower case letters. This is because there is a single representative ﬁrm, and Yt therefore denotes aggregate output produced within a period. Kt is the aggregate stock of capital owned by old households in period t and leased to the ﬁrm, while Nt is the 1Formally, as we discuss in more depth in Chapter 9, there are both income and substitution eﬀects associated with changes in Rt+1. A higher Rt
+1 makes current consumption expensive relative to consumption in old age, which works to make saving higher. This is what is called a substitution eﬀect. On the other hand, since a household must do some positive saving in its youth to allow it to consume in its old age, a higher Rt+1 also eﬀectively endows the household with more income in the future, which makes it want to consume more in the present. This is what is called an income eﬀect. Since the income and substitution eﬀects go in opposite directions, it is not possible in general to determine how Rt+1 impacts saving. When wt changes, in contrast, there is only a positive income eﬀect which makes the household desire to consume more both in its youth as well as old age, the latter of which necessitates saving more while young. 153 total number of young households, each of whom supply unit of labor inelastically. F (⋅) is an increasing and concave function with constant returns to scale; these are exactly the same assumptions made for the Solow model. At is an exogenous productivity variable. While it is exogenous and can hence change, we will not consider diﬀerences between its value in t and subsequent periods (i.e. if we consider any change in At it will be permanent). Hence, for notational ease we shall henceforth drop the t subscript on A. The ﬁrm is a price-taker and chooses Kt and Nt to maximize its proﬁt each period. The problem is static and the same each period. It is: max Kt,Nt Πt = AF (Kt, Nt) − wtNt − RtKt (8.9) As previously encountered in the Solow model, the optimality conditions are to equate marginal products to factor prices: AFK(Kt, Nt) = Rt AFN (Kt, Nt) = wt (8.10) (8.11) Because of the assumption of constant returns to scale in F (⋅), the ﬁrm will earn zero proﬁt, and we therefore need not worry about to whom proﬁt in the ﬁrm accrues. 8.1.3 Equilibrium and Aggregation In period t, there are Nt−1 old
households who, in aggregate, supply Nt−1st−1 units of capital to the representative ﬁrm. Denote this by Kt. Because this capital was chosen prior to period t, we can treat the initial aggregate stock of capital as exogenous. Total capital available for the ﬁrm in t + 1 will be Nt times the saving of young households, or: Kt+1 = Ntst (8.12) To derive the aggregate resource constraint, let both (8.1) and (8.2) hold with equality and sum across the number of agents. Doing so, we obtain: Ntcy,t + Ntst = Ntwt Nt−1c0,t = [Rt + (1 − δ)] Nt−1st−1 (8.13) (8.14) Making use of (8.12) and deﬁning Cy,t = Ntcy,t and C0,t = Nt−1c0,t as aggregate consumption of young and old households, respectively, (8.13)-(8.14) may be written: 154 Cy,t + Kt+1 = Ntwt C0,t = [Rt + (1 − δ)] Kt Summing (8.15)-(8.16) together, we get: Cy,t + C0,t + Kt+1 − (1 − δ)Kt = wtNt + RtKt (8.15) (8.16) (8.17) As discussed in the Solow model, via the assumption of constant returns to scale wtNt + RtKt = Yt. Furthermore, Kt+1 − (1 − δ)Kt = It, or aggregate investment. In eﬀect, young households do total investment of Kt+1, whereas old households do disinvestment of (1 − δ)Kt (i.e. they consume their leftover capital before dying). Aggregate investment is the sum of investment by each generation of household. The same is true for aggregate consumption. (8.17) then reduces to an entirely conventional aggregate resource constraint: Ct + It = Yt (8.18) All told, the equilibrium of the economy is characterized by the following equations simultaneously holding: st = s(wt, Rt+1) K
t+1 = Ntst It = Kt+1 − (1 − δ)Kt Yt = Ct + It Yt = AF (Kt, Nt) Rt = AFK(Kt, Nt) wt = AFN (Kt, Nt) Ct = Cy,t + Co,t Cy,t = Ntwt − Kt+1 (8.19) (8.20) (8.21) (8.22) (8.23) (8.24) (8.25) (8.26) (8.27) (8.19)-(8.27) feature nine endogenous variables (st, wt, Rt, Kt+1, It, Yt, Ct, Cy,t, and C0,t) in nine equations. Kt, Nt, and A are exogenous. Let us assume that the size of the young population evolves exogenously according to: 155 Nt = (1 + n)Nt−1 (8.28) That is, the growth rate of the youth population is given by n ≥ 0. n = 0 would mean that the same number of households are born each period as die, so that the total population is constant. n > 0 means that more young households are born each period than old households die, so that the total population would be growing. Because of potential growth in the population, it is convenient to re-write these equations in per capita terms, just as we did in the Solow model. wt and Rt are factor prices and do not depend on the size of the population, and st is already in expressed in per capita terms. For other variables, let lowercase variables denote the variable expressed relative to the size of the young population (which equals the workforce), i.e. kt = Kt, and so Nt on. An exception is the consumption of the old population, for which we deﬁne c0,t = C0,t Nt−1 since there are Nt−1 old households alive in period t., cy,t = Cy,t Nt, yt = Yt Nt Divide both sides of (8.20) by Nt+1 to get: Kt+1 Nt+1 = Nt Nt+1 st (8.29) Using (8.28) and our per capita notation,
this is: kt+1 = st 1 + n Divide both sides of (8.23) by Nt and make use of the assumption that F (⋅) has constant (8.30) returns to scale: As we did in the Solow model, deﬁne f (kt) = F (kt, 1), allowing us to write: Yt Nt = AF (Kt Nt, Nt Nt ) yt = Af (kt) (8.31) (8.32) As was discussed in Chapter 5, the assumption that F (⋅) is constant returns to scale implies that factor prices may be expressed in terms of the capital-labor ratio as: Rt = Af ′(kt) wt = Af (kt) − Aktf ′(kt) (8.33) (8.34) The other equations are relatively straightforward to express in per worker terms. We are left with: 156 st = s(wt, Rt+1) kt+1 = st 1 + n it = kt+1(1 + n) − (1 − δ)kt yt = ct + it yt = Af (kt) Rt = Af ′(kt) wt = Af (kt) − Aktf ′(kt) ct = cy,t + co,t 1 + n cy,t = wt − kt+1(1 + n) (8.35) (8.36) (8.37) (8.38) (8.39) (8.40) (8.41) (8.42) (8.43) (8.35)-(8.43) are the same as (8.19)-(8.27), except they are written in per worker terms and Nt and has been eliminated using (8.28). The key endogenous variable in (8.35)-(8.43) is kt+1; the rest of the endogenous variables are extraneous, since once kt+1 is determined these are all determined. (8.35), (8.40), and (8.41) can be combined together to yield: kt+1 = s (Af (kt) − Aktf ′(kt), Af ′(kt+1)) 1 + n (8.44) is the central equation of the OLG model
. It is a diﬀerence equation implicitly relating kt+1 to kt, A, and parameters. It only implicitly forms this relationship because kt+1 appears on both the left and right hand sides. This caveat aside, (8.44) is not fundamentally diﬀerent than the central equation of the Solow model – given kt, A, and parameters, kt+1 is determined from one equation. The main diﬀerence relative to the Solow model is that rather than assuming a simple saving rule, we have instead derived one based on dynamic intertemporal optimization. (8.44) The downside of having derived a saving function from ﬁrst principles, however, is that without having a speciﬁc functional form for s(⋅), it is diﬃcult to say much about the properties of this diﬀerence equation, such as whether there exists a point where kt+1 = kt and, if so, whether such a point is unique. In the next section, we will make functional form assumptions on u(⋅) and F (⋅) which lead to a particularly simple form of s(⋅) with a number of desirable properties. 157 8.2 Cobb-Douglas Production and Logarithmic Utility Let us suppose that the ﬂow utility function of household is the natural log, i.e. u(⋅) = ln(⋅). With this speciﬁcation of preferences, (8.5) may be written: 1 wt − st = β [Rt+1 + (1 − δ)] 1 [Rt+1 + (1 − δ)] st (8.45) simpliﬁes nicely: Simplifying further so as to isolate st on the left hand side, we obtain: 1 β = wt − st st st = β 1 + β wt (8.45) (8.46) (8.47) < 1, it says that young households save a (8.47) has a clean interpretation. Since β 1+β constant fraction of their income earned in youth. This is similar to the Solow model, except that the saving rate has been derived from an optimization problem. We can see that the bigger is β, the bigger the share of its income a young household will choose to save.
