Mehdi Senouci
email: senouci@pse.ens.fr

Technical change in a two-sector model of optimal growth

Keywords: Sector-specic Technical Change ; Optimal Growth, Capital Accumulation, Golden Rule, Two-sector Models Sector-specific technical change, optimal growth, capital accumulation, golden rule, two-sector models. JEL codes: C62, E22, O39, O41

This paper investigates into the consequences of sector-specific technological progress in a two-sector, optimal growth model. In accordance with existing theory, we find that consumptionspecific Hicks-neutral technical shocks increase consumption but leave other parameters unchanged. Hicks-neutral, investment-specific technical shocks increase the wage-rental ratio, and increase steady-state consumption by a factor equal to the macroeconomic ratio of capital share to labor share. If the elasticity of substitution is equal to one in the long run, the growth regime with only investment-specific technical change is sustainable and asymptotically balanced.

Introduction

Since the seminal study by [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF], considerable effort has been devoted by growth economists to distinguish between consumption-specific and investment-specific technical change. Figure 1 represents the post-war evolution of the ratio of real investment to real consumption in the USA. It emerges clearly from this picture that real investment has grown more rapidly than real consumption on average for more than sixty years 1 .

The case for predominant investment-specific technical progress is even more obvious for the last thirty years. As figure 2 shows, with the beginning of the 'Great Moderation' the deflator of investment has started to continuously decrease relatively to the deflator of consumption. Consistently with these observations, several empirical studies have concluded that investmentspecific technical change was responsible for a part of GDP growth that far outweighs the share of investment in GDP 2 . Firm theoretical foundations about the consequences of sector-specific technical change are however rare. Meade (1962) describes the growth path of an economy enjoying steady consumption-specific and investment-specific technological progress when production functions are of Cobb-Douglas type. [START_REF] Kimball | Proof of Consumption Technology Neutrality[END_REF] has shown that, in an economy governed by a social planner with a logarithmic or power utility function, the optimal path was independent of the path of consumption-specific productivity. This neutrality result is also evoked in [START_REF] Fisher | Relative Prices, Complementarities and Co-movement among Components of Aggregate Expenditures[END_REF] and [START_REF] Basu | Sector-Specific Technical Change[END_REF]. Still, there does not exist -to our knowledgeany theoretical study of the consequences of investment-specific technical change. As figures 1 and 1 Still, notably, the ratio of nominal investment to nominal consumption is stationary. See [START_REF] Whelan | New Evidence on Balanced Growth, Stochastic Trends, and Economic Fluctuations[END_REF]. 2 See Gort, [START_REF] Gort | How Much of Economic Growth is Fueled by Investment-Specific Technological Progress?[END_REF], [START_REF] Cummins | Investment-Specific Technical Change in the United States (1947-2000): Measurement and Macroeconomic Consequences[END_REF], [START_REF] Pakko | Investment-Specific Technology Growth: Concepts and Recent Estimates[END_REF] and Ngai and Samaniego 2 show, this analysis is crucial in understanding the growth pattern that prevailed in the USA at least for the last thirty years.

This paper is an attempt to partially bridge this gap. We analyze the consequences of technical shocks on the two-sector golden rule steady state. We presume this method is best suited to analyze first-order effects of technical progress, that stem from the expansion of the production possibilities of the economy. Indeed, when technical change is continuously expected, many effects intertwine. For example, suppose that an economy is expecting some positive technical shock (in any of the industries) to happen in the future. Then, in general, the agents will attempt to consume some of the future consumption benefits today. To do this, they must (temporarily) allocate more inputs to the consumption sector for some time. This strategy does not reflect optimal long-term calculation, it just merely represents some inter-temporal substitution of consumption. At least theoretically, these are second-order effects. Here we concentrate on the first-order effects which are the only ones that we observe when the economy moves from one steady state to another 3 .

Consistently with existing results, we show that consumption-specific technical shocks are neutral; they do not influence the wage-rental ratio, the macroeconomic capital-labor ratio, nor the distribution of inputs between sectors: they merely make consumption increase one-for-one. Investment-specific technical shocks have richer consequences. They are shown to increase the steady-state wage-rental ratio and to increase final consumption by a factor equal to the macroeconomic ratio of capital share to labor share. The sustainability of a growth regime led by investmentspecific productivity growth thus crucially depends on the elasticity of substitution between capital and labor inputs in the whole economy. When the elasticity is strictly greater than one, the con-tribution of investment-specific productivity growth to consumption growth increases over time, even if investment-specific productivity growth is constant. On the contrary, when this elasticity is strictly less than one, this contribution eventually vanishes as time tends to infinity, and consumption growth cannot be sustained by investment-specific technical change alone.

The rest of the paper is organized as follows. Section 1 presents the two-sector golden rule in the absence of technical progress. Section 2 analyzes the consequences of consumption-specific technical shocks. Section 3 does the same for investment-specific technical shocks. Section 4 discusses the issue of the sustainability of the growth path with only investment-specific technical change. Section 5 concludes.

1 The two-sector golden rule

Technology and optimization problem

We suppose in this section that there is no technical change. Time is discrete. Population is constant at L and the economy is initially endowed with a stock of capital K 0 . Capital depreciates by a factor δ from one period to another.

