Growth Theory and Convergence Endogenous growth

Transcription

1 Growth Theory and Convergence Endogenous growth Joanna Tyrowicz University of Warsaw, Faculty of Economics 02-09/03/2011 Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

2 Exogenous growth models - main conclusions In the steady state there is no endogenous (=created from inside of the economy) growth in the long run Capital will stick to its steady state level because it is optimal for firms In this class of models both firms and consumers may optimise benefits from development decrease with development and growth is totally exogenous The rate of economic growth is usually given by something like : γ = f (ˆk) δ ρ n σ This is very similar to what economists call AK model (1) Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

3 Basics of the AK model Production function: Y = AK α and α 1 Standard consumption function c(t) = c(0)e γ C, γ C = g δ ρ n σ Standard captial accumulation function k(t) = k(0)e γ L, γ K Growth given by a standard function γ = g δ ρ n σ No covergence in this model, no sudden growth episodes, smooth increase in GDP per capita Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

4 How can we get some endogenous growth? Constantly improving productivity via 1 raising the labour input - labour more productive per se [human capital] 2 diminishing the inefficiencies - better environment of the production process [government in the model] 3 raising the labour input - labour better equipped with capital [different capital goods in production process, two stages of production, intermediate sector in models] 4 raising the quality of capital input - labour equipped with capital of better productivity [same capital good is becoming more productive] Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

5 How can we get endogenous growth? Human capital - Uzawa (1966) and Lucas (1988) Some labour engaged in production of the final good, but some also engaged in the indirectly productive activity of developing human capital (second sector) Different intensities of using human capital between sectors (more in the educational sector) Three types of assets: financial, physical capital and human capital By no arbitrage condition - each should give the same return. Whatever the conditions (i.e. specific production function) - the ratio of human to physical capital is constant in the equilibrium (because of the no arbitrage condition). Consequently, no endogeneity in the steady state! [imagine a shortfall of H - what would be the outcome?] If production function assumes constant returns to scale, this whole thing reduces to a simple AK model Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

6 How can we get some endogenous growth? Government spending in the model In a Ramsey model with no effect on productivty, government will have a budget constraint: g = τk α transfers + τ L Assume G (by means of τ permanently k lower in the longrun, but initially welfare effect dominates Assume G (by means of τ L permanently k lower even initially, no welfare effect Problem: at aggregate level reduces to a standard AK model, with no endogenous reasons for growth But you can try to relax CRS assumption and try to put government spending in the production function (e.g. Y = K 1 α L α G 1 α ) G would reflect positive effect of government spending on productivity (e.g. roads, infrastructure, etc.) You can also try to put in coupled with labour (e.g. Y = K α L 1 α G 1 α ). This would mean combining positive effect on production with negative effect on consumption under some conditions it can mean higher growth rate Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

7 Government spending - explicitly on the model Government spending in the model: Assume Y = L 1 α K α G 1 α (the component of K α G 1 α reflects positive effect of τy on production) Of course standard identity will have to hold: Y = C + I + G This will transform = (1 τ)αl1 α K α 1 G 1 α δ ρ n σ = (1 τ)α(τl) 1 α α σ γ Y = (1 τ)mpk δ ρ n σ This is because on a firm level, firm choses L i and K i, but cannot choose G (exogenous) In other words, on can combine two effects (bad on consumption but good on productivity) and this way get a positive effect of government spending on growth But Still no endogeneity in steady state. To get the government, you need to tax keep in mind that proportional taxation is never optimal, so G/Y has to be a specific proportion (derived from the model) but never a proportional function! [lump sum taxes best] This is called a difference between decentralised equilibrium and a centralised one (central planner) Decentralised equilibrium is only optimal if τ is optimally chosen, i.e. lump sum with an adequate size γ Y = 1 α τl α δ ρ n γ Y = σ 1 α (1 τ)α(τl) α σ (2) Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