This is quite intuitive – the bigger is β, the more patient the household is, and, other things being equal, the more it ought to want to save. Finally, related to our discussion above, note that the saving of young households does not depend on Rt+1 with this particular speciﬁcation of preferences. In eﬀect, with logarithmic utility the income and substitution eﬀects associated with Rt+1 exactly cancel out. If the production function is Cobb-Douglas (i.e. F (Kt, Nt) = K α t N 1−α t, with 0 < α < 1), the real wage paid to young households is: This can be written in terms of capital per capita as: wt = (1 − α)AK α t N −α t wt = (1 − α)Akα t (8.48) (8.49) Note that with Cobb-Douglas production the per worker production function is simply: yt = Af (kt) = Akα t (8.50) Hence, the real wage is simply proportional to output, with wt = (1 − α)yt. And since saving of young households is just proportional to the real wage, saving of these households is 158 therefore simply proportional to total output per worker, which is again similar to the Solow model. (8.49) can be combined with (8.47) in conjunction with (8.36) to derive the central equation of the OLG model with these functional form assumptions: kt+1 = β(1 − α)Akα t (1 + β)(1 + n) (8.51) The diﬀerence equation in (8.51) has similar properties to the central equation in the Solow model. In particular, the slope is: dkt+1 dkt = αβ(1 − α)A (1 + β)(1 + n)kα−1 t ≥ 0 (8.52) t t This slope is positive, so kt+1 is increasing in kt. Furthermore, when kt → 0, kα−1 → ∞, so it starts out very steeply sloped. But when kt → ∞, kα−1 → 0, so when kt gets very large the slope goes to zero. Similarly to what we did in the Solow model, we can plot
kt+1 as a function of kt. It starts in the origin with a very steep slope, increases at a decreasing rate, and asymptotes to a slope of zero. Because it starts with a slope greater than one and ends with a slope of zero, and because it is continuous and starts in the origin, it must cross a 45 degree line showing points where kt+1 = kt exactly once away from the origin. This means that there exists a steady state in which kt+1 = kt = k∗. Furthermore, because the curve lies above the line for kt < k∗ and below the 45 degree line when kt > k∗, the steady state is stable. This is depicted graphically in Figure 8.1 below: 159 Figure 8.1: Plot of the Central Equation of the OLG Model Algebraically, the steady state can be solved for as: k∗ = ( β(1 − α)A (1 + β)(1 + n) ) 1 1−α (8.53) The OLG model with these functional form assumptions will have similar properties to the Solow model. Suppose that, in period t, the old generation is endowed with a capital stock that is less than the steady state capital stock, kt < k∗. This is depicted graphically in Figure 8.2 below. The next period’s capital stock per worker, kt+1, can be determined oﬀ the curve given the initial kt. We observe that kt+1 > kt. We can then iterate forward through time by reﬂecting oﬀ of the 45 degree line. In period t + 1, the economy will start with kt+1 which is still less than k∗. This means that kt+2 > kt+1, and the process will continue until the capital stock per worker settles down to the steady state. Something similar would happen, but in reverse, if the economy were initially endowed with more than the steady state capital stock. 160 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝛽𝛽
(1−𝛼𝛼)(1+𝛽𝛽)(1+𝑛𝑛)𝐴𝐴𝑘𝑘𝑡𝑡𝛼𝛼 𝑘𝑘∗ 𝑘𝑘∗ Figure 8.2: Dynamics in the OLG Model We consider a couple of quantitative experiments to see how the model works in practice. Suppose that β = 0.95 and α = 1/3. Suppose that n = 0.05 (so that the number of young households born increases by 5 percent each period). Suppose A = 1. Then the steady state capital stock works out k∗ = 0.172. Figure 8.3 shows the dynamic trajectories of capital per worker from three diﬀerent initial starting values in period 0 – one where the economy starts in the steady state, another where it starts 50 percent above the steady, and another where it starts 50 percent below the steady state. 161 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝛽𝛽(1−𝛼𝛼)(1+𝛽𝛽)(1+𝑛𝑛)𝐴𝐴𝑘𝑘𝑡𝑡𝛼𝛼 𝑘𝑘∗ 𝑘𝑘∗ 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+2 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+2 Figure 8.3: Quantitative Convergence in the OLG Model We can observe that if the economy starts in the steady state, it stays there. If it starts above, it converges down to the steady state, and similarly from below. This is qualitatively similar to the Solow model. There are a couple of interesting diﬀerences worth highlighting,
however. First, for similar parameter values the steady state capital stock is much lower in the OLG model in comparison to the Solow model. Second, the economy converges to the steady much quicker in the OLG model in comparison to the Solow model. To understand why these diﬀerences arise, it is useful to return to the expression for the steady state capital stock per worker in the augmented Solow model from Chapter 6. See, in particular, (6.63). Setting the growth rate of labor augmenting technology, z, equal to zero, the expression for the steady state capital stock per worker in that model can be written: 1 1−α k∗ = ( sA δ + n ) (8.54) We can interpret (8.54) similarly to (8.53). The eﬀective saving rate in the OLG economy β(1−α). Given values of β, α, and n, one can choose the saving rate in the Solow model is (1+β)(1+n) to be the same, i.e. s =. For the parameter values used above, the comparable saving rate in the Solow model would be about 0.3. What is diﬀerent between the OLG and Solow economies is that the steady state capital stock per worker does not depend on capital’s β(1−α) (1+β)(1+n) 162 00.050.10.150.20.250.30123456789101112HorizonConvergence to Steady Statek0 = kk0 = 0.5kk0 = 1.5k* depreciation rate, δ, in the OLG model. Indeed, the two economies are essentially identical if δ → 1 in the Solow model, regardless of what the value of δ is in the OLG economy. What is going on is the following. In the OLG economy, capital eﬀectively depreciates completely each period, regardless of what the actual depreciation rate on capital is. This is because there is no intergenerational transfer of capital – each period, the old generation simply consumes any leftover capital before dying. As a result, the capital stock available for production in any period is newly created each period. Because of this, capital does not accumulate in the OLG economy in the way that it does in the Solow economy where there is a
single representative household that lives in perpetuity. The steady state capital stock is consequently much smaller in the OLG economy and transition dynamics take place much faster for otherwise similar parameter values in comparison to the Solow model. All that said, there is a subtle issue in comparing the two models related to the interpretation of a unit of time. In the Solow model, a household lives forever, and a unit of time can be interpreted however one pleases, with a year perhaps a natural starting point. In the OLG economy, a household lives only two periods, and it is therefore most appropriate to think of a unit of time as corresponding to roughly a generation, so thirty or so years. For example, a depreciation rate at an annual frequency of 10 percent corresponds to depreciation over a thirty year period of about 95 percent (i.e. (1 − δ)30 ≈ 0.05 when δ = 0.1, so 95 percent of the capital stock would become obsolete after 30 years). Furthermore, a discount factor of β = 0.95 at an annual frequency corresponds to a discount factor over a generation of about 0.2 (i.e. β30 ≈ 0.2 for β = 0.95). Finally, the growth rate of the population, n, ought to be thought of as the growth rate per generation, not per year. If the population grows at 1 percent per year, then over thirty years it ought to grow at (1 + n)30 ≈ 35. Taking these subtle issues into account, the steady state and dynamics of the OLG and Solow models do not look as diﬀerent as one might conclude in the paragraph above. As in the Solow model, it is possible to examine the dynamic eﬀects of changes in A or other parameters (e.g. β, which would inﬂuence how much young households choose to save). We leave these as exercises, only noting that the eﬀects are similar to what would obtain in the Solow model. 8.3 The Golden Rule and Dynamic Ineﬃciency In the Solow model, we introduced the concept of the Golden Rule and put forth the possibility that an economy could be saving too much. In this sense, an economy could be dynamically ineﬃcient in that it could increase consumption in the present and all subsequent periods by simply saving a smaller fraction of output each period. We now return to this 163
