Neither Solow, RCK, nor Diamond provide satisfying answers to the question: Why does economic growth occur? Their principle result has been that capital's earnings reflect its contribution to output, thus capital accumulation does not account for most long-run growth or cross-country differences. Furthermore, the only determinant of income in the models other than capital is A, the "effectiveness of labor," whose behavior is exogenous. This chapter focuses on the accumulation of knowledge-- supposedly the impetus of A's technological change. 3.1 Framework and AssumptionsTo model the accumulation of knowledge, a separate sector of the economy where ideas are developed needs to be introduced. Then, we have to model how resources are developed between the conventional output sector and the new one, R&D. Building on the position that devoting more resources to research yields more discoveries, this production function will be largely standard. Besides, in the long run, the randomness of the process of discovery has smaller effect. Two major simplifications enable this model. First, both goods and R&D production functions are Cobb-Douglas, meaning they are power functions, but the sum of exponents is not restricted to 1. Secondly, the fraction of output saved and the fractions of the labor force and capital stock used in R&D are exogenous and constant. SpecificsThere are four main variables: \(L, K, A, & Y\). The model is set in continuous time, with \(a_L\) being the labor force used in the R&D sector and \(1 - a_L\) being the portion used in the goods-producing sector. Likewise for \(a_K\). Because knowledge is nonexclusive, both sectors use the full stock of A. The quantity of output produced at time (t) is thus:\(Y(t) = [(1 - a_K)K(t)]^\alpha [A(t) (1 - a_L) L(t)]^{1 - \alpha}\), where \(0 < \alpha < 1\)Note once more that there are constant returns to capital and labor: with a given technology, doubling the inputs doubles the amount that can be produced. The production of new ideas depends on the quantities of capital and labor engaged in research and on the current level of technology, thus: \(\dot A(t) = B[a_K K(t)]^\beta [a_L L(t)]^\gamma A(t) ^\theta\), where \(B > 0, \beta \geq 0, \gamma \geq 0\)B is a shift parameter. Notice that the production function for knowledge are not assumed to have constant returns to scale to capital and labor (different exponents). CRS allows for replication, but technological progress must always be new. Thus, there might be diminishing (or increasing!) returns to R&D. The parameter \(\theta\) reflects the effect of existing stock of knowledge on the success of R&D, which can operate in either direction. Past discoveries might augment further exploration (positive), or all the easiest discoveries have been made first, the low-hanging fruit argument (negative). As in Solow, saving is exogenous and constant, and depreciation is zero for simplicity's sake:\(\dot K(t) = sY(t)\)Likewise, population growth is exogenous, constant, and assumed to be positive:\(\dot L(t) = nL(t)\)Because there are two endogenous variables now (capital and knowledge) the behavior is complex. First, we will consider the model without capital to showcase the model's central messages about R&D. 3.2 The Model without CapitalDynamics of Knowledge AccumulationWhen there is no capital in the model, the production function for output becomes:\(Y(t) = A(t) (1 - a_L) L(t)\)Similarly, the production function for new knowledge is defined as:\(\dot A(t) = B[a_L L(t)]^\gamma A(t) ^\theta\)The top equation implies that the growth of output is proportional to A, so the growth rate of output per worker equals the growth rate of A. Therefore we will focus on the dynamics of A:\(g_A(t) \equiv \frac{\dot A(t)}{A(t)}\)\(= Ba_L^\gamma L(t) ^\gamma A(t) ^{\theta - 1}\)Taking logs of both sides and differentiating with respect to time gives the expression for the growth rate of g (that is, the growth rate of the growth rate of A):\(\frac{\dot g_A(t)}{g_A (t)} = \gamma n + (\theta - 1) g_A (t)\)Now multiplying both sides by \(g_A(t)\) yields:\(\dot g_A(t) = \gamma n g_A(t) + (\theta - 1) [g_A(t)]^2\)So, the initial values of labor and knowledge determine the initial value of the