Key Ideas A model is a simplified description of reality. Economists use data to evaluate the accuracy of models and understand how the world works. Correlation does not imply causation. Experiments help economists measure cause and effect. Economic research focuses on questions that are important to society and can be answered with models and data. Getting Started A. The Big Picture Chapter 2 emphasizes the importance of using real-world data to test hypotheses and decide which of the competing theories is most compelling.
For illustrative purposes, the authors construct a simple model of the returns to education. For each option chosen, there is a runner-up choice; we refer to the value of the best alternative forgone. Optimizing economic agents identify and weigh the relevant costs and benefits for each option, and choose the option with the highest net benefit.
When two or more economic agents are optimizing, we may find an equilibrium: a situation in which no party has an incentive to unilaterally change its behavior. Economists study small pieces of the economy microeconomics or the entire. Thousands of like-minded potential tenants compete to rent scarce apartments, generating equilibrium rental rates.
Bush promoted ethanol in his State of the Union Address, the number of new plants under construction or expansion rose and the number of ethanol plants rose; experimental evidence confirms that production subsidies encourage production. Here are the essential elements: Introduce key question: Is college worth it?
Empirical Issues: argument by anecdote, correlation versus causation, omitted variables, reverse causality [15 mins. In the past few years, we have read more and more headlines about the rising cost of higher education and whether it adequately prepares college graduates for the working world. This chapter helps frame that discussion and prepares the student for Chapter 11, which covers hiring decisions and the relationship between labor productivity and wages.
This issue is complicated by the fact that college financial aid officers are skilled at using first-degree perfect price discrimination, which is covered in Chapter It could be the case that the 30 students in a class are each being charged a different price for their college education.
Chapter Outline 2. Economists expand the models that seem better at explaining or predicting outcomes, while reworking or discarding the inferior models, and then repeat the entire process with new and improved models. Teaching Ideas: At some point in the term one might compare economics and physics, which are arguably the most rigorous of the natural and social sciences, respectively.
Models and Data—The authors use the topic of navigation to weigh the advantages and disadvantages of using two-dimensional maps, like those shown in Exhibits 2.
Determining an optimal flight path is better done with a three-dimensional globe than a two-dimensional world map, but the two-dimensional subway map is very useful for subway commuters. Models omit unnecessary details, but what is unnecessary varies from user to user. One can imagine whether a single city map could help one identify city parks for the noisy kids in the back seat , low-traffic bike routes, post office drop. Interestingly, Google Maps can accomplish many of these tasks, though it is not a standard printed map.
Alternative Teaching Examples: One might ask a class to imagine trying to model the trajectory of a stuffed toy e. Could we replace Elmo with a sphere or cube with the same mass?
Does wind speed matter? An Economic Model—The simple returns-to-education model assumes that investing in one extra year of education increases your future wages by 10 percent; this model predicts that those with higher educational attainment will have higher income, on average.
This model is an approximation that ignores many subtleties about the labor market but does generate a prediction that can be tested empirically if one has sufficient data on years of education and wages.
Alternative Teaching Examples: One could launch a fruitful discussion by asking some of the following questions: o Will your wages rise 10 percent if you stick around for the fifth year of college? Would all employers do this?
This exercise suggests that the economic rationale of this condition may not apply to problems in which the objective value has no discounting see also Part e of Exercise 7. The current-value Hamiltonian is. We claim that there exists a path c t , r t that satises Eqs.
More formally, consider the standard phase diagram in the c, r space for the dierential equations I7. Note that since q. This path satises the initial condition for r 0 as well as Eqs. Moreover, it also satises the transversality condition I7. We next claim that this path is optimal. Hence Theorem 7. We have shown that the unique optimal plan is characterized by I7. It follows by Theorem 7. Suppose, to reach a contradiction, that it is feasible. Third, we get a minor issue out of the way.
Such paths do not satisfy the law of motion of the unconstrained problem I7. But these paths are clearly sub-optimal since the household is better o by consuming the investment that goes to waste without aecting the accumulation of capital. It follows that we can ignore these paths without loss of generality, and any remaining paths feasible for the constrained problem are also feasible for the unconstrained problem.