8 How can we get some endogenous growth? Technology - take 1: 1 Jones and Manrelli (1991) 2 Keep the long-term properties of AK models and add a neoclassical technology term Y = AK + βk α (3) 3 This yields MPK = A + αβk α 1. 4 For small K, neoclassical component dominates (good for analysing accumulation and development in the beginning ) 5 For larger K, AK model makes sense 6 Growth is there, it changes depending on the level of developmetn and some conditional catching up is possible Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

9 How can we get some endogenous growth? Technology - take 2: R&D based growth models Dates back to Schumpeter (creative destruction) vertical and horizontal differentiation Introducing ipad does not destroy the market for computers (creation not destructive horizontally), but introducing Pentium killed the market for 486 (destructive vertically) So may have two types of approches: variety of products (non-rival, innovation does not destroy the old one) or a quality ladder of products (rival, innovation destroys automatically the past one) Need more than one sectors in the economy For investment in R&D to make sense, you need to expect some gain: abnormal profits because you need to cover the innovation costs depart from perfect competition (monopoly power due do patent protection) Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

10 Endogenous growth models with technology Three ways of getting inventions 1 As a by-product of R&D spending (inventions arrive if you spend certain fixed amount of money or at Poisson rate, based on accumulated expenses) There are some externalities (individual expenses sum to less than generally optimal expenses so room for centralised equilibrium and central planner) Final good becomes irrelevant We get intuition on how we get endogenous growth - via increasing the variety of intermediate products or raising the qualit of these intermediate products These intermediate products are always used in final good production (so to say: always demanded, except of course for the case of destruction) 2 Through saving labour Technology saves labour through learning of workers Technological process is common knowledge (need no imperfect competition, no patent protection, no monopolies) As earlier, profit maximisation happens with reference to labour and capital, but not learning - firms do not internalise learning, so positive externality (room for central planner). 3 As a purposeful activity This is how you get trade, FDI, institutions, and all that stuff in the growth models... (to appear later) Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

11 Endogenous growth models - increasing the variety of products (1) There are two types of manufacturing activities: production of the final consumption good, the output production of the physical units of the capital goods of the highest quality in a given moment. Third activity, research and development (R&D), designs new intermediate goods. Some important assumptions concerning these processes: both manufacturing activities - production of consumption goods and production of intermediate goods - use the same Cobb-Douglas production function, Y can be used in a perfectly substitutable manner for consumption/investment (later also exports). Y produced by all firms is physically identical and the aggregate of the output Y i = L 1 α i N j=1 x α ij (4) Intuition: assume just for a second that all components (all j s are identical), then x i,j = x i,j and Y i = L 1 α i N j=1 x α ij = L 1 α i (Nx i ) α = L 1 α i Increasing the variety of N is the source of endogenous growth x α i N 1 α (5) Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

12 Endogenous growth models - increasing the variety of products (2) Savings/investment proportional to income S I = s Y, C = (1 s) Y (keep it simple, could have a consumer choice and few more equations/parameters too) Labour force does not grow either, just to keep it simple If invested, one unit of the good can be transformed into given type of capital good, in accordance with a design embodied in a patent for that type. Entrepreneurs try to invent new types of capital goods. If successful, they produce the capital good and sell it a price P N. As a result the number of types of capital good (N) rises over time. In the final good sector, profit is given by: Π i = Y i 1 w L i N p j x ij (6) j=1 This yields demand for x ij from the final producers to be: δπ i δx ij = 0 x i,j = L i ( α p j ) α 1 α (7) Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

13 Endogenous growth models - increasing the variety of products (3) In the intermediate sector, profit is given by: Π j = p j x j Y i 1 x j = (p j 1) L i ( α p j ) α 1 α (8) Remember that we assumed it: it takes one unit of intermediate good to transform it to the demanded amount of this particular intermediate product Producer of the intermediate product will set price such that δπ j δx j = 0 To compute it, one needs to replace x j = L i ( α p j ) α 1 α from (7) in (8) From the optimisation of (8) we get that 1,2,...j p j = 1 α > 1 (9) Though prices for all j s are the same, this is not perfect competition, because p j > MC j We did not distinguish really why are there many j s, so we do not specify why they should differ, so this is why we have the same prices for all j s Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