growth rate. The continual growth is characterized by the above equation. Now we will examine particular cases for more insight. Case 1: \(\theta > 1\)The figure below shows the dynamics of the growth rate of knowledge when the influence of the existing stock of knowledge (theta) is below zero:Thus, the rate of change of growth i positive for small values and negative for large. Solving the earlier equation for this gives:\(g_A * = \frac{\gamma}{1 - \theta} n\)Implying that no matter the economy's initial conditions, the growth rate of knowledge will converge to the steady state. In contrast to Solow, Ramsey, and Diamond models, the long-run growth rate of output per worker is determined within the model rather than by exogenous technological progress. However, one of the troubling aspects may be that this growth is a function of the rate of population growth, n, as it seems that most countries with high population growth don't in fact have high growth rate of output per worker. The equation above also implies that the fraction of the labor force engaged in R&D does not affect long-run growth, which also may be surprising. However, if we analyze a growth in \(a_L\)within the model, the increase in the growth rate would not be sustained. Intuitively, this speaks to the fact that the contribution of additionally knowledge to the production of new knowledge is not strong enough to be self-sustaining. Like a rise in savings within Solow, it has a level, not growth, effect:Case 2: \(\theta > 1\)This corresponds to a case where production of new knowledge rises more than proportionally with the existing stock. This implies that \(\dot g_A\) is always positive:The significance is great-- economies never converges to a balanced growth path but rather is ever-growing. Intuitively, knowledge is so useful in the production of new knowledge that each marginal increase in the level of knowledge develops into a larger contribution, thus the growth rate of knowledge rises. The impact of an increase in the fraction of labor engaged in R&D is now dramatic, since it will increase the growth rate even more. Case 3: \(\theta = 1\)Here, the existing knowledge is just productive enough at generating new knowledge that it is proportional. The expressions simplify to:\(g_A (t) = Ba_L ^\gamma L(t) ^\gamma\)\(\dot g_A (t) = \gamma n g_A (t)\)The phase diagram is shown below:Thus if population growth is positive, the growth rate of A is increasing over time, but if it is negative, then it will be constant. Thus, there is no adjustment for steady state since the economy immediately exhibits steady growth. In this case, the growth rates of knowledge, output, and output per worker are equal to \(Ba_L ^\gamma L ^\gamma\). Changes in the growth of labor effect long-run growth of the economy. In this instance, since we have removed saving and the only activity that remains is consumption, we can see that the fraction of society's resources devoted to knowledge actually acts as a savings rate since it will increase the output of goods in the future. Thus, the case of \(\theta = 1; n = 0\) is very similar to the model where the saving rate impacts long-run growth and is called a linear growth model. Importance of Returns to Scale to Produced FactorsThe disparity of ramifications above is rooted in the decreasing, increasing, or constant (respectively) returns to scale to produced factors of production. We have eliminated capital from the model for simplicity and the growth of labor is exogenous, so knowledge is the only produced factor. Essentially, the returns are so important because of compound growth. Importance of Population GrowthThis rests on the natural assumption that when more people are around, more discoveries are made. When more discoveries are made, the stock of knowledge grows faster and so does the output per person. The proper interpretation of this feature relies upon noting that A represents knowledge that can be shared across the world. The more people, the more discoveries (regardless of cross-country disparities).3.3 The General CaseNow we reintroduce capital into the model and note how this modifies our previous analysis. The Dynamics of Knowledge and CapitalAgain, there are