This also implies that the optimal value of the constrained problem is weakly lower than the optimal value of the unconstrained problem. For strictly concave utility functions, the alternating policy suggested in the hint would result in rst order utility losses and would not approximate the unconstrained optimum policy.
More specically, we consider the path [i t [. Upon integrating Eq. We next claim that lim. To see this, note that using I7. Since, lim. Using this in Eq. Using lim. As the limit of the rst term in Eq. The last equation and Eq. Suppose, to reach a contradiction, there is. The limiting path of consumption does not exist since consumption jumps innitely often in any given interval and it does not have a piecewise continuous limit. As long as the limiting path is well dened and feasible, it would also be optimal and the optimum would be attained.
Note that the constrained problem satises Assumption 7. Essentially, Theorem 7. Hence, as long as we nd an interior solution that is optimal for the constrained problem, it will be feasible and optimal for the unconstrained problem since the latter is a concave problem.
We claim that there exists - 0 such that this optimal plan satises c t - for all t. We prove this in three steps. Second, we claim that c t 0 for all t. Third, we claim that there exits - 0 such that the optimal plan satises c t - for all t. We are interested in the problem 1 , but 1 does not necessarily t into the optimal control framework of Chapter 7, hence we instead analyze lim ao 1 :.
Solutions Manual for Introduction to Modern Economic Growth 81 For any 1 : with nite :, note that the investment function is strictly convex therefore the results in Section 7. Then, taking the limit of Eq. Next, note that since the objective function in I7. In fact, 1 does not have a continuous optimal solution, but the optimal solution is approximated arbitrarily closely by [1 a t , 1 a t [ t as : increases.
To see this, we use the implicit function theorem and dierentiate Eq. Hence, 1 1 is indeed decreasing over the range it is dened and its plot in Figure 7. We rst claim that the system in I7. To study the local behavior, we linearize the system around this steady state.
Since only one eigenvalue is negative, Theorem 7. This proves our claim that the system in I7. We next claim that the saddle path plan [1 t , 1 t [ t is the unique optimal plan, which in turn shows that the optimal investment plan will converge to the steady state. To show this, we verify that the conditions of Theorem 7. The rst-order and feasibility conditions are satised by construction. Then, we invoke Theorem 7. We have shown that the optimal plan is the saddle path stable plan, hence the statement in this exercise follows if we show the saddle path is downward sloping.
We rst claim that the linearized system in I7. Suppose, to reach a contradiction, that 1 1 , 2 1 have the same signs, and suppose that they are both positive the proof for the negative case is symmetric.
This yields the desired contradiction, proving that the eigenvector has components with dierent signs and the saddle path for the linearized system is downward sloping. We assume that the adjustment cost of installing capital 1 when the current capital is 1 is given by 1c 1,1 , so the total cost of installing 1 is 1 1 c 1,1. The Hamiltonian is given by. Moreover, Theorem 7. We next compare the steady state characterized by Eq. Rewriting Eq. Intuitively, the marginal product of capital is lower in this case and hence the capital level is higher, since investment has the additional benet of lowering future investment costs in view of the functional form c 1,1.
As we have shown in Exercise 7. Then the plan [1 t , 1 t [ t that solves the unconstrained problem satises the above conditions and is also the solution for the constrained problem. First note that this plan satises all the feasibility constraints in I7. Second, note that it also satises all of the optimality conditions in I7. Moreover, note also that the complementary slackness condition in I7.
Then, the second equation in I7. To see this formally, rst note that in a neighborhood t [. It follows that the complementary slackness condition in Eq. Together with I8. Again use I8. Furthermore, the resource ow constraint I8. Exercise 8. We prove this result by contradiction. Substituting into I8.
By construction this plan satises I8. Hence, for any a. This shows that the household will choose a consumption plan where the corresponding asset holdings are arbitrarily negative for all t. In order to show that such an allocation will violate feasibility, we have to analyze what eects such a behavior would have in equilibrium recall that the analysis above was entirely from the households point of view taking wages and interest rates as given and acting as if assets were in innite supply.
In equilibrium, per capita assets have to be equal to the economys per capita capital stock, i. Hence, an allocation as in Part b would require that the economys capital stock will be arbitrarily negative.