discussion in the context of the OLG economy with logarithmic utility and Cobb-Douglas production. We use the model to document exactly how and why the economy might be dynamically ineﬃcient as well as how a benevolent government might be able to employ intergenerational transfers to restore steady state eﬃciency. In the Solow model, the Golden Rule was deﬁned as the saving rate which maximized steady state consumption per worker (or per eﬃciency unit of labor in the augmented model). It is somewhat trickier to deﬁne such a concept in the OLG economy for two reasons. First, there is no explicit saving rate in an OLG economy; rather, the saving rate is a function of deep parameters related to preferences and technology. Second, at any point in time, there are two kinds of households alive. A level of saving which maximizes the steady state consumption of one generation may not do so for the other. We will therefore conceptualize the Golden Rule in terms of the steady state capital stock per worker (rather than a saving rate) which maximizes aggregate consumption per worker, given above in (8.43). To the extent to which n > 0, this implicitly puts more weight on the consumption of young households because there are more young than old households at any point in time. For the general functional form, assuming a steady state exists, we can solve for steady state investment per worker from (8.37) as: Steady state total consumption per worker is then: i∗ = (δ + n)k∗ In terms of the per worker production function, this is simply: c∗ = y∗ − (δ + n)k∗ c∗ = Af (k∗) − (δ + n)k∗ The Golden Rule capital stock maximizes c∗. Hence, it must satisfy: dc∗ dk∗ = 0 ⇔ Af ′(k∗) = δ + n (8.55) (8.56) (8.57) (8.58) Note that (8.58) is exactly the same condition (assuming z = 0) implicitly characterizing the Golden Rule in the augmented Solow model (see (6.67)). Using the Cobb-Douglas functional form assumption (8.58) can be written: αAk∗α−1 = δ + n
(8.59) This means that the Golden Rule capital stock satisﬁes: 164 k∗,gr = ( αA δ + n ) 1 1−α (8.60) Nothing guarantees that (8.60) coincides with (8.53). For the steady state capital stock to be consistent with the Golden Rule, the following must be satisﬁed: α(1 + β)(1 + n) β(1 − α) = δ + n (8.61) We can deﬁne an economy as being dynamically ineﬃcient if dc∗ dk∗ < 0. If this were the case, consumption could be higher by reducing the steady state capital stock. Reducing the steady state capital stock would entail immediately increasing consumption of young households, and so aggregate consumption per capita could increase both in the present and in the future by simply accumulating less capital. Let us examine the parameter values for which a situation of dynamic ineﬃciency might arise. The derivative of steady state consumption withe respect to the steady state capital stock being negative would require that: α(1 + β)(1 + n) β(1 − α) < δ + n (8.62) The economy is most likely to be dynamically ineﬃcient when (i) δ is large, (ii) β is large, (iii) α is small, or (iv) n is large. What is the intuition for why an economy could potentially be dynamically ineﬃcient, and why does this possibility depend on these parameters in the way described above? In the OLG economy, the only way for young households to provide for their consumption in old age is to save. They must save regardless of whether the steady state return to saving, R∗ + (1 − δ), is high or low if they wish to consume in old age. Suppose that δ is very high, for example. Then the steady state return to saving is comparatively low. If this is the case, saving is a relatively ineﬃcient way to transfer resources intertemporally. Both generations could conceivably be better oﬀ if there were a way to instead directly transfer resources across generations – i.e. to subsidize the consumption of the old and ﬁnance this with a tax on saving of the young. A similar result would obtain if α is small (
which other things being equal makes R∗ small) or β is large (a large β increases k∗ and reduces R∗. If n is very large, there are many more young households than old at a given point in time. One therefore needs to tax young households at a comparatively low rate to subsidze consumption of the old. As we shall show in the next section, if an economy ﬁnds itself in the dynamically ineﬃcient range, it is conceivably possible for a benevolent government to make all generations better oﬀ through a tax and transfer system. 165 8.3.1 Government Intervention Suppose that there is a government. This government does no consumption, but it can tax saving via the constant rate τ. The ﬂow budget constraint for a young household in period t becomes: cy,t + (1 + τs)st ≤ wt (8.63) In (8.63), if τ > 0 then saving is expensive compared to consuming. The government can use the proceeds from the tax on saving to subsidize the capital income old households. In particular, suppose that it does so at rate (1 + τk). The ﬂow budget constraint of an old household is: c0,t+1 ≤ [(1 + τk)Rt+1 + (1 − δ)] st (8.64) For the government’s tax and transfer system to be consistent with a balanced budget (i.e. we do not allow for the possibility that the government may issue debt), it must be that total revenue raised by the tax on saving equals the total cost of the capital tax subsidy in any period: τsNtst = τkRtNt−1st−1 (8.65) Preferences of a household are the same. With the new budget constraints, (8.63)-(8.64), the ﬁrst order optimality condition for a young household making a saving decision is: The ﬁrst order optimality condition for the household is similar to (8.5): (1 + τs)u′ (wt − (1 + τs)st) = β [(1 + τK)Rt+1 + (1 − δ)] u′ ([(1 + τK)Rt+1 + (1 − δ)] st) (8.66) If we once again assume log utility, (8
.66) simply becomes: 1 + τs wt − (1 + τs)st = β 1 st (8.67) Solving (8.67) for st, we obtain: β (1 + β)(1 + τs)wt Note that (8.68) reduces to (8.47) when τs = 0. Quite naturally, the bigger is τs, the smaller will be st for a given wt. If we once again assume Cobb-Douglas production, then (8.68) st = 166 wt = (1 − α)Akα t. Then the central equation of the OLG model, (8.36), may be written: kt+1 = β(1 − α)Akα t (1 + β)(1 + τs))(1 + n) The steady state capital stock per worker can be solved for as: k∗ = ( β(1 − α)A (1 + β)(1 + τs)(1 + n) ) 1 1−α (8.69) (8.70) If the government wishes to implement the Golden Rule capital stock, it can set τs so that (8.70) coincides with the Golden Rule steady state capital stock, (8.60). Doing so requires that: Simplifying terms: β(1 − α) (1 + β)(1 + τs)(1 + n) = α δ + n β(1 − α)(δ + n) α(1 + β)(1 + n) = (1 + τs) After some algebraic simpliﬁcation, we can solve for τs as: τs = β(1 − α)(δ + n) − (1 + β)α(1 + n) α(1 + β)(1 + n) (8.71) (8.72) (8.73) Note that the τs necessary to achieve the Golden Rule could be positive or negative. It will be positive when: β(1 − α)(δ + n) > (1 + β)α(1 + n) (8.74) Note that (8.74) is exactly the same condition for the economy to be dynamically ineﬃcient derived above, (8.62). In other words, what (8.73) tells us is that if the economy is dynamically ineﬃcient, saving should
be taxed (i.e. τs > 0). In contrast, if the economy is below the Golden Rule, to get to the Golden Rule saving should be subsidized (i.e. τs < 0). Let us see what τk must be for the government to balance its budget under a plan to implement the Golden Rule. Combine (8.68) with (8.65), while noting that Nt−1st−1 = Kt, to obtain: τsβ (1 + β)(1 + τs)Ntwt = τkRtKt (8.75) Because of Cobb-Douglas production, we must have Ntwt/(RtKt) = 1−α α. Hence, τk must satisfy: 167 τk = 1 − α α τsβ (1 + β)(1 + τs) (8.76) Let us do a couple of quantitative experiments to examine how a government might be able to improve total welfare in an economy by implementing taxes. Let us ﬁrst consider the parameter conﬁguration in which there are no taxes – i.e. τs = τk = 0. Suppose that α = 0.2, β = 0.95, δ = 0.1, and n = 0.5. These parameters are slightly diﬀerent than those considered in the example above, but we choose them to ensure that consumption of neither generation of agent ever goes negative. With these parameters, the economy is not dynamically ineﬃcient. Referencing (8.74), we have β(1 − α)(δ + n) = 0.456, whereas (1 + β)α(1 + n) = 0.585. The steady state capital stock per worker comes out to k∗ = 0.186. Steady state total consumption = 0.465. Steady state and consumption by generation are: c∗ = 0.602, c∗ y lifetime utility for a young household is U = ln(c∗ y Suppose that the economy sits in this steady state from period 0 to period 2. Then, in period 3, a government decides to implement a tax system to move the economy to the Golden Rule; i.e. it sets τk according to (8.76) and τs according to (8.73). This requires setting τs = −0.22 and τk = −
0.55 – in other words, saving is subsidized and capital income is taxed. The economy will eventually converge to a new steady state with higher capital (k∗ = 0.253), but it takes a few periods to get there. = 0.293, and c∗ 0 ) = −1.99.2 ) + β ln(c∗ 0 Figure 8.4 plots the dynamic trajectories of capital per worker (upper left), consumption (both aggregate consumption and consumption of each generation, upper right), and utility (lower left). For the utility graph, we show lifetime utility for young agents – i.e. U = u(cy,t) + βu(co,t+1) – whereas for old agents we simply show utility from current consumption – i.e. u(co,t). 2Note that utility is an ordinal concept and there is nothing wrong with utility being negative, as we discuss later in the book. 168 Figure 8.4: Implementing Taxes to Hit the Golden Rule As we can see, subsidizing saving results in more capital accumulation. It also results in more consumption for young households (both in the present and in all future periods). But because subsidizing saving requires taxing capital income, the old generation is hurt in period 3 when the tax system is implemented – its consumption falls. Aggregate consumption initially falls, but ultimately ends up higher than where it started once the economy settles to its new steady state. This is conceptually similar to the Solow model – increasing saving when starting from below the Golden Rule results in an initial fall in consumption but higher consumption in the long run. Does the implementation of this tax system make households better oﬀ? To see this, focus on the lower left plot. The current utility of the old generation alive at the time the tax system is implemented falls from −0.76 to −1.77. It is rather obvious old households would be hurt because their consumption is lower. Furthermore, even though consumption is higher for young households, lifetime utility for these households actually falls (both immediately 169 0.150.170.190.210.230.250.270123456HorizonCapital per Worker00.10.20.30.40.50.60.70123456HorizonConsumptionccyco-2.5-2-1.5-1-0.500123456HorizonUtilityLifetime Utility of YoungCurrent Utility of Old when the tax system is