two endogenous state variables, A and K. Thus we focus on the growth rates of these two factors. Substituting the production function into the expression for capital accumulation yields:\(\dot K(t) = s(1 - a_K)^\alpha (1 - a_L) ^ {1 - \alpha} K(t) ^\alpha A(t) ^{1 - \alpha} L(t) ^{1 - \alpha}\)Dividing both sides by K(t) and defining \(c_K = s(1 - a_K)^\alpha (1 - a_L) ^ {1 - \alpha}\) gives us:\(g_K(t) \equiv \frac{\dot K(t)}{K(t)}\)\(= c_K [\frac{A(t) L(t)}{K(t)}] ^{1 - \alpha}\)Now taking logs and differentiating with respect to time:\(\frac{\dot g_K(t)}{g_K (t)} = (1 - \alpha) [g_A (t) + n - g_K (t)]\)Therefore, the growth of capital will always be rising if the bracketed argument is positive. The figure below summarizes this information:And now finding the growth rate of A, dividing both sides of \(\dot A = B(a_K K) ^\beta (a_LL)^\gamma A^\theta\) by A yields: \(g_A (t) = c_A K(t) ^\beta L(t) ^\gamma A(t) ^{\theta - 1}\)Where \(c_A \equiv BA_K ^\beta a_L ^\gamma\). Minus the capital term, this is essentially the same equation we came up with before. Taking logs and differentiating with respect to time gives us:\(\frac{\dot g_A(t)}{g_A (t)} = \beta g_K (t) + \gamma n + (\theta - 1) g_A (t)\)Thus, the set of points where the growth of A is constant has an intercept of \(- \frac{ \gamma n}{\beta}\) and a slope of \(\frac{(1 - \theta)}{\beta}\). The graph below displays this: Though the production function for output exhibits constant returns to scale in the two produced factors (capital and knowledge) the overall returns to scale depends upon the particular functions of those factors. As we can see in the equations, the degree of returns to scale in both A and K is \(\beta + 1\), meaning if we increase both K and A by a factor of X, then the growth rate of A increases by the factor \(X ^{\beta + 1}\). Now we will analyze cases of different growth rates. Case 1: \(\beta + \theta < 1\)The figure below shows that regardless of where the growth of K and growth of A begin, they will converge at Point E, where the rate of growth of both variables is equal to zero. Thus this point must satisfy:\(g_A * + n - g_K * = 0\)and\(\beta g_K * + \gamma n + (\theta - 1) g_A * = 0\)rewriting this with the definition of \(g_K * \) in the first equation and substituting yields:\(g_A * = \frac{\beta + \gamma}{1 - (\theta _ \beta)} n\)Again, we know that \(g_K * = g_A * + n\). The final equation then implies that when A and K are growing at these rates, output is growing at the rate of growth of capital. Output per worker is therefore growing at the rate of growth of knowledge. This is often referred to as a semi-endogenous growth model, as long-run growth arises from population growth and the parameters of knowledge as well as endogenous generation.Case 2: \(\beta + \theta = 1 ; n = 0\)Having seen that locus where \(g_K * = 0 \) is given by \(g_K * = g_A * + n\), when the above parameters are set, we simply have \(g_K = g_A\). Both are given by this 45 degree line:The growth rates are constant and the economy is on a balanced growth path. However, because changes in \(a_L , a_K\) involve shifts of resources between goods production and R&D, the results are ambiguous. This is known as a fully endogenous growth model since long-run growth depends solely on the range of parameters. 3.4 The Nature of Knowledge and the Determinants of the Allocation of Resources to R&DOverviewWhat determines \(a_L, a_K\)? It's necessary to define what kind of knowledge we're relying upon-- from basic scientific knowledge like germ theory or the Pythagorean theorem to how to start a particular lawn mower on a cold morning. Consequently, we have no reason to assume the determinants of the accumulation of these different types of knowledge will be the same. But, there are two characteristics that all knowledge shares: nonrivalry and excludability. This means that it is not used up, though it can be rationed. The dispersion through communication and the degree of excludability therefore play a large role in development of all knowledge. Support for Basic Scientific ResearchFrom medieval monasteries to modern universities, basic scientific knowledge has always been made available relatively freely. It has