We will rst characterize the interior solution c t [-, using Theorem 7. Then we will show that the solution is actually the global optimum using the relationship between Theorem 7. This will then imply that the restriction c t - does not aect the solution as Theorem 7. Let us start with the maximization problem of the household. Let us rst consider Assumption 7. As n t c t 0, the monotonicity of the utility function is satised. This shows the rst part. For the third part we need our restriction that c t [-,.
As marginal utility is nite for all c t 0, I8. Hence, Assumption 7. In light of this we can use Theorem 7. So suppose there is a solution to this restricted problem which satises c t -. The analysis in Chapter 7 established that such a solution is characterized by the rst-order conditions of the current-value Hamiltonian. To show that the restriction c t [-, is not restrictive, we will now use Theorem 7. Note that Theorem 7. If the concavity of ' a t , j t is strict, the solution is unique.
Note that we also need to check that the state variable a t is chosen from a convex set, but a t R. Hence, the last thing we have to show is, that in equilibrium we will have r t : 0. Equilibrium interest rates are given by the net marginal product of capital, i.
From I8. Hence, in the steady state, interest rates will be higher than :. Hence in equilibrium interest rates will indeed exceed the population growth rate :. For further details we refer to Exercise 7.
Therefore, even though Theorem 7. We claim however that a slight modication of Arrows theorem can be used to establish uniqueness for the household problem I8. Consider Solutions Manual for Introduction to Modern Economic Growth 91 any admissible path [a t , c t [ t that attains the optimal value for the representative house- hold.
We will show that this path must be the same as [ a t , c t [ t , proving uniqueness. Then, using Eqs. The critical step of the proof is the observation in Eq. This leads to the uniqueness of the optimal path as established above. The dynamics of consumption and capital in the neoclassical growth model are depicted in Figure 8. So suppose that initial consumption c 0 started above the stable arm.
From Figure 8. The behavior of the capital stock is a little more complicated. However, as consumption will steadily increase, there will be t such that c. Hence, for all t t, consumption will still be increasing and the capital stock will decrease.
This implies that the capital stock will be zero in nite time, i. At this allocation however, feasibility will be violated. To see this, note that the dynamic behavior of consumption will still be given by the Euler equation, i. Now suppose initial consumption is too low, i. From the phase diagram in Figure 8. This is an important result, because it shows that such a path cannot solve the problem. But then, the conjectured path could not have been optimal.
Alternatively, we can also argue that such a path will violate the transversality conditions see the discussion following Proposition 8. In this exercise, we linearize the system in I8. A rst-order approximation of the system in Eq. Hence the local behavior of system I8. Then, the solution to the linearized system I8. Without loss of generality, we assume. This establishes that one of the eigenvalues, 1 is negative and the other one, 2 , is positive.
Hence, had the linear approximation in Eq. Hence, the fact that the equilibrium path is stable implies that j 2 corresponding to the equilibrium path must be close to zero, that is j 2 - 0.
We now assume that Eq. Hence j 1 is uniquely determined from the initial value of capital. We next explicitly calculate 1 and see how it responds to the exogenous parameters. Recall that 1 is the negative solution to. The solutions are given by the quadratic formula. The smaller and the negative real root, 1 , is given by. Recall that the higher [ 1 [, the faster the convergence. The intuition generalizes to solving non-linear systems with one initial and one end value constraint.
Therefore we should take these comparative statics as suggestive. Intuitively, the more inelastic the substitution between capital and labor, the faster the economy faces diminishing returns and the faster the convergence to steady state note that this eect is also present in the Solow model. That oI. The per capita consumption level in the steady state is given by see 8.
In the steady state, the marginal product of capital has to be such that there is no consumption growth see 8. Solutions Manual for Introduction to Modern Economic Growth 97 This shows that the steady state level of consumption will always be decreasing in the dis- count rate.
The reason why there cannot be oversaving in the neoclassical growth model in contrast to the Solow model is simply that equilibrium has to be consistent with con- sumer maximization. But any plan which would have had the property that by saving less, consumption could be increased could not have been optimal in the rst place as such a plan was clearly available by simply consuming more to begin with.
