Endogenous growth through R&D — Empirical issues and extensions Carl-Johan Dalgaard Lecture notes May 2 00 7
A SIMPLE VERSION OF THE R&D MODEL Consider the model from B&S Ch. 6 with a slight simpli fi cation to fascilitate the discussion. That is, we have the following structure. First, equilibrium production of each variety: 2 1 x = α ¯ 1 − α A 1 − α L, Second, total output is, in symmetrical equilibrium: 1 2 α x α L 1 − α N = A Y = A ¯ 1 − α α 1 − α NL Third, we have the resource constaint of the economy x + η ˙ Y = C + N ¯ N, 2
A SIMPLE VERSION OF THE R&D MODEL so new ideas are produced assuming a lab-equipment formulation N = Y R η ≡ s R ˙ η Y, where s R is endogenous. Finally, here is the simpli fi cation: C = (1 − s ) Y, i.e. in stead of Ramsey-consumers, we have "Solowian" consumption behavior. Inserting the consumption function into the resource con- staint, and rearrangeing, gives us h i s − N ¯ x Y x + η ˙ N ⇔ ˙ Y = (1 − s ) Y + N ¯ N = Y η 3
A SIMPLE VERSION OF THE R&D MODEL Finally, inserting for ¯ x and Y leaves us with " # Y ≡ s ∗ s − α 2 ˙ R N = η Y. η Hence, in this version of the model the growth rate of the economy is simply µ Y ¶ ∗ 1 2 α N = s ∗ ˙ N A 1 − α α 1 − α L = s ∗ R R η N η where obviously g N = g Y , and due to C = (1 − s ) Y , it clearly follows that g C = g N = g Y . That is, balanced growth prevails. We are therefore left with a couple of testable predictions. (1) All other things remaining equal, a higher investment share in R&D should lead to faster growth. (2) A larger labor force should lead to faster growth. 4
INVESTMENTS IN R&D AND PROSPERITY Given the lab-equipment R&D model we essentially have developed some microfoundations for an assumption like ˙ N = s R Y. That is, new ideas — or technology — are the result of investments in R&D. Nonneman and Vanhoudt (1996) use this formulation to provide a (further) augmentation of the Solow model, thus taking the empirical work of Mankiw, Romer and Weil (1992) slightly further. In their model s R is exogenous. Q: is higher s R associated with faster growth? Also another motiva- tion for re-visiting MR W: The human capital augmented solow model does not seem to do a particularly good job in explaining productivity di ff erences in the OECD. 5
INVESTMENTS IN R&D AND PROSPERITY Figure 1: MRW p. 426. The growth regression neither physical capital nor human capital are signi fi cant at 5%. In Levels-regressions, the fi t becomes progressivly poorer as the sample is limited. 6
INVESTMENTS IN R&D AND PROSPERITY Nonneman and Vanhoudt simply adds “knowledge capital" to the list of inputs Y = K α H β N γ ( AL ) 1 − α − γ − β . “A" is an exognous source of technological progress. Stricktly speaking, therefore, the model suggests that changes in s R (for example) only leads to level e ff ects. Growth e ff ects are interpreted as transitional dynamics. A property of the steady state is worth fl eshing out. Since ˙ N Y N = s R N, and assuming exogenous growth of x percent, labor force growth of n ³ ´ ∗ ³ ´ ∗ Y Y = n + x percent implies that, in steady state, n + x = s R ⇔ s R . N N Under competitive markets, the “Return to R&D" is γ Y N . Hence R = γn + x r ∗ . s R 7
INVESTMENTS IN R&D AND PROSPERITY Figure 2: Nonneman and Vanhoudt (1996), p. 950 * s R does seem to be associated with faster growth (under the model — in transition to steady state). * In general the augmentation improves the fi t. α is estimated to about 1 / 3 . 8
INVESTMENTS IN R&D AND PROSPERITY * Compared with MR W the impact from HC is lowered ( β ≈ . 15 ) * Also, they fi nd γ ≈ . 085 . Plausible? Taking this fi nding seriously allows for a consistency check. Consider the US: n = . 025 , x = 0 . 02 and s R = 0 . 025 gives r ∗ R = . 085 . 045 0 . 025 ≈ 0 . 153 (or about .20 if we also added a depreciation rate). Is this plausible? * There is a literature which attempts to estimate r R directly , using industry data. An idea that goes back to Grilliches (1979, Bell Journal of Economics). To illustrate. First step, di ff . the production tech wrt time g Y = αg K + βg H + γg N + (1 − α − β − γ ) ( x + n ) . 9
INVESTMENTS IN R&D AND PROSPERITY Now suppose indeed ˙ N/N = g N = s R Y/N. Substitute back into the above equation g Y = αg K + βg H + γs R Y/N + (1 − α − β − γ ) ( x + n ) Since, in theory, γ = r R N/Y we now have g Y = αg K + βg H + r R s R + (1 − α − β − γ ) ( x + n ) , where s R are R&D investments in value added. We can think of r R as a paramter to be estimated (not without problems). Grilliches fi nd r R , for the US, to be around 20%; which is roughly consistent with Nonneman and Vanhoudts fi ndings. 10