a positive externality, and thus academics are sympathetic to the view that it should be subsidized. With the RCK model, one could even solve for the optimal research subsidy. Private Incentives for R&D and InnovationMany innovations are marginal. In order to result in economic incentives, R&D must be excludable. There are three distinct externalities from R&D: consumer-surplus effect, business-stealing effect and the R&D effect. The first means that the firms using the innovation generate positive surpluses. The second refers to creative destruction, thus is negative. The last means that subsequent research will be aided by the discovery; it is positive. Alternative Opportunities for Talented IndividualsBaumol (1990) takes a historical perspective when he argues that rent-seeking has had detrimental effects on encouraging innovation. Murphy, Shleifer, and Vishny (1991) discuss the motivations behind talented individuals' decisions to pursue socially productive activities. They emphasize three factors: the size of relevant market, the degree of diminishing returns, and well-functioning capital markets. Learning-by-DoingThe final determinant of knowledge accumulation is somewhat different in character. The central idea from Arrow (1962) is that as individuals produce goods they will inevitably think of ways to improve the production process. This represents a positive externality, spontaneous order, from any economic activity. This means that all inputs are engaged in production:\(Y(t) = K(t) ^\alpha [A(t) L(t)] ^{1 - \alpha}\)The simplest case would be when learning is a side effect of the production of new capital. This means that the stock of knowledge is a function of the stock of capital, thus:\(A(t) = B K(t) ^\phi\)The dynamics of this economy would be given by:\(\dot K(t) = sB ^{1 - \alpha} K(t) ^\alpha K(t) ^{\phi (1 - \alpha)} L(t) ^{1 - \alpha}\)In words, this means that we can think of there being only one productive input, capital. A case that has received particular attention is \(\phi = 1, n = 0\). Here we would find the production function and capital accumulation becomes:\(Y(t) = bK(t)\) where \(b \equiv B^{1 - \alpha} L^{1 - \alpha}\)\(\dot K(t) = sbK(t)\)The dynamics are straightforward; K grows steadily at rate sb. Once again, long-run growth is endogenous and depends on the saving rate. 3.5 The Romer ModelOverviewThis model explains the allocation of resources as built up by microeconomic foundations: endogenous technological change. R&D is undertaken by profit-maximizing economic factors, fueling growth, in turn fueling more R&D. As we know, if knowledge is sold at marginal cost, the creators of the knowledge earn negative profits. However, Romer assumes that knowledge consists of ideas, which are distinct and imperfect substitutes to the actual inputs in production that embody these ideas. He also assumes that the developer of an idea has monopoly rights over the idea use (patents). Thus the developer can charge above marginal cost and we have incentives for R&D. To keep things simple, we will utilize our case where \(\theta = 1 , n = 0\). There are no transition dynamics, thus the model will immediately jump to new steady states. The Ethier Production Function and the Returns to Knowledge CreationThe first step is describing how knowledge creators have market power. We look at ideas as within the range from 0 to A, though the combination of ideas for specialized inputs into production is infinite. When an idea is available, the input into production that transforms the idea into practice can be seen as technology that transforms labor one-for-one into input. \(L(i)\) will denote both the quantity of labor devoted to producing input as well as the quantity of input that goes into final-goods production. This was proposed by Ethier (1982):\(Y = [\int_{i = 0} ^A L(i) ^\phi di] ^{1/\phi} , 0 < \phi < 1\)Denoting \(L_Y\) as the total number of workers producing inputs, \(Y = [A (\frac{L_Y}{A}) ^\phi] ^{1/ \phi} \Longrightarrow A^{(1 - \phi)/ \phi} L_Y\)This expression has two critical implications. First, there are constant returns to \(L_Y\), meaning that holding the stock of knowledge constant, doubling the inputs into production doubles the