In this exercise we consider a neoclassical economy where tech- nological progress is not Harrod neutral, but capital-augmenting. Besides the dierent technology, this is just the standard economy with technological progress described in Chapter 8. Hence the competitive equilibrium is dened as in Denition 8. The household maximization problem follows exactly along the same lines as in Chapter 8. In the steady state, consumption has to be constant, so that from I8.
As I8. The reason is the following: 0 is the inverse of the intertemporal elasticity of substitution, i.
But this economy does not experience growth in the steady state as the technology is constant. Hence, consumption is constant over time so the consumers preferences about intertemporal substitution do not matter once the steady state is reached. Note that 0 matters of course for the transitional dynamics, in particular for the speed of convergence. Let us now allow for technological progress, i. It is clear that this economy will not have a steady state where consumption and output are constant.
For consumption growth to be constant, I8. Let us dene. This can be seen from I8. And as t grows at an exponential rate,. Rearranging terms yields the dierential equation. Hence, this economy does only admit a BGP equilibrium if the production function indeed takes the Cobb-Douglas form. As t and I t grow at the same rate, I I I is constant.
This shows that consumption grows at rate j I along the BGP. Secondly note that now 0 does matter as it determines. The reason is that now there is consumption growth on the BGP so that consumers preferences about substituting consumption intertemporally do matter.
In particular, note that I8. To understand this result, note that per capita consumption grows at rate q j so that I8.
The level of 0 governs the consumers willingness to intertemporally substitute consumption. Intuitively, if 0 is higher, interest rates also have to be higher to convince consumers to have consumption growing at rate q j. But as interest rates equal the net of depreciation marginal product of capital and has decreasing returns, the normalized level of the capital-labor ratio t will have to be lower.
An increase in the discount rate j and an increase in the depreciation rate c will both reduce the economys normalized per capita capital stock. This is also intuitive. If consumers discount the future more, there will be less capital accumulation so that the capital stock will be lower. Similarly, if the depreciation rate is higher, more savings are needed to preserve a given capital stock. This will also reduce capital accumulation. Then we get that T1. This is exactly the innite horizon budget constraint requiring that the net present value of consumption cannot exceed the net present value of wages plus initial assets.
The necessary condition of maximizing I8. Hence, the capital stock has to grow at rate q. Using I8. Hence, c t t is constant too, i. The Euler equation I8. Dierentiating I8. As c t is growing along the BGP in particular consumption is growing at rate q , we can rewrite I8. Hence, utility of the CRRA form is the only utility function which is consistent with balanced growth if technological progress is labor-augmenting.
That we recover the Euler equation is not surprising - it is just a consequence of the First Welfare Theorem. Additionally we have the resource constraint I8. Using the denition of the gross interest rate in I8. Substituting this into I8. Hence, consumption grows at rate q. But then we can use I8. To ensure that such an equilibrium is well dened, we nally need to make appropriate parametric assumptions to satisfy the transversality condition.
Hence the transversality condition in I8. To prove these properties in this economy we will show that we can transform the problem so that it coincides with the optimal growth problem of the neoclassical growth model without technological progress.
First of all note that the First Welfare Theorem applies to the economy of this exercise. To make this problem isomorphic to the canonical optimal growth problem without technological progress, note that I8. As 0 10 and the last term are just positive transformations which do not aect the maximization, we can drop those terms. Note that we dropped the second constraint I8. But the problem in transformed variables contained in I8. The only thing we have to ensure is, that the problem is well dened, i.
We rst consider the economy with a heterogenous set of households H. Similarly, the asset evolution equations I8. Moreover, given the path of prices [r t , n t [ t , Theorem 7. This completes the characterization of the equilibrium with heterogenous agents. The analysis in this section is identical to the baseline analysis in Chapter 8. It follows that the aggregate per capita variables are identical in the two economies.
This exercise then establishes that there is a representative consumer for the neoclassical economy when the preferences are CES and when we take the no-Ponzi scheme condition as the appropriate borrowing restriction for the household. This is not surprising since we have shown in Section 5. Theorem 5. In the above analysis, Gormans aggregation theorem best manifests itself at the step that allows us to go from the individual rst-order conditions I8.
Our goal is to construct example economies in which this seemingly small dierence can generate dierent equilibrium paths. Consider a two household economy, i. H , 1, in which the initial conditions are given by a. Here, household is the poor graduate student with no assets and household 1 is another agent in this economy.