implemented as well as when the economy settles down to its new steady state). The extra consumption in youth is not enough to compensate them for the lower consumption in old age. Since both young and old households end up with lower utility, implementing the Golden Rule via a tax system starting from the initial steady state described above would evidently not be a desirable thing for a government to do. Suppose instead that the economy initially sits in a steady state but is dynamically ineﬃcient in the sense of being above the Golden Rule capital stock. In particular, keep the same parameters as above but instead assume that δ = 0.5. This makes β(1 − α)(δ + n) = 0.76, whereas (1 + β)α(1 + n) remains at 0.585. Assuming no taxes, the initial steady state capital stock is k∗ = 0.186. Suppose that the economy sits in this steady state from period 0 to period 2. Then, in period 3, the government implements the tax system described above to achieve the Golden Rule. It must therefore set τs = 0.3 and τk = 0.448. In other words, it is taxing saving and using the proceeds from this tax to subsidize capital income. Figure 8.5 below plots dynamic trajectories of selected variables after this change. 170 Figure 8.5: Implementing Taxes to Hit the Golden Rule: Starting from Dynamic Ineﬃciency Because the government switches to taxing saving, capital accumulation falls and the economy converges to a lower steady state capital stock. The consumption of young households falls both initially as well as in the new steady state, but the consumption of old households rises. Furthermore, aggregate consumption initially rises and then declines, but it ultimately always remains higher than where it began. This is again similar to the Solow model, where decreasing the saving rate from a position initially above the Golden Rule results in higher consumption both in the present and at every subsequent date. In the lower left plot we again show utility of the two types of agents. The old generation alive at the time the tax system is implemented naturally beneﬁts in the form of higher utility – its consumption is higher, and so that generation is better oﬀ. It is also better oﬀ in all subsequent periods. What is interesting is that the young generation also beneﬁts – even though its consumption immediately falls, the higher consumption it gets to enjoy in old age is more than enough
to make up for it. The lifetime utility of young households is higher at 171 0.10.110.120.130.140.150.160.170.180.190.20123456HorizonCapital per Worker00.10.20.30.40.50.60.70123456HorizonConsumptionccyco-2.5-2-1.5-1-0.500123456HorizonUtilityLifetime Utility of YoungCurrent Utility of Old every subsequent date after the implementation of the tax system. It is instructive to compare the results shown in Figures 8.5 and 8.4. In Figure 8.5, all generations (both the young and old generations at the time the tax is implemented, as well as future generations yet to be born) are better oﬀ after the implementation of the tax system. Since it is possible to improve the welfare of all generations, the initial situation of dynamic ineﬃciency is what economists call Pareto ineﬃcient. An allocation is said to be Pareto ineﬃcient if it is possible to redistribute resources in such a way as to make at least some agents better oﬀ and no agents worse oﬀ. This is the case depicted in Figure 8.5. This is not the case in Figure 8.4. In that case, implementing a tax system to achieve the Golden Rule actually makes all generations worse oﬀ. That result is somewhat sensitive to the particulars of the parameterization – it is conceivable that implementing the Golden Rule starting from a steady state which is not dynamically ineﬃcient could improve the welfare of some generations and hurt it for others, but it is not possible to make all generations better oﬀ by implementing such a tax system if the economy is initially not dynamically ineﬃcient. In contrast, if the economy is initially dynamically ineﬃcient, implementation of a tax and transfer system like the one described above can be unambiguously welfare-enhancing. 8.4 Incorporating Exogenous Technological Growth In the OLG model, an economy (might) converge to a steady state in which it does not grow. We include “(might)” because without saying something more speciﬁc about preferences and production it is possible a steady state does not exist. Let us assume, however, that a steady state does exist
(as it does with the preference and production assumptions considered throughout this chapter). If that is the case, then over long horizons this economy will exhibit no growth. Similarly to the Solow model, this counterfactual prediction can be remedied by re-writing the model to allow for exogenous growth in a technology variable. Once again let Zt denote the level of labor augmenting technology. While similar to At, Zt directly multiplies labor input in the aggregate production function. Formally: Yt = AF (Kt, ZtNt) (8.77) For notational ease we once again treat At = A as constant. The real wage equals the marginal product of labor, which in this case is: The rental rate on capital is again just the marginal product of capital: wt = AZtFN (Kt, ZtNt) (8.78) 172 Rt = AFK(Kt, ZtNt) (8.79) One can see the diﬀerence between A and Zt in (8.78)-(8.79). Whereas A directly impacts the marginal product of both factors, Zt only directly impacts the marginal product (and hence factor price) of labor input. We shall assume that Zt grows at an exogenous and constant rate, z ≥ 0: Zt = (1 + z)Zt−1 (8.80) Other than these additions, the general OLG model in the levels of variables is identical to what is presented above, (8.19)-(8.27). To make progress in analyzing the model, let us assume that ﬂow utility is the natural log and that production is Cobb-Douglas. Rather than re-writing the equilibrium conditions in per worker terms, let us deﬁne ̂xt = Xt as a variable ZtNt per eﬃciency unit of labor, for some variable Xt. The equilibrium conditions in levels taking account of this growth, and making these functional form assumptions, are: wt st = β 1 + β Kt+1 = Ntst It = Kt+1 − (1 − δ)Kt Yt = Ct + It Yt = AK α t (ZtNt)1−α Rt = αAK α−1 t (ZtNt)1−α wt = (1 − α)AZtK α t (
ZtNt)−α Ct = Cy,t + Co,t Cy,t = Ntwt − Kt+1 (8.81) (8.82) (8.83) (8.84) (8.85) (8.86) (8.87) (8.88) (8.89) With this preference and production speciﬁcation there exists a balanced growth path. Per worker variables, like st, grow at the rate of labor augmenting technology. Therefore, deﬁne ̂st = st. The real wage, as in the Solow model, will also grow at the rate of labor Zt augmenting technology. Therefore deﬁne ̂wt = wt Zt. (8.81) can be therefore be written: ̂st = β 1 + β ̂wt 173 (8.90) Divide both sides of (8.82) by ZtNt; Kt+1 ZtNt = Ntst ZtNt (8.91) Multiply and divide the left hand side of (8.91) by Zt+1Nt+1, and make use of the fact that Zt+1Nt+1 ZtNt = (1 + z)(1 + n), to write this as: ̂kt+1 = ̂st (1 + z)(1 + n) (8.92) Note that (8.92) reduces to (8.36) when z = 0. (8.83) and (8.84) can be similarly transformed to: ̂it = (1 + z)(1 + n)̂kt+1 − (1 − δ)̂kt ̂yt = ̂ct +̂it (8.93) (8.94) The production function and expression for the real wage are straightforward to transform into per eﬃciency unit variables: ̂yt = Âkα ̂wt = (1 − α)Âkα t t (8.95) (8.96) Note that the rental rate on capital is stationary along a balanced growth path, and needs no transformation: The aggregate resource constraint is straightforward to transform: Rt = αÂkα−1 t Divide both sides of (8.88
) by ZtNt: ̂yt = ̂ct +̂it Ct ZtNt = Cy,t ZtNt + Co,t ZtNt (8.97) (8.98) (8.99) As before, we wish to scale consumption of the old generation by eﬃciency units of labor at the time of its birth. Hence, multiply and divide the last term by Zt−1Nt−1: ̂ct = ̂cy,t + Co,t Zt−1Nt−1 Zt−1Nt−1 ZtNt (8.100) 174 Note that (8.100) may be written as follows, which of course reduces to (8.42) when z = 0: ̂ct = ̂cy,t + co,t (1 + z)(1 + n) Finally, turn to (8.89). Divide both sides by ZtNt: Cy,t ZtNt = Ntwt ZtNt − Kt+1 ZtNt ̂cy,t = ̂wt − Kt+1 Zt+1Nt+1 Zt+1Nt+1 ZtNt Which can be written: Which is just: (8.101) (8.102) (8.103) ̂cy,t = ̂wt − (1 + z)(1 + n)̂kt+1 (8.104) of course reduces to (8.43) when z = 0. All told, the equilibrium conditions of the (8.104) model re-written in stationary form are given below: ̂wt ̂st = β 1 + β ̂st (1 + z)(1 + n) ̂kt+1 = ̂it = (1 + z)(1 + n)̂kt+1 − (1 − δ)̂kt ̂yt = ̂ct +̂it ̂yt = Âkα Rt = αÂkα−1 ̂wt = (1 − α)Âkα ̂co,t (1 + z)(1 + n) ̂ct = ̂cy,t + t t t �
�cy,t = ̂wt − (1 + z)(1 + n)̂kt+1 (8.105) (8.106) (8.107) (8.108) (8.109) (8.110) (8.111) (8.112) (8.113) The central equation of the OLG model can be derived by combining (8.105) and (8.111) with (8.106): ̂kt+1 = β(1 − α)Âkα (1 + β)(1 + z)(1 + n) t 175 (8.114) (8.114) is similar to (8.51) and reduces to it when z = 0. The allowance for z ≥ 0 does not change the fact that a steady state capital stock per eﬃciency unit of labor exists. At this steady state, variables expressed in per eﬃciency terms are constant, and per worker variables grow at rate z, while level variables grow at approximately z + n. The proof of this is exactly as in Chapter 6 and is not repeated here. We simply wish to note that the steady state / balanced growth path properties of the OLG economy are identical to the Solow model and hence consistent with the time series stylized facts. 8.5 Summary • As opposed to their representative agent counterparts, overlapping generation models feature economies where people are diﬀerentiated by age. • Under some functional form assumptions, there is a unique and stable steady state. Increases in productivity and the discount factor raise steady-state capital and output. An increase in the growth rate of the population reduces them. • The competitive equilibrium is usually not Pareto eﬃcient. The reason is that there is a missing market between agents alive and those agents yet to be borne. In equilibrium, the economy might accumulate too much or too little capital relative to the eﬃcient benchmark. • A benevolent government can correct this market failure by issuing debt. Since the government is inﬁnitely lived, this essentially solves the problem of missing markets. • Since government debt aﬀects equilirbium prices and quantities, Ricardian Equivalence does not hold. Key Terms • Questions for Review 1. Recall that factor prices with Cobb Douglas production are wt = A(1 − α)kα t Rt = Aαkα−