output. Second, output is increasing in A, since raising the stock of knowledge raises output; a new idea creates value. Now we will introduce the market structure where the monopolist charges a constant price for each unit of the input and output is produced when firms take these inputs at their prices. Competition causes the firms to sell output at marginal cost. Consider the cost-minimization problem of the representative output producer. Let \(p(i)\) denote the price charged by the patent-holder for each unit of the input embodying the idea. The Lagrangian for producing one unit of output at minimum cost is:\(\mathcal{L} = \int _{i = 0} ^A p(i) L(i) di - \lambda [(\int _{i = 0} ^A L(i) di) ^{1/\phi} - 1]\)The first-order condition for an individual L(i) is then\(p(i) = \lambda L(i) ^{\phi - 1}\)This now implies that\(L(i) = [\frac{\lambda}{p(i)}] ^{\frac{1}{1 - \phi}}\)This shows that the holder of the patent faces a downward-sloping demand curve, since L(i) is a smoothly decreasing function of p(i). When \(\phi\) is closer to 1, the marginal product of an input declines more slowly as the quantity rises, thus the elasticity of demand is greater when the inputs are closer substitutes. Given this, the firms earn zero profits since marginal cost equals average cost. The Rest of the ModelSince population is fixed, workers can be employed either in producing intermediate inputs or in R&D. Letting \(L_A(t)\) denote the number of workers engaged in R&D at time t, then the equilibrium of the labor market requires:\(L_A(t) + L_Y(t) = \bar L\)The production function for new ideas is linear in the number of workers employed in R&D and proportional to the existing stock of knowledge:\(\dot A(t) = BL_A (t) A(t)\)Now, the assumptions concerning household maximizing behavior. The representative individual's lifetime utility is:\(U = \int _{t = 0} ^\infty e^{- \rho t} \ln C(t) dt, \rho > 0\)As in RCK, the individual's budget constraint is that the present value of lifetime consumption cannot exceed his initial wealth plus the present value of lifetime labor income:\(\int _{t = 0} ^\infty e^{- \rho t} \ln C(t) dt \leq X(0) + \int _{t = 0} ^\infty e^{- \rho t} w(t) dt\)Where X(0) is the initial wealth per person, r is the interest rate, and w(t) is the wage at t. The third set of assumptions illustrates the microeconomic foundations of R&D. This includes free entry, that the patent-holder chooses how much input embodies his idea, the price, but has to take the wage as given. Suppose idea i is created at time t, letting \(\pi (i, \tau)\) represent the profits earned by the creator at time \(\tau\):\(\int _{t = 0} ^\infty e^{- r( \tau - t)} \pi (i, \tau) d \tau = \frac{w(t)}{BA(t)}\)Finally, we assume general equilibrium. This means that the wages paid in R&D and input producers are equal. Second, the only asset in the economy is patents with initial wealth being the present value of the future profits from already-invented ideas. Lastly, the only use of the output good is for consumption. Thus the equilibrium in the goods market at time t requires:\(C(t) \bar L = Y(t)\) Solving the ModelThe fact that the aggregate level of the economy resembles a linear growth model suggests that in equilibrium, the allocation of labor between R&D and production of inputs is not likely to change over time. We will look for the equilibrium where \(L_A, L_Y\) are constant. Specifically, this will mean exploring the implications that a fixed labor in ideas has upon the present value of the profits from the creation of an idea and the cost. The first step is considering the price that the patent-holder would charge for his input. The standard result from microeconomics is that the profit-maximizing price of a monopolist is \(\frac{\eta}{(\eta - 1)}\) times the marginal cost, where \(\eta\) signifies the elasticity of demand. We already know (from the cost-minimizing equation) that the elasticity of demand is constant and equal to \(\frac{1}{1 - \phi}\). Also, since one unit of input can be produced from one unit of labor, the marginal cost of supplying the input is w(t). Each monopolist therefore charges:\(\frac{[1/(1 - \phi)]}{[1/(1 - \phi)] -1} \cdot w(t) \Longrightarrow \frac{w(t)}{\phi}\)Going