INVESTMENTS IN R&D, SCALE AND PROSPERITY Going back to our expression for the growth rate: µ Y ¶ ∗ 1 2 α N = s ∗ ˙ N A 1 − α α 1 − α L = s ∗ R g Y = R η N η N&V provide some evidence in favor of the prediction that s ∗ R ↑ goes along with γ Y ↑ ; at least in the OECD (and at least in transition). But the model also suggest that L ↑ should lead to accelerating growth. In more “general R&D models", this would be "R&D labor", not just the labor force. Still, eventually the two would be proportional. This implication is heavily criticized by Jones (1995), and started the "scale controversy". 11
INVESTMENTS IN R&D, SCALE AND PROSPERITY Figure 3: Jones, 1995; p. 517. Basic point: R&D labor has increased, but g Y is US (and other places) have remained stationary. So can we come up with an alternative model? 12
A “SEMI-ENDOGENOUS" GROWTH MODEL We keep basically the entire Romer framework, leading to : 1 2 α A 1 − α α 1 − α L g Y = g N = s ∗ . R η But, assume now that η ( N ) = φN σ , φ > 0 , σ > 0 . That is, suppose it progressively becomes more and more di ffi cult to shift the frontier (to innovate), and therefore more costly to get the next good idea. 1 . We now have 1 2 α A 1 − α α 1 − α L g N = s ∗ . R φN σ 1 A version with Ramsey savings is discussed in B&S ch. 6.1.8. 13
A “SEMI-ENDOGENOUS" GROWTH MODEL Is constant per capita growth feasible? Yes if L increases. Di ff . the growth rate wrt time g N ˙ = n − σg N = 0 , g N which therefore requires N = 1 g ∗ σn > n for σ < 1 . Key implications : 1. Per capita growth rate is µ 1 − σ ¶ g ∗ Y − n = g ∗ N − n = n. σ Hence constant growth in labor force (science input) is associated with constant growth in GDP per capita. n is exogenous, but tech. change endogenous (“semi") 14
A “SEMI-ENDOGENOUS" GROWTH MODEL 2. Policies (which matter for e.g. s R ) does not matter for the growth rate; but for the level of income per capita. To see this, note that on a balanced growth path µ L ¶ ∗ 1 2 α σn = s ∗ R A 1 − α α N = 1 1 − α g ∗ N σ φ so the level of N 1 /σ 1 2 α s ∗ R A 1 − α α 1 − α N ∗ = L 1 /σ . φ 1 σ n so 1 /σ µ Y ¶ ∗ 1 2 α s ∗ R A 1 − α α 1 − α 1 2 α 1 2 α 1 − α N ∗ = A L (0) 1 /σ e ( n/σ ) t . = A 1 − α α 1 − α α 1 − α φ 1 L σ n 15
A “SEMI-ENDOGENOUS" GROWTH MODEL 3. Scale still matters, but in a more subtle way: more L implies a higher level of GDP per worker. 4. worrisome prediction: If n declines (or fall), eventually growth in GDP per worker should move in the same direction, since g ∗ y = ³ ´ 1 − σ n. σ As a matter of transitional dynamics: If n falls the growth rate in N (thus Y/L ) should be either montonically declining, or follow a hump shaped path (Phase diagram). 16
CONTRASTING EVIDENCE: HA AND HOWITT (2005) Figure 4: Ha and Howitt, 2005; p. 11 Observation 1: From 1950-2000 TFP growth has remained stationary. 17
CONTRASTING EVIDENCE: HA AND HOWITT (2005) Figure 5: Ha and Howitt, 2005, p. 14 Observation 2: Growth in R&D input (however measured) has not remained constant. 18
CONTRASTING EVIDENCE: HA AND HOWITT (2005) Figure 6: H & H (2005), p. 17 Observation 3: The share of R&D in GDP (i.e. s ∗ R ) has remained constant. Obs. 1 -3 hard to reconsile with semi-endogenous growth theory 19
A SIMPLE MODEL WHICH IS CONSISTENT WITH THE EVIDENCE (Dalgaard and Kreiner, 2001) Final output, Y t , is produced using human capital augmented labor in- put, H t = h t L t , ideas, N t , and some fi xed factor of production denoted by Z t Z 1 − α N β Y t = H α t , Z ≡ 1 , 0 < α < 1 , 0 < β ≤ 1 Ideas are produced using units of fi nal output (the ’lab-equipment’ framework): ˙ N t = s R Y t , N 0 given The parameter s R denotes the share of total output invested in R&D. 20
A SIMPLE MODEL WHICH IS CONSISTENT WITH THE EVIDENCE Assume: ( a ) H t = h t L t , ˙ L t /L t = n , and h t endogenously growing. ( b ) β < 1 and α + β = 1 . Does perpetual human capital accumulation make sense? - Human capital is not just quantity of information but also quality - Complementarity between human capital and scienti fi c knowledge ⇒ If science continues to progress, i.e. ˙ N/N > 0 , quality of knowledge may continue to expand. 21
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