14 Endogenous growth models - increasing the variety of products (4) The finding that 1,2,...j p j = 1 α may be plugged into both demand for x j... x j = L( α ) 1 α α = Lα 1 α 2α (10) p j... and final output function Y = LNα 2α 1 α (11) We clearly see that higher N (variety of products) means more output (more growth) But this is only once we get new research - we are not having it yet! Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

15 Endogenous growth models - increasing the variety of products (4) Define η to be the cost of research (e.g. no of units of the final good to be devoted to R&D) We will also need the benefits of research (actually, present value of these benefits) Benefits follow from the fact, that I enjoy a monopoly power over my product for a certain (potentially infinit) horizon V = Π j e r(s t) ds = (p j 1)x j e r(s t) ds = ( 1 1)... (12) t t t α When solved (we can actually compute an integral from each of these expressions), this yields V = 1 α α Lα 1 α 2 1 r Who is going to pay for research? Consumers! In a standard way, from Ramsey-Keynes rule ċ c = r ρ σ by plugging the demanded interest rate (by the innovators) to supply interest (by the consumers), we get γ C = ċ 1 α N c = α η α 2 1 α ρ σ Everybody is in equilibrium (consumers, firms and innovators), but is this the best equilibrium? Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26 (13) (14) (15)

16 Endogenous growth models - increasing the variety of products (5) Central planner would like to solve the following maximisation problem max c,x j 0 c 1 σ 1 σ e ρt dt (16) s.t. (1) Y = L 1 α (Nx j ) α This yields a Hamiltonian (2) Y = C + Nx j + ηṅ Ṅ = 1/η[L1 α (Nx j ) α C Nx j ] H = c1 σ λ 1 σ η [L1 α (Nx j ) α C Nx j ] (17) We do not have to solve the whole Hamiltonian, to see that the optimum is different than the one solved by the entrepreneurs and consumers separately. More speciically: δh δx j = 0 = αl 1 α (Nx j ) α 1 N (18) x CP j = Lα 1 1 α x j = Lα 2 1 α Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

17 Endogenous growth models - increasing the variety of products (6) Monopolistic competition ushers an inefficiency (static, so always the same), so the central planner optimisation yields higher growth. Policy solution here is not to tax the monopolists (counter-effective, drive profits down, so lower rate of innovation). The best policy is to actually subsidisise the monopolist, by subsidising the price of intermediate products in the production of the final output (drive the demand for x j s up, Π j N : optimal price for each x j is 1, not 1/α, so the goverment pays 1 α leaving the final good producer(s) to pay α. Summary Economic growth is a consequence of innovation - not capital accumulation (as previously) This model does not take into account that innovators could have to compete for capital - they are actually the only reason why the consumers should save at all (no other use of savings, so zero price for savings otherwise) Decentralised equilibrium will always be worse than central planner. The only reason why there is any innovation at all, is that there is profit, so there has to be a market imperfection to have innovation This is against the typical thinking of the industrial policy, but if you look closely, in most (all?) countries it is illegal to abuse the monopoly power preventing competition, but not actually to charge a monopoly price... In fact, there are many blends of the endogenous growth model which heavily rely on market imperfections Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

18 Endogenous growth through increasing the quality of products (1) Specialised firms produce each variety of capital goods, and rent it out to the final good producers at a rental price P j,kj There is a certain fixed N, the number of capital goods used in the production of the final product. We will assume N does not change at all. There is an improvement in quality, though. Each x j has quality level q k j. Investment in innovation raises q k j to q k j +1 Improvement of the quality of intermediate good is costly, η In order to motivate an entrepreneur to engage in R&D activity, the successful innovator is granted a patent for the capital good invented. The leading-edge producer acts as a monopolist in this environment. This is to say that they maximise profit function with respect to price level using a demand function. The demand for each variety of quality-adjusted intermediate good, x j,kj, follows from the optimality condition that equates the rental rate to the marginal productivity, as given by the equation: P j,kj = dy dx j = αah (1 α) q k j α x j α 1 (19) Demand for the intermediate good of quality k j in sector j then takes the following form: x j,kj = H(αA) 1/(1 α) q k j α/(1 α) P 1/(1 α) j,k j Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