back to our quantity of inputs equations:\(L_A(t) + L_Y(t) = \bar L \Longrightarrow \frac{(\bar L - L_A)}{A(t)}\)Knowing the price and quantity of inputs allows us to calculate the profits:\(\pi (t) = \frac{(\bar L - L_A)}{A(t)} [\frac{w(t)}{\phi} - w(t)]\)\(= \frac{1 - \phi}{\phi} \frac{\bar L - L_A}{A(t)} w(t)\)To determine the present value of profits from an invention, hence the incentive to innovate, we need to determine the economy's growth rate and interest rate. Since \(L_A, L_Y\) are both assumed constant, the growth rate of Y is \(\frac{1 - \phi}{\phi}\) times the growth rate of A, thus:\(\dot Y = \frac{1 - \phi}{\phi} BL_A\)Both wages and consumption grow at the same rate of output, since all output is consumed and there are constant returns to scale with competition. Now combining all these parts we find that the profits from a given invention grow at the rate:\(\frac{1 - 2\phi}{\phi} BL_A\)Once we know the growth rate of consumption, the interest rate is straightforward. Recall that consumption growth for a household with constant-relative-risk-aversion utility is \(\frac{\dot C(t)}{C(t)} = \frac{[r(t) - \rho]}{\theta}\) with logarithmic utility, theta is 1. Thus equilibrium requires:\(r(t) = \rho + \frac{\dot C(t)}{C(t)} \)\(= \rho + \frac{1 - \phi}{\phi} BL_A\)Therefore, the profits earned from the discovery of a new idea at the time t simplifies down to:\(\pi (t) = \frac{1 - \phi}{\phi} \frac{\bar L - L_A}{\rho + BL_A} \frac{w(t)}{A(t)}\)We are now in a position to find the equilibrium value of \(L_A\) by adding in the cost of an invention:\(\frac{1 - \phi}{\phi} \frac{\bar L - L_A}{\rho + BL_A} \frac{w(t)}{A(t)} = \frac{w(t)}{BA(t)}\)\(L_A = (1 - \phi) \bar L - \frac{\phi \rho}{B}\)Finally, modifying the equation to keep profits positive and adding in the growth rate of output will give us:\(\frac{\dot Y(t)}{Y(t)} = max [\frac{(1 - \phi)^2}{\phi} B \bar L - (1 - \phi) \rho, 0]\)We have succeeded in describing how long-run growth is determine by the underlying microeconomic environment. ImplicationsThe model has two major sets of implications. The first concerns the determinants of long-run growth: \(\rho\): when individuals are less patient (discount rate is higher) and fewer workers engage in R&D, which aligns with our understanding of R&D as investment. \(\phi\): an increase in the substitutability among inputs (phi) also reduces growth since fewer workers engage in R&D since patent-holders' market power is lower with each input embodying idea contributing less to output. \(B\): an increase in R&D productivity (B) raises growth, drawing more workers to that sector. \(\bar L\): an increase in the size of population raises long-run growth since more workers are engaged in R&D (same fraction) and the market expands, so the patent-holder can reach more people. The second set of implications revolves around equilibrium and optimal growth. Since the economy is not perfectly competitive, there is no reason to assume decentralized equilibrium will be socially optimal. By solving the individual's lifetime utility, we can see that the socially optimal level of growth is given by:\(L_A ^{OPT} = max [\bar L - \frac{\phi}{1 - \phi} \frac{\rho}{B} , 0]\)Comparing this to the original equilibrium, we find: \(L_A ^{EQ} = (1 - \phi) L_A^{OPT}\)Because of the model's three externalities-- consumer-surplus effect, business-stealing effect, and R&D effect-- we can see that the equilibrium number of workers engaged in R&D is always less than the optimal number, thus growth is always inefficiently low. ExtensionsThree of the most significant extensions are: variants of capital embodying ideas, Jones's evidence regarding long-run growth, and innovation taking the from of discrete improvements in the inputs (quality-ladder model). 3.6 Empirical Application: Time-Series Tests of Endogenous Growth ModelsAre Growth Rates Stationary?The Magnitudes and Correlates of Changes in Long-Run Growth Discussion3.7 Empirical Application: Population Growth and Technological Change since 1 Million B.C. A Simple ModelResultsDiscussionPopulation Growth versus Growth in Income per Person over the Very Long Run 3.8 Models of Knowledge Accumulation and the Central Questions of Growth Theory