That is, household will initially go into debt if and only if he consumes more than he earns at time 0. We next describe two scenarios in which this is possible. As the rst example, consider the case in which a 1 is very large think of 1 as Bill Gates so that the initial capital-labor ratio is large.
Since interest rate is low early on, would like to borrow and consume more early on, which creates a force that increases c. This creates a wealth eect which may make agent save early on rather than borrow.
In general, it is not clear which force dominates and whether Eq. However, when 0 is very small intertemporal substitution is suciently elastic , it can be shown that has a lot of incentives to tilt consumption to earlier dates, the rst force dominates, and Eq. Even though is in debt in steady state, he pays interest on his debt so the level of the debt does not grow. Hence, is not running a Ponzi scheme and Bill Gates is willing to lend him money since he is getting a fair rate of return from him.
In particular, wages n t also increase over time. Assume this time that 0 is high, so that would like a atter consumption prole. Given that faces an increasing wage prole, he has an incentive to borrow early on and smooth his consumption over time. Intuitively, might borrow in equilibrium to buy his Ferrari as a graduate student! In a closed economy, the equilibrium asset level of the representative household is equal to the level of the capital stock, which is always positive.
As our analysis in part a shows, replacing the no-Ponzi scheme condition with a no-borrowing constraint does not change the equilibrium path in a representative household model. But as this exercise demonstrates the equilibrium path might change once we have heterogenous agents. The exercise also shows that the Gorman aggregation theorem does not necessarily apply to the neoclassical economy if we assume the no-borrowing constraint. Hence the no-Ponzi condition is the right borrowing restriction if we are studying issues not related to credit constraints, since it enables us to study the simpler representative household economy without loss of generality.
Changing the utility function does not change anything in the denition of an equilibrium. Hence, to characterize the equilibrium in this economy we have to derive the system of dierential equations characterizing the entire evolution of these two variables. The diculty is that and 1 share the same 0, face the same wages, and they both have somewhat low levels of initial wealth. This makes it dicult to get the eect in equilibrium. Nevertheless, there exists parameterizations such that this happens.
In this exercise this term is not constant but depends on the level of consumption c t. Prot maximization by competitive rms implies that the marginal product of capital net of deprecation is equal to the real interest rate, i.
The three equations contained in I8. The implied path for per capita consumption and the eective capital-labor ratio is the desired equilibrium path for these variables. This concludes the characterization of the equilibrium. To see why, recall that along the BGP the capital-output ratio is constant. Hence c t t has to be constant along the BGP, i. This proves that this economy does not admit a BGP with a positive growth rate.
For given interest rates r j, consumption growth will therefore be increasing in the level of consumption. Intuitively, the higher the level of consumption, the more willing the consumer to tilt his consumption schedule as the subsistence level loses in importance. Hence, the growth rate of consumption will be a function of the level of consumption and consumption growth is not constant. This is inconsistent with balanced growth. We will show below however, that this economy will feature balanced growth asymptotically.
The transversality condition was given in I8. In particular we show that asymptotically per capita con- sumption will grow at the constant rate q and that the eective capital-labor ratio will be Solutions Manual for Introduction to Modern Economic Growth constant. Using the Euler equation I8. Substituting this in I8. This can only be satised if j 1 0 q I8. Now let us think about the transitional dynamics of this economy.
To stress the similarity between this economy and the baseline model with labor-augmenting technological progress analyzed in Section 8. Now consider the original economy where t 1 is not absent. In particular, the economy will also be saddle path stable so that there Solutions Manual for Introduction to Modern Economic Growth is one stable arm and the solution will be on this arm and converges to the steady state.
In particular I8. This shows that growth will be balanced asymptotically as claimed in the analysis in Part d above. Although the steady state of the system is the same as in the economy where the t 1. Hence asymptotically as t tends to innity, this economy is characterized by exactly the same equations as the baseline model. Therefore it is also intuitive that the required parametric restriction in I8.
Note in particular that the saddle path will also be a function of time. The system however will still be saddle path stable, i. The rest of the analysis is exactly analogous to the case considered above. Hence, the economy will again have a BGP asymptotically and this BGP is exactly the same as the one characterized above and therefore also the same as in the baseline model.