19 Endogenous growth through increasing the quality of products (2) We can simplify the demand of intermediate good j to: x j,kj = HA 1/(1 α) q k j α/(1 α) α 2/(1 α) (20) The profit equation for the producer of the intermediate good is given by: Π j = x j,kj (P j,kj 1), where the production cost of an intermediate good of any quality is constant and normalised to one. The price of intermediate good takes form of constant mark-up on production costs: Π j P j,kj = 0 P = P j,kj = 1 α. The monopoly price of intermediate good is constant over time and sectors. Using these results, the overall production of intermediate inputs, (x j s), and total output, (Y ), can be determined for the whole economy. Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

20 Endogenous growth through increasing the quality of products (3) Summing up over the N given sectors, we obtain the following expressions: N N Y = AH (1 α) q k j α xj α = HA 1/(1 α) α 2α/(1 α) j=1 j=1 where Q is defined as an aggregate quality in all sectors index: q k j α/(1 α) = HA 1/(1 α) α 2α/(1 α) Q, (21) N X = X j,kj = HA 1/(1 α) α 2/(1 α) Q, Q = N j=1 j=1 q k j α/(1 α) The innovation in the model takes form of increasing the quality of the existing capital good, that is of an increase in k j. The k j s innovator in sector j raises the quality from q k j 1 to q k j. This innovator is able to set the monopoly price and sell the quantity of intermediate goods given by market equilibrium. The flow of profit per the unit of time associated with the quality rung k j takes the form of: Π j,kj = x j ( 1 α α ) = HA 1/(1 α) ( 1 α α ) α 2/(1 α) q k j α/(1 α) The inventor of k j quality rung intermediate goods keeps their monopolistic position only for a certain period of time. Once Joanna Tyrowicz a higher (WNE quality UW) intermediate good L2. Endogenous in sector growth j is introduced, the previous 02-09/03/2011 innovator 20 / 26 (22) (23)

21 Endogenous growth through increasing the quality of products (4) Innovations are standardly assumed to arrive randomly with a Poisson arrival rate υ j,kj, which depends primarily on the flow of resources to R&D in sector j. Specifically, we assume that: υ j,kj = Z j,kj φ(k j ), (24) where Z j,kj denotes the resources spent on R&D in sector j, and φ(j) describes the productivity of used inputs, with the following form: φ(k j ) = 1 η q (k j +1)α/(1 α), (25) where η > 0 represents the costs of the research. Naturally, the higher η is the lower the probability of innovation for given values of Z j,kj and k j. The second term on the right-hand side of equation (25) indicates the negative effect of how advanced is the quality in sector j on the probability of yet new innovation. The higher rung at the quality ladder, represented by k j + 1, requires more resources to be devoted to keep the probability, υ j,kj, on the same level. Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

22 Endogenous growth through increasing the quality of products (5) The probability of success in the innovation process determines for how long the monopolist enjoys the stream of profit. The present value of the profit from the k j th innovation in sector j is given by: V j,kj = Π j,kj [1 e ( rt j,k j ) ]/r (26) r is the interest rate and T j,kj = t j,kj +1 t j,kj is the time interval over which the k j th innovation is in the forefront (before new innovation arrives). Taking into account the Poisson distribution of innovation, the expected present value of profit computed at time t kj is given by the expression: E(V j,kj ) = Π j,kj /r (1 e rτ ) υ j,kj e υ j,k τ j dτ (27) 0 Firms decide to engage into R&D activity if the expected profits are at least as high as cost of improving the existing intermediate good j. The expected reward per unit of time for pursuing the (k j + 1)th innovation is υ jkj E(V j,kj +1). The expected flow of net profit from research in a sector j at quality rung k j is given by: Π j,kj = p j,kj E(V j,kj +1) Z j,kj. Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