1 t (a) Assuming log utility, solve for the steady-state wage and rental rates. 176 (b) Plot the responses over time to a permanent increase in A. (c) Plot the responses over time to a permanent increase in n. 2. Suppose the lifetime utility function of a household is Ut = − 1 c1−σ y,t 1 − σ + β − 1 c1−σ o,t+1 1 − σ with σ ≥ 0. The budget constraints are the same. cy,t + st = wt co,t+1 = Rt+1st (a) Substitute the constraints into the objective function and take the ﬁrst order condition for st. (b) Solve for the optimal st as a function of wt, Rt, and parameters. (c) Is the derivative of the optimal st with respect to Rt unambiguous? How about with respect to wt? Explain the intuition of this. 3. Suppose that we include government spending in the model. Total govern- ment spending is given by Gt = Ntτ where τ is a lump sum tax collected from all young agents. This means that per capita government spending, gt = Gt Nt is constant and given by τ. (a) The utility maximization problem is Ut = ln cy,t + β ln co,t+1 max cy,t,co,t+1,st s.t. cy,t + st = wt − τ co,t+1 = stRt+1. Solve for the optimal st as a function of factor prices, τ, and other parameters. (b) The proﬁt maximization problem is given by Πt = max Kt,Nt AK α t N 1−α t − wtNt − RtKt Derive the ﬁrst order conditions for labor and capital. 177 (c) The capital accumulation equation in per capita terms is kt+1 = st 1 + n. Using your answers from the ﬁrst two parts, write the accumulation equation as a function of kt, exogenous variables and parameters. (d) Plot the capital accumulation line against the 45 degree line. Clearly label the steady state and show that it is stable. (e) Plot the eﬀects of an increase in g. 178 Part III The Microeconomics of Macroeconom
ics 179 All economics is microeconomics. Essentially every economic problem contains some person or ﬁrm maximizing an objective subject to constraints. Essentially every economic problem contains some notion of equilibrium in which the maximization problem of various market participants are rendered mutually consistent with market clearing conditions. As we discussed in Chapter 3, a major achievement in economics over the last forty years has been to incorporate these microeconomic fundamentals into models designed to answer macroeconomic questions. In this section, we cover each optimization problem in detail and how they come together in equilibrium. Macroeconomics is focused on dynamics – i.e. the behavior of the aggregate economy across time. For most of the remainder of the book, we focus on a world with two periods. Period t is the present and period t + 1 represents the future. Two periods are suﬃcient to get most of the insights of the dynamic nature of economic decision-making. By virtue of being the largest component in GDP, consumption is covered in two chapters, 9 and 10. The key microeconomic insight is that consumption is a function of expected lifetime income rather than just current income. We also discuss how elements such as taxes, wealth, and uncertainty aﬀect consumption decisions. In Chapter 11, we introduce the idea of a competitive equilibrium. A competitive equilibrium is a set of prices and allocations such that everyone optimizes and markets clear. In an economy without production, the market clearing conditions are straightforward and therefore represent an ideal starting point. In Chapter 12 we derive the solution to the household’s problem when it chooses both how much to consume and how much to work. We also derive the ﬁrm’s optimal choice of capital and labor. Chapters 13 and 14 bring a government sector and money into the economy. Finally, we close in Chapter 15 by showing that the competitive equilibrium is Pareto Optimal. A key implication of this is that activist ﬁscal or monetary policy will not be welfare enhancing. Chapter 17 studies the determinants of unemployment. Throughout the rest of the book, we are silent on unemployment and instead focus on hours worked as our key labor market indicator. In this Chapter we show some facts concerning unemployment, vacancies, and job ﬁnding rates. We then work through a stylized version of the Diamond-Mortenson-Pissarides (DMP) search and matching model of unemployment. 180 Chapter 9 A Dynamic Consumption-Saving Model Modern macroeconomics is dynamic. One of the cornerstone dynamic
models is the simple two period consumption-saving model which we study in this chapter. Two periods (the present, period t, and the future, period t + 1) is suﬃcient to think about dynamics, but considerably simpliﬁes the analysis. Through the remainder of the book, we will focus on two period models. The key insights from two period models carry over to models with multiple future periods. In the model, there is a representative household. There is no money in the model and everything is real (i.e. denominated in units of goods). The household earns income in the present and the future (for simplicity, we assume that future income is known with certainty, but can modify things so that there is uncertainty over the future). The household can save or borrow at some (real) interest rate rt, which it takes as given. In period t, the household must choose how much to consume and how much to save. We will analyze the household’s problem both algebraically using calculus and using an indiﬀerence curve - budget line diagram. The key insights from the model are as follows. First, how much the household wants to consume depends on both its current and its future income – i.e. the household is forward-looking. Second, if the household anticipates extra income in either the present or future, it will want to increase consumption in both periods – i.e. it desires to smooth its consumption relative to its income. The household smooths its consumption relative to its income by adjusting its saving behavior. This has the implication that the marginal propensity to consume (MPC) is positive but less than one – if the household gets extra income in the present, it will increase its consumption by a fraction of that, saving the rest. Third, there is an ambiguous eﬀect of the interest rate on consumption – the substitution eﬀect always makes the household want to consume less (save more) when the interest rate increases, but the income eﬀect may go the other way. This being said, unless otherwise noted we shall assume that the substitution eﬀect dominates, so that consumption is decreasing in the real interest rate. The ultimate outcome of these exercises is a consumption function, which is an optimal decision rule which relates optimal consumption to things the household takes as given – current income, future income, and the real interest rate. We will make use of the consumption function derived in this chapter
throughout the rest of the book. 181 We conclude the chapter by consider several extensions to the two period framework. These include uncertainty about the future, the role of wealth, and borrowing constraints. 9.1 Model Setup There is a single, representative household. This household lives for two periods, t (the present) and t+1 (the future). The consumption-saving problem is dynamic, so it is important that there be some future period, but it does not cost us much to restrict there to only be one future period. The household gets an exogenous stream of income in both the present and the future, which we denote by Yt and Yt+1. For simplicity, assume that the household enters period t with no wealth. In period t, it can either consume, Ct, or save, St, its income, with St = Yt − Ct. Saving could be positive, zero, or negative (i.e. borrowing). If the household takes a stock of St into period t + 1, it gets (1 + rt)St units of additional income (or, in the case of borrowing, has to give up (1 + rt)St units of income). rt is the real interest rate. Everything here is “real” and is denominated in units of goods. The household faces a sequence of ﬂow budget constraints – one constraint for each period. The budget constraints say that expenditure cannot exceed income in each period. Since the household lives for two periods, it faces two ﬂow budget constraints. These are: Ct + St ≤ Yt Ct+1 + St+1 ≤ Yt+1 + (1 + rt)St. (9.1) (9.2) The period t constraint, (9.1), says that consumption plus saving cannot exceed income. The period t + 1 constraint can be re-arranged to give: Ct+1 + St+1 − St ≤ Yt+1 + rtSt. (9.3) St is the stock of savings (with an “s” at the end) which the household takes from period t to t + 1. The ﬂow of saving (without an “s” at the end) is the change in the stock of savings. Since we have assumed that the household begins life with no wealth, in period t there is no distinction between saving and savings. This is not true in period t + 1.