Although we know that this locus will converge to its counterpart of the baseline model, there is no reason why it should shift up over time as in Part e above. Note that Problem I8. Note that the rst condition I8. Then Theorem 7. Note that, after substituting the competitive market prices for r t and n t from Eq. Integrating Eq. Note that the intratemporal rst-order condition in Eq. Recall that c t and t grow at the same constant rate q.
In this case, integrating Eq. This is not entirely correct. Actually, the only restriction we will get will be Eq. The only restriction we get is Eq. Given that is constant at. Solutions Manual for Introduction to Modern Economic Growth Intuitively, the interest rate is constant only if the intertemporal elasticity of substitution remains constant as c grows, which explains why the utility function must be CES when viewed as a function of c. For the intratemporal trade-o, there are three economic forces.
First, income and hence consumption is growing at rate q hence the marginal utility of con- sumption is shrinking at rate 0q, which creates a force towards more leisure the income eect. Second, wages are growing at rate q hence the marginal return to labor is growing at rate q, which creates a force towards more labor the substitution eect.
Third, marginal benet to leisure might also be changing as consumption grows, depending on whether con- sumption or leisure are complements or substitutes. In particular, when 0 1, we need the leisure and consumption to be substitutes with the functional form in I8. Note that we did not explicitly consider the labor supply of the representative household as labor will be supplied inelastically. The corresponding current- value Hamiltonian for this problem is given by. As usual, this equation describes the consumers intertemporal consumption behavior.
This however now takes the tax sequence the consumer faces into account. If the tax schedule is increasing over time, i. Hence, an increasing tax schedule acts like a higher interest rate, as the returns of investing today are higher than doing so tomorrow.
To study the optimal steady state tax rate, suppose the economy is in the steady state. Using this and I8. Although this tax rate maximizes the steady state utility of the representative consumer, it will not maximize the utility of the representative household if the economy starts away from the steady state. In particular taxes therefore determine the speed of adjustment to the steady state capital stock and the consumption level during the transitional dynamics.
This is not taken into account when taxes are chosen to maximize the steady state utility of the representative consumer. Hence, let us study the social planners problem to characterize the equilibrium allocation. In Chapter 7 we introduced costs of adjustment by assuming that those costs are represented by a function c 1 which is continuously dierentiable, strictly increasing and strictly convex. As in the analysis in Chapter 7, these constraints show that the costs of adjustment c i t just represent a loss of resources without adding to either consumption or capital accumulation.
The corresponding current-value Hamiltonian is given by. Additionally we can dierentiate I8. Let us rst look for a steady state where consumption and capital are constant, i.
From the capital accumulation equation I8. This is intuitive, as for capital to be constant, investment has to be exactly high enough to replace the depreciated capital stock.
Hence investment will be positive but constant. Finally we have to show that the transversality condition is satised on the path that leads to the steady state.
This shows that the economy with adjustment costs has a unique steady state. Furthermore we can see from I8. As is strictly concave, the steady state level of capital will be lower. Secondly, the capital stock is lower and given that the capital stock in neoclassical growth model without adjustment costs was already below the golden rule level, a lower level of capital will unambiguously decrease consumption.
Let us now turn to the transitional dynamics. We will just provide the intuition. As in Chapter 7 it is seen from I8. Hence, in this case the transitional dynamics are very similar to the ones of the neoclassical growth model. The reason is that with a linear adjustment cost function there are no incentives to smooth investment expen- ditures.
If adjustment costs are convex i. Hence, adjustment costs introduce a second force which calls for slow path of capital accumulation. Not only tends capital accumulation to be slow because of the consumption smoothing eect, but investment will also be smooth to reduce investment costs. Hence, if there are adjustment costs of investing, capital accumu- lation will be slowed down. As the steady state will be similar to the standard neoclassical growth model, adjustment costs of investment are often introduced as a explanation why the transition to the steady state might not occur as fast as the standard neoclassical growth model predicts.
Consider the budget constraint of the representative house- hold. Let us rst analyze the case where there are ' separate assets.
Although this economy uses ' capital goods, the resource constraint C t A. Hence let us normalize the prices to unity, i. We're sorry! We don't recognize your username or password. Please try again.
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