23 Endogenous growth through increasing the quality of products (6) Algebraic manipulations over Π j,kj, E(V j,kj ) yield: Π j,kj E(V j,kj ) = r + υ j,kj ( ) 1 α = HA 1/(1 α) α 2/(1 α) q k j α/(1 α) /(r + υ j,kj ) (28) α If the set-up costs for introducing a new quality of capital good exceed the present value of future profits then there is no economic reason to spend the resources on R&D. Similarly, if the set-up costs are smaller than the expected profits, there are strong incentives to engage in the innovation activity. (r + υ j ) = ( 1 ( ) 1 α ζ ) HA1/(1 α) α 2/(1 α). (29) α The right-hand side of equation (29) represents the expected rate of return from research. As this flow of profit last only until the next innovation, the rate of return must cover the ordinary rate of return, r, plus the premium for the probability, p, per unit of time, corrected for the extra cost of external financing. Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

24 Endogenous growth through increasing the quality of products (7) Followingly, the probability of successful innovation per unit of time is given by the following equation: υ j = ( 1 ( ) 1 α ζ ) HA1/(1 α) α 2/(1 α) r. (30) α The growth rate of the economy will be equal to the aggregate quality growth rate. Recall that definition of Q, is in (22) In sector j, the term q k j α/(1 α) does not change if no innovation occurs, but rises to q (k j +1)α/(1 α) in the case of research success. Since υ j, which is defined in equation (30), varies across sectors, the expected proportionate change in Q per unit of time takes be expressed in the following form: E( Q/Q) = N p j ψ(j) [q α/(1 α) 1]. (31) j=1 In this case ψ(j) accounts for the simple weightening mechanism with the interpretation that the innovation has occurred precisely in sector j. Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

25 Endogenous growth through increasing the quality of products (8) Substituting for p j, the growth rate of aggregate quality, Q: N ( ( ) r) γ Q = 1 α ψ(j) ζ 1 HA 1/(1 α) α 2/(1 α) [q α/(1 α) 1]. (32) α j=1 The only remaining unknown remaining the interest rate, r. As in every general equilibrium model these are the households who provide the capital, basing on the intertemporal consumption-savings choice. Ramsey-Keynes rule for household optimisation: γ C = ċ c = 1 r ρ)σ (33) ( By plugging r from (32) to γ C we solve the model Y and X are constant multiples of Q, so they grow at the same rate as Q. Although Z cannot be represented in this way, the growth rate of the aggregate R&D spending is exactly the same as total output and aggregate quality. To see this, note that except q k j all parameters on the right-hand side in the Z equation are constant over time. Consequently, the growth rate of consumption can be determined to be equal to the growth rates of Y, X, and Z: γ C = γ Y = γ X = γ Z = γ Q Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26

26 Endogenous growth models - summary Model solutions In the equilibrium we have positive growth rates with internal reasons for growth Growth positively depends on the variety of products (no destruction) or on the quality upgrading (innovation through creative destruction ) If innovation is by expanding the variety of products, the growth is unconstrained If innovation is by improving the quality, there will be so-to-say decreasing returns to scale the more innovation a country has already had, the more difficult it is (less probable) to get new ones Bottom line: Growth rate depends on: the amount of resources devoted to innovation activity (R&D) the degree to which technology can be privately appropriated (degree or duration of monopoly, also the intellectual property protection) the potential to internalise externalities and the degree of patience of investors (time preference for consumption always there!). Except for ρ and σ, main reason for growth (the extensiveness of innovative activity) are determined endogenously in the model! High growth implies high growth in physical capital, but as a result and not the cause of technological progress. Joanna Tyrowicz (WNE UW) L2. Endogenous growth 02-09/03/ / 26