St+1 is the stock of savings the household takes from t + 1 to t + 2. St+1 − St is its saving in period t + 1 – the change in the stock. So (9.3) says that consumption plus saving (Ct+1 + St+1 − St) cannot exceed total income. Total income in period t + 1 has two components –Yt+1, exogenous ﬂow income, and interest income on the stock of savings brought into period t, rtSt (which could be negative if the household borrowed in period t). We can simplify these constraints in two dimensions. First, the weak inequality constraints 182 will hold with equality under conventional assumptions about preferences – the household will not let resources go to waste. Second, we know that St+1 = 0. This is sometimes called a terminal condition. Why? St+1 is the stock of savings the household takes into period t + 2. But there is no period t + 2 – the household doesn’t live into period t + 2. The household would not want to ﬁnish with St+1 > 0, because this would mean “dying” without having consumed all available resources. The household would want St+1 < 0 – this would be tantamount to dying in debt. This would be desirable from the household’s perspective because it would mean borrowing to ﬁnance more consumption while alive, without having to pay oﬀ the debt. We assume that the ﬁnancial institution with which the household borrows and saves knows this and will not allow the household to die in debt. Hence, the best the household can do is to have St+1 = 0. Hence, we can write the two ﬂow budget constraints as: Ct + St =Yt Ct+1 =Yt+1 + (1 + rt)St. (9.4) (9.5) St shows up in both of these constraints. We can solve for St from one of the constraints and then plug it into the other. In particular, solving (9.5) for St: St = Ct+1 1 + rt − Yt+1 1 + rt. Now, plug this into (9.4) and re-arrange terms. This yields: Ct + Ct+1 1 + rt = Yt + Yt+1 1 + rt. (9.6) (9
.7) We refer to (9.7) as the intertemporal budget constraint. In words, it says that the present discounted value of the stream of consumption must equal the present discounted value of the stream of income. The present value of something is how much of that thing you would need in the present to have some value in the future. In particular, how many goods would you need in period t to have F Vt+1 goods in period t + 1? Since you could put P Vt goods “in the bank” and get back (1 + rt)P Vt goods in the future, the P Vt = F Vt+1. In other words, Ct+1 1+rt 1+rt is the present value of period t + 1 consumption and Yt+1 is the present value of period t + 1 1+rt income. The intertemporal budget constraint says that consumption must equal income in a present value sense. Consumption need not equal income each period. Having discussed the household’s budget constraints, we now turn to preferences. We assume that lifetime utility, U, is a weighted sum of ﬂow utility from each period of life. In particular: 183 U = u(Ct) + βu(Ct+1), 0 ≤ β < 1. (9.8) Here, U refers to lifetime utility and is a number, denominated in utils. Utility is an ordinal concept, and so we don’t need to worry about the absolute level of U. All that matters is that a higher value of U is “better” than a lower value. u(⋅) is a function which maps consumption into ﬂow utility (so u(Ct) is ﬂow utility from period t consumption). β is a discount factor. We assume that it is positive but less than one. Assuming that it is less than one means that the household puts less weight on period t + 1 utility than period t utility. This means that we assume that the household is impatient – it would prefer utility in the present compared to the future. The bigger β is, the more patient the household is. We sometimes use the terminology that lifetime utility is the present value of the stream of utility ﬂows. In this setup, β is the factor by which we discount future utility ﬂows, in a way similar to how 1 is the factor by which we discount future ﬂ
ows of goods. So sometimes we 1+rt is the goods discount factor.1 Finally, will say that β is the utility discount factor, while we assume that the function mapping consumption into ﬂow utility is the same in periods t and t + 1. This need not be the case more generally, but is made for convenience. 1 1+rt We assume that the utility function has the following properties. First, u′(⋅) > 0. We refer to u′(⋅) as the marginal utility of consumption. Assuming that this is positive just means that “more is better” – more consumption yields more utility. Second, we assume that u′′(Ct) < 0. This says that there is diminishing marginal utility. As consumption gets higher, the marginal utility from more consumption gets smaller. Figure 9.1 plots a hypothetical utility function with these properties in the upper panel, and the marginal utility as a function of Ct in the lower panel. Below are a couple of example utility functions(Ct) = θCt, u(Ct) = Ct − θ 2 u(Ct) = ln Ct u(Ct) = C 1−−. (9.9) (9.10) (9.11) (9.12) 1It is sometimes useful to use the terminology of discount rates, particularly if and when one is working in “continuous time” (we are working in “discrete time”). In particular, rt is the goods discount rate and one over one plus this, i.e., is the goods discount factor. We could deﬁne ρ as the utility discount rate, implicitly deﬁned by β = 1 1+ρ. Assuming 0 < β < 1 means assuming ρ > 0. 1 1+rt 184 Figure 9.1: Utility and Marginal Utility The utility function in (9.9) is a linear utility function. It features a positive marginal utility but the second derivative is zero, so this utility function does not exhibit diminishing marginal utility. The second utility function is called a quadratic utility function. It features diminishing marginal utility, but it does not always feature positive marginal utility – there exists a satiation point about which utility is decreasing in consumption. In particular, if Ct > 1/θ, then marginal utility is negative. The third utility function is the log utility function. This utility function is particularly attractive because it is
easy to take the derivative and it satisﬁes both properties laid out above. The ﬁnal utility function is sometimes called the isoelastic utility function. It can be written either of the two ways shown in (9.12). Because 1−σ is included or not. If σ = 1, then this utility is ordinal, it does not matter whether the −1 utility function is equivalent to the log utility function. This can be shown formally using L’Hopital’s rule. Note that nothing guarantees that utility is positive – if Ct < 1 in the log utility case, for example, then u(Ct) < 0. There is no problem with the level of utility being negative, since utility is ordinal. For example, suppose you are considering two values of consumption, C1,t = 0.9 and C2,t = 0.95. With the log utility function we have ln 0.9 = −0.1054, which is negative. We have ln(0.95) = −0.0513, which is also negative, but less negative than utility with C1,t = 0.9. Hence, C2,t is preferred to C1,t. Mathematical Diversion How can we show that the isoelastic utility function, (9.12), is equivalent to the 185 𝐶𝐶𝑡𝑡 𝐶𝐶𝑡𝑡 𝑢𝑢(𝐶𝐶𝑡𝑡) 𝑢𝑢′(𝐶𝐶𝑡𝑡) log utility function when σ = 1? If we evaluate (9.12) with σ = 1, we get 0 0, which is undeﬁned. L’Hopital’s Rule can be applied. Formally, L’Hopital’s rule says that if: f (x) g(x) = 0 0, lim x→a where a is some number and x is the parameter of interest, then: f (x) g(x) lim x→a = lim x→a f ′(x) g′(x). (9.13) (9.14) In other words, we can evaluate the function as the ratio of the ﬁrst derivatives
evaluated at the point x = a. In terms of the isoleastic utility function, x = σ, a = 1, f (x) = C 1−σ − 1, and g(x) = 1 − σ. We can write C 1−σ = exp((1 − σ) ln Ct). The derivative of this with respect to σ is: d exp((1 − σ) ln Ct) dσ = − ln Ct exp((1 − σ) ln Ct). (9.15) In (9.15), − ln Ct is the derivative of the “inside” with respect to σ, and exp((1 − σ) ln Ct) is the derivative of the “outside”. The derivative of 1 − σ with respect to σ is -1. If we evaluate these derivatives at σ = 1, we get − ln Ct for the f ′(xt) and −1 for g′(xt). The ratio is ln Ct, meaning that as σ → 1, the isoelastic utility function is simply the log utility function. 9.2 Optimization and the Euler Equation The household faces an optimization problem in which it wants to pick Ct and St in period t to maximize lifetime utility, (9.8), subject to the two ﬂow budget constraints, (9.4) and (9.5). We already know that St can be eliminated by combining the two ﬂow budget constraints into the intertemporal budget constraint (9.7). We can then think about the problem as one in which the household chooses Ct and Ct+1 in period t. Formally: max Ct,Ct+1 U = u(Ct) + βu(Ct+1) subject to: Ct + Ct+1 1 + rt = Yt + Yt+1 1 + rt. 186 (9.16) (9.17) This is a constrained optimization problem, with (9.17) summarizing the scarcity that the household faces. The household acts as a price-taker and takes rt as given. To solve a constrained optimization problem, solve the constraint for one of the two choice variables (it does not matter which one). Solving for Ct+1, we get: Ct+1 = (1 + rt)(Yt − Ct) + Yt+1. (9
.18) Now, plug (9.18) into the lifetime utility function for Ct+1. This renders the problem of a household an unconstrained optimization problem of just choosing Ct: max Ct U = u(Ct) + βu ((1 + rt)(Yt − Ct) + Yt+1). (9.19) To characterize optimal behavior, take the derivative with respect to Ct: ∂U ∂Ct = u′(Ct) + βu′ ((1 + rt)(Yt − Ct) + Yt+1) × −(1 + rt). (9.20) In (9.20), the u′ ((1 + rt)(Yt − Ct) + Yt+1) is the derivative of the “outside” part, while −(1+ rt) is the derivative of the “inside” with respect to Ct. The term inside u′ ((1 + rt)(Yt − Ct) + Yt+1) is just Ct+1. Making that replacement, and setting the derivative equal to zero, yields: u′(Ct) = β(1 + rt)u′(Ct+1). (9.21) Expression (9.21) is commonly called the consumption Euler equation. In economics, we often call dynamic ﬁrst order optimality conditions Euler equations. This condition is a necessary, though not suﬃcient, condition for the household optimization problem. It says that, at an optimum, the household should pick Ct and Ct+1 so that the marginal utility of period t consumption, u′(Ct), equals the marginal utility of period t + 1 consumption, βu′(Ct+1), multiplied by the gross real interest rate (i.e. one plus the real interest rate). What is the intuition for why this condition must hold if the household is behaving optimally? Suppose that the household decides to consume a little bit more in period t. The marginal beneﬁt of this is the extra utility derived from period t consumption, u′(Ct). What is the marginal cost of consuming a little more in period t? If the household is consuming a little more in t, it is saving a little less (equivalently, borrowing a little more). If it saves a little bit less in period t, this means it has to forego 1 + rt units of consumption in t
+ 1 (since it has to pay back interest plus principle). The lost utility in period t + 1 from consuming a little less is βu′(Ct+1). The total loss in utility is this times the decline in consumption, so β(1 + rt)u′(Ct+1) represents the marginal cost of consuming a little more in period t. At an optimum, the marginal beneﬁt of consuming a little more in period t must equal the 187 marginal cost of doing so – if the marginal beneﬁt exceeded the marginal cost, the household could increase lifetime utility by consuming more in t; if the marginal beneﬁt were less than the marginal cost, the household could increase lifetime utility by consuming a little less in period t. The Euler equation, (9.21), can be re-arranged to be written: u′(Ct) βu′(Ct+1) = 1 + rt. (9.22) The left hand side of (9.22) is what is called the marginal rate of substitution (MRS) between period t and t + 1 consumption. The MRS is simply the ratio of the marginal utilities of Ct and Ct+1. The right hand side is the price ratio between period t and period t + 1 consumption. In particular, getting an additional unit of period t consumption requires giving up 1 + rt units of t + 1 consumption (via the logic laid out above). In this sense, we often refer to the real interest rate as the intertemporal price of consumption – rt tells you how much future consumption one has to give up to get some more consumption in the present. At an optimum, the MRS is equal to the price ratio, which ought to be a familiar result to anyone who has taken intermediate microeconomics. Example Suppose that the utility function is the natural log. Then the Euler equation can be written: This can be re-arranged: Ct+1 Ct = β(1 + rt). 1 Ct = β(1 + rt) 1 Ct+1. (9.23) (9.24) The left hand side of (9.24) is the gross growth rate of consumption between t and t + 1. Hence, the Euler equation identiﬁes the expected growth rate of consumption as a function of the degree of impatience, β, and the real interest rate, rt. It does not identify the
levels of Ct and Ct+1 – (9.24) could hold when Ct and Ct+1 are both big or both small. Other factors held constant, the bigger is β, the higher will be expected consumption growth. Likewise, the bigger is rt, the higher will be expected consumption growth. β < 1 means the household is impatient, which incentivizes consumption in the present at the expense of the future (i.e. makes Ct+1/Ct less than one, other things being equal). rt > 0 has the opposite eﬀect – it incentivizes deferring consumption to the future, which makes Ct+1/Ct greater than one. If β(1 + rt) = 1, these two eﬀects oﬀset, and the 188 household will desire Ct+1 = Ct. Example Suppose that the utility function is the isoelastic form, (9.12). Then the Euler equation can be written: C −σ t = β(1 + rt)C −σ t+1. Take logs of (9.25), using the approximation that ln(1 + rt) = rt: − σ ln Ct = ln β + rt − σ ln Ct+1. This can be re-arranged to yield: ln Ct+1 − ln Ct = 1 σ ln β + 1 σ rt. (9.25) (9.26) (9.27) Since ln Ct+1 − ln Ct is approximately the expected growth rate of consumption between t and t + 1, this says that consumption growth is positively related to the real interest rate. The coeﬃcient governing the strength of this relationship is 1/σ. The bigger is σ (loosely, the more concave is the utility function) the less sensitive consumption growth will be to changes in rt, and vice-versa. 9.3 Indiﬀerence Curve / Budget Line Analysis and the Consump- tion Function The Euler equation is a mathematical condition that is necessary if a household is behaving optimally. The Euler equation is not a consumption function, and it does not indicate how much consumption in the present and future a household should have if it is behaving optimally. The Euler equation only indicates how much relative consumption the household should do in the future versus the present, as a function of the real
interest rate. We would like to go further and determine the levels of period t and t + 1 consumption. In so doing, we will be able to discern some features of the consumption function. We will ﬁrst proceed graphically, using an indiﬀerence curve / budget line diagram. The budget line is a graphical representation of the intertemporal budget constraint, (9.7). It graphically summarizes the scarcity inherited by the household. Let’s consider a graph with Ct+1 on the vertical axis and Ct on the horizontal axis. The budget line will show all combinations of Ct and Ct+1 which exhaust resources – i.e. which make the intertemporal budget constraint 189 hold. Solving for Ct+1 in terms of Ct: Ct+1 = (1 + rt)(Yt − Ct) + Yt+1. (9.28) Given Yt, Yt+1, and rt, the maximum period t + 1 the household can achieve is Ct+1 = (1 + rt)Yt + Yt+1. This level of consumption can be achieved if the household saves all of its period t income (consumption cannot be negative). Conversely, the maximum period t consumption the household can achieve is Ct = Yt + Yt+1. This involves consuming all of 1+rt period t income and borrowing the maximum amount possible, Yt+1, to ﬁnance period t 1+rt consumption. Yt+1 is the borrowing limit because this is the maximum amount the household 1+rt can pay back in period t. These maximum levels of Ct and Ct+1 form the horizontal and vertical axis intercepts of the budget line, respectively. The budget line must pass through the “endowment point” where Ct = Yt and Ct+1 = Yt+1. Consuming its income in each period is always feasible and completely exhausts resources. Finally, the slope of the budget line is = −(1 + rt), which does not depend on Ct or Ct+1. Hence the budget line is in fact a dCt+1 dCt line, because its slope is constant. Figure 9.2 plots a hypothetical budget line. Points inside the budget line are feasible but do not exhaust resources. Points beyond the budget line are infeasible. Figure 9.2: Budget Line An indiﬀerence curves shows combinations of Ct and Ct+1 (
or “bundles” of period t and t + 1 consumption) which yield a ﬁxed overall level of lifetime utility. There will be a diﬀerent indiﬀerence curve for diﬀerent levels of lifetime utility. In particular, suppose that a household 190 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 (1+𝑟𝑟𝑡𝑡)𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+1 𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+11+𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 Feasible Infeasible Slope: −(1+𝑟𝑟𝑡𝑡) has a bundle (C0,t, C0,t+1) which yields overall utility level U0: U0 = u(C0,t) + βu(C0,t+1). (9.29) Consider simultaneous changes in Ct and Ct+1 of dCt and dCt+1 (these are changes relative to C0,t and C0,t+1). Take the total derivative of (9.29): dU = u′(C0,t)dCt + βu′(C0,t+1)dCt+1. (9.30) Since an indiﬀerence curve shows combinations of Ct and Ct+1 which keep lifetime utility lifetime utility ﬁxed, these hypothetical changes in Ct and Ct+1 must leave dU = 0 (i.e. unchanged). Setting this equal to zero, and solving for dCt+1 dCt, we get: dCt+1 dCt = − u′(C0,t) βu′(C0,t+1). (9.31) In other words, (9.31) says that the slope of the U = U0 indiﬀerence curve at (C0,t, C0,t+1) is equal to the negative of the
ratio of the marginal utilities of periods t and t + 1 consumption. Since both marginal utilities are positive, the slope of the indiﬀerence curve is negative. That the indiﬀerence curve is downward-sloping simply says that if the household increases period t consumption, it must decrease period t + 1 consumption if lifetime utility is to be held ﬁxed. Given that we have assumed diminishing marginal utility, the indiﬀerence curve will have a “bowed in” shape, being steepest when Ct is small and ﬂattest when Ct is big. If Ct is small, then the marginal utility of period t consumption is relatively high. Furthermore, if Ct is small and lifetime utility is held ﬁxed, then Ct+1 must be relatively big, so the marginal utility of period t + 1 consumption will be relatively small. Hence, the ratio of marginal utilities will be relatively large, so the the indiﬀerence curve will be steeply sloped. In contrast, if Ct is relatively big (and Ct+1 small), then the marginal utility of period t consumption will be relatively small, while the marginal utility of period t + 1 consumption will be large. Hence, the ratio will be relatively small, and the indiﬀerence curve will be relatively ﬂat. Figure 9.3 plots some hypothetical indiﬀerence curves having this feature. Note that there is a diﬀerent indiﬀerence curve for each conceivable level of lifetime utility. Higher levels of lifetime utility are associated with indiﬀerence curves that are to the northeast – hence, northeast is sometimes referred to as the “direction of increasing preference.” Indiﬀerence curves associated with diﬀerent levels of lifetime utility cannot cross – this would represent a contradiction, since it would imply that the same bundle of periods t and t + 1 consumption yields two diﬀerent levels of utility. Indiﬀerence curves need not necessarily be parallel to one another, however. 191 Figure 9.3: Indiﬀerence Curves We can think about the household’s optimization problem as one of choosing Ct and Ct+1 so as to locate on the “highest” possible indiﬀerence curve without violating the budget constraint. Figure 9.4 below shows how to think about this. There
is a budget line and three diﬀerent indiﬀerence curves, associated with utility levels U2 > U1 > U0. Diﬀerent possible consumption bundles are denoted with subscripts 0, 1, 2, or 3. Consider ﬁrst the bundle labeled (0). This bundle is feasible (it is strictly inside the budget constraint), but the household could do better – it could increase Ct and Ct+1 by a little bit, thereby locating on an indiﬀerence curve with a higher overall level of utility, while still remaining inside the budget constraint. Consumption bundle (1) lies on the same indiﬀerence curve as (0), and therefore yields the same overall lifetime utility. Consumption bundle (1) diﬀers in that it lies on the budget constraint, and therefore exhausts available resources. Could consumption bundle (1) be the optimal consumption plan? No. (0) is also feasible and yields the same lifetime utility, but via the logic described above, the household could do better than (0) and hence better than bundle (1). The bundle labeled (2) is on the highest indiﬀerence curve shown, and is hence the preferred bundle by the household. But it is not feasible, as it lies completely outside of the budget line. If one were to continue iterating, what one ﬁnds is that consumption bundle (3) represents the highest possible indiﬀerence curve while not violating the budget constraint. This bundle occurs where the indiﬀerence curve just “kisses” the budget line – or, using formal terminology, it is tangent to it. Mathematically, at this point the indiﬀerence curve and the budget line are tangent, which means they have the same slope. Since the slope of the budget line is −(1 + rt), and the slope of the indiﬀerence curve 192 𝐶𝐶𝑡𝑡 𝐶𝐶𝑡𝑡+1 𝑈𝑈=𝑈𝑈0 𝑈𝑈=𝑈𝑈1 𝑈𝑈=𝑈𝑈2 𝐶𝐶0.𝑡𝑡 𝐶𝐶0.�
�𝑡+1 Slope: −𝑢𝑢′(𝐶𝐶0,𝑡𝑡)𝛽𝛽𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1) 𝑈𝑈2>𝑈𝑈1>𝑈𝑈0 is − u′ βu′ derived above. (Ct) (Ct+1), the tangency condition in this graph is no diﬀerent than the Euler equation Figure 9.4: An Optimal Consumption Bundle Having established that an optimal consumption bundle ought to occur where the indiﬀerence curve just kisses the budget line (i.e. the slopes are the same), we can use this to graphically analyze how the optimal consumption bundle ought to change in response to changes in things which the household takes as given. In particular, we will consider exogenous increases in Yt, Yt+1, or rt. We will consider varying one of these variables at a time, holding the others ﬁxed, although one could do exercises in which multiple variables exogenous to the household change simultaneously. In the text we will analyze the eﬀects of increases in these variables; decreases will have similar eﬀects but in the opposite direction. Consider ﬁrst an increase in current income, Yt. Figure 9.5 analyzes this graphically. In the ﬁgure, we use a 0 subscript to denote the original situation and a 1 subscript to denote what happens after a change. In the ﬁgure, we suppose that the original consumption bundle features C0,t > Y0,t, so that the household is borrowing in the ﬁrst period. Qualitatively, what happens to Ct and Ct+1 is not aﬀected by whether the household is saving or borrowing prior to the increase in current period income. Suppose that current income increases from Y0,t to Y1,t. Nothing happens to future income or the real interest rate. With the endowment point on the budget line shifting out to the right, the entire budget line shifts out horizontally, with no change in the slope. This is shown with the blue line. The original consumption bundle, (C0,t, C0,t+1), now lies
inside of the new budget line. This means that the household can locate on a higher indiﬀerence curve. In the new optimal consumption bundle, labeled 193 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 (1+𝑟𝑟𝑡𝑡)𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+1 𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+11+𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 𝑈𝑈=𝑈𝑈0 𝑈𝑈=𝑈𝑈1 𝑈𝑈=𝑈𝑈2 𝐶𝐶0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡 𝐶𝐶1,𝑡𝑡+1 𝐶𝐶2,𝑡𝑡 𝐶𝐶2,𝑡𝑡+1 𝐶𝐶3,𝑡𝑡 𝐶𝐶3,𝑡𝑡+1 (0) (1) (2) (3) (C1,t, C1,t+1) and shown on the blue indiﬀerence curve, both current and future consumption are higher. We know that this must be the case because the slope of the indiﬀerence curve has to be the same at the new consumption bundle as at the original bundle, given that there has been no change in rt and the indiﬀerence curve must be tangent to the budget line. If only Ct or only Ct+1 increased in response to the increase in Yt, the slope of the indiﬀerence curve would change. Similarly, if either Ct or Ct+1 declined (rather than increased), the slope of the indi�
��erence curve would change. Figure 9.5: Increase in Yt From this analysis we can conclude that Ct increases when Yt increases. However, since Ct+1 also increases, it must be the case that Ct increases by less than Yt. Some of the extra income must be saved (equivalently, the household must decrease its borrowing) in order to ﬁnance more consumption in the future. This means that 0 < ∂Ct < 1. An increase in ∂Yt Yt, holding everything else ﬁxed, results in a less than one-for-one increase in Ct. We often refer to the partial derivative of Ct with respect to current Yt as the “marginal propensity to consume,” or MPC for short. This analysis tells us that the MPC ought to be positive but less than one. In Figure 9.5 the household is originally borrowing, with C0,t > Y0,t, so S0,t < 0. As we have drawn the ﬁgure, this is still the case in the new consumption bundle, so S1,t < 0. However, graphically one can see that S1,t > S0,t – the household is still borrowing, but is borrowing less. This is a natural consequence of the analysis above that shows that St must increase in response to an increase in Yt – the household consumes some of the extra income and saves the rest, so saving goes up. If the increase in income is suﬃciently big, the household could 194 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡+1 𝑌𝑌1,𝑡𝑡 𝐶𝐶0,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶𝐶1,𝑡𝑡 𝐶𝐶1,𝑡𝑡+1 Original endowment Original consumption bundle New consumption bundle New endowment switch from borrowing to saving, with S1,t > 0. We have not drawn the ﬁgure this way, but it is a
possibility. Consider next an increase in Yt+1, holding everything else ﬁxed. The eﬀects are shown in Figure 9.6. The increase in Yt+1 from Y0,t+1 to Y1,t+1 pushes the endowment point up. Since the new budget line must pass through this point, but there has been no change in rt, the budget line shifts out horizontally in a way similar to what is shown in Figure 9.5. Figure 9.6: Increase in Yt+1 As in the case of an increase in Yt, following an increase in Yt+1 the original consumption bundle now lies inside the new budget line. The household can do better by locating on an indiﬀerence curve like the one shown in blue. In this new consumption bundle, both current and future consumption increase. This means that saving, St = Yt − Ct, decreases (equivalently, borrowing increases), because there is no change in current income. The results derived graphically in Figures 9.5 and 9.6 reveal an important result. A household would like to smooth its consumption relative to its income. Whenever income increases (or is expected to increase), the household would like to increase consumption in all periods. The household can smooth its consumption by adjusting its saving behavior. In response to an increase in current income, the household saves more (or borrows less) to ﬁnance more consumption in the future. In response to an anticipated increase in future income, the households saves less (or borrows more), allowing it to increase consumption in the present. The household’s desire to smooth consumption is hard-wired into our assumptions on preferences. It is a consequence of the assumption of diminishing marginal utility, mathematically characterized by the assumption that u′′(⋅) < 0. With u′′(⋅) < 0, the 195 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡+1 𝐶𝐶0,𝑡𝑡 𝐶𝐶1,𝑡𝑡 𝐶𝐶0,𝑡𝑡+1 𝐶
𝐶1,𝑡𝑡+1 𝑌𝑌0,𝑡𝑡+1 Original consumption bundle New consumption bundle Original endowment New endowment household would prefer to increase consumption by a little bit in both periods in response to a change in income (regardless of the period in which that income increase occurs), as opposed to increasing consumption by a lot only in the period in which that increase in income occurs. The example below makes this clear. Example √ t = 0.5C −0.5 Suppose that the utility function is u(Ct) = Ct. Suppose further that β = 1 and rt = 0 (both of which substantially simplify the analysis). The Euler equation is t+1, which requires that Ct = Ct+1. Suppose that, originally, then 0.5C −0.5 Yt = Yt+1 = 1. Combining the Euler equation with the intertemporal budget constraint (with β = 1 and rt = 0) then means that Ct = 0.5 and Ct+1 = 0.5 is the optimal consumption bundle. Lifetime utility is 1.4142. Suppose that current income increases to 2. If the household chooses to spend all of the additional income in period t (so that Ct = 1.5 and Ct+1 = 0.5), then lifetime utility increases to 1.9319. If the household chooses to save all of the additional income, spending it all in the next period (so that Ct = 0.5 and Ct+1 = 1.5), then lifetime utility also increases to 1.9319. If, instead, the household increases consumption by 0.5 in both periods, saving 0.5 more in period t, then lifetime utility increases to 2. This is better than either of the outcomes where consumption only adjusts in one period or the other. Next, consider the eﬀects of an increase in the interest rate, rt. Because this ends up being a bit messier than a change in Yt or Yt+1, it is easier to begin by focusing on just how the budget line changes in response to a change in rt. Consider ﬁrst the budget line associated with r0,t, shown in black below. Next consider an increase in the interest rate to r1,t. The budget line always must always pass through the endowment point, which is unchanged. A higher interest
rate reduces the maximum period t consumption the household can do (because it can borrow less), while it increases the maximum period t + 1 consumption the household can do (because it can earn more on saving). These changes have the eﬀect of “pivoting” the new budget line (shown in blue) through the endowment point, with the horizontal axis intercept smaller and the vertical axis bigger. The slope of the new budget line is steeper. Eﬀectively, the budget line shifts inward in the region where Ct > Yt and outward in the region where Ct < Yt. 196 Figure 9.7: Increase in rt and Pivot of the Budget Line Now, let us consider how an increase in rt aﬀects the optimal consumption choices of a household. To do this, we need to use the tools of income and substitution eﬀects, and it matters initially whether the household is borrowing (i.e. Ct > Yt) or saving (i.e. Ct < Yt). Consider ﬁrst the case where the consumer is initially borrowing. This is shown below in Figure 9.8. The initial consumption bundle is C0,t and C0,t+1, and the household locates on the black indiﬀerence curve. Figure 9.8: Increase in rt: Initially a Borrower 197 𝐶𝐶𝑡𝑡+1 𝐶𝐶𝑡𝑡 �1+𝑟𝑟0,𝑡𝑡�𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+1 𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+11+𝑟𝑟0,𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡+1 �1+𝑟𝑟1,𝑡𝑡�𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+1 𝑌𝑌𝑡𝑡+𝑌𝑌𝑡𝑡+

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