At each period, a benevolent social planner allocates available capital (K) and labor (L) resources in the consumption sector and the investment sector: K C , L C goes to the former and K I , L I to the latter. Inputs are costlessly transferable from one sector to another at each date. Consumption technology is represented by the production function

C t = F K C t , L C t
and investment technology by I t = G(K I t , L I t ). Both production functions satisfy usual neoclassical properties: F and G are C 2 , show constant returns to scale, strictly decreasing marginal productivity with respect to both inputs and verify Inada conditions. The planner's intertemporal preferences are summarized by a well-behaved instantaneous utility function v(C) and by the rate of preferences for the present β = 1/(1 + θ), where θ > 0 denotes the subjective rate of impatience. The problem he solves is:

max (K C t ,K I t ,L C t ,L I t ) t≥0 ∞ t=0 1 (1+θ) t v (C t ) s.t. C t ≤ F K C t , L C t I t ≤ G K I t , L I t K C t + K I t ≤ K t L C t + L I t ≤ L K t+1 = (1 -δ)K t + I t (1)
Let's denote by (p C t , p I t , r t , w t , q t ) t≥0 the vector of lagrangian multipliers. This optimization problem we treated in [START_REF] Senouci | Optimal Growth and the Golden Rule in a Two-Sector Model of Capital Accumulation[END_REF]. When both goods are produced, first-order equations are:

                           u (c t ) = p C t q t = p I t p C t ∂F ∂K C t = r t p I t ∂G ∂K I t = r t p C t ∂F ∂L C t = w t p I t ∂G ∂L I t = w t q t = 1 1+θ (r t+1 + (1 -δ)q t+1 ) (2)
The exposition is made easier with per-capita variables.

Let k = K/L, c = C/L, i = I/L, k C = K C /L C , k I = K I /L I , u(c) = v(Lc)/L, f k C = F k C , 1 , g k I = G k I ,
1 and let ω = w/r denote the wage-rental ratio. Static efficiency requires that, at all dates:

ω = f k C f (k C ) -k C = g k I g (k I ) -k I (3)
Because functions f (x) f (x) -x and g(x) g (x) -x are both strictly increasing, equation (3) determines each sector's capital per worker ratio as two strictly increasing functions of the wage-rental ratio, k C (ω) and k I (ω):

∂k C ∂ω = - f 2 f f k C > 0 (4)
And:

∂k I ∂ω = - g 2 gg k I > 0 (5)
Notice that the relationships k C (ω) and k I (ω) are not altered by Hicks-neutral technology shocks.

Let's define l C = L C /L and l I = L I /L the fractions of the labor force respectively employed in the consumption and in the investment sector. Since resources are fully employed, it holds that:

l C k C + l I k I = k l C + l I = 1 (6)
Thus, if k C = k I , we have that:

l C = k-k I k C -k I = l C (k, ω) l I = k C -k k C -k I = l I (k, ω). (7)

The quasi-golden rule

Because of decreasing marginal productivity of capital in both industries, consumption cannot grow asymptotically in the absence of technical progress4 . We have shown elsewhere5 that, under a few conditions, the economy governed by a social planner eventually converges to a "quasi-golden rule" steady state independent of the initial conditions and of the instantaneous utility function, but dependent on the subjective discount factor, and characterized by 6 :

       g k I = δ + θ f (k C ) f (k C ) -k C = g(k I ) g (k I ) -k I = ω l I g k I = δk ⇔ k C -k k C -k I g k I = δk (8)
We call ω * θ the quasi-golden rule wage-rental ratio, and k * θ the quasi-golden rule capital-labor ratio, for each θ > 0.

The first equation in [START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF] has the following interpretation. Suppose that the optimal steady state is reached, so that the social planner plans to set k C , k I , l C , l I (and, so, k) constant at their quasi-golden rule value for the infinite future. Suppose further that the social planner unexpectedly receives a free unit of capital at date t = τ. If he installs this marginal unit in the consumption sector from t = τ on, the resulting intertemporal benefit is of:

∞ t=τ 1 1 + θ t-τ (1 -δ) t-τ f k C (ω * θ ) u (c * θ ) = 1 + θ δ + θ f k C (ω * θ ) u (c * θ ) (9)
Another strategy is to put this additional unit of capital in the investment sector at t = τ and, then, to install the windfall quantity of capital 1 -δ + g k I in the consumption sector for t ≥ τ + 1. The increase in the objective function of the social planner seen from t = τ is:

1 -δ + g k I (ω * θ ) ∞ t=τ +1 1 1+θ t-τ (1 -δ) t-τ -1 f k C (ω * θ ) u (c * θ ) = 1-δ+g (k I (ω * θ )) δ+θ f k C (ω * θ ) u (c * θ ) (10)
Because of intertemporal optimality, these two strategies must yield the same benefits. Equality of ( 9) and ( 10) then reduces to the first equation in [START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF]. The quasi-golden rule is thus centered on the technology of the investment sector. Notably, ω * θ is independent of the production function for consumption f. This is logical, as investment can be thought as a production roundabout, which profitability only depends on the productivity of the investment sector.

The second equation in [START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF] reflects equalization of technical marginal rates of substitution across the two sectors, and the third equation in [START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF] reflects the condition that net investment must be zero at steady state.

The pure golden rule

When the rate of impatience θ tends to zero, steady-state approaches the pure golden rule defined by:

       g k I = δ f (k C ) f (k C ) -k C = g(k I ) g (k I ) -k I = ω k C -k k C -k I g k I = δk (11)
which also corresponds to the maximization of steady-state consumption problem. Let's call ω * and k * the pure golden rule values of ω and k. Clearly, for all θ > 0, ω * > ω * θ and k * > k * θ .

Some remarkable identities, already presented in [START_REF] Cremers | Dynamic Efficiency in the Two-Sector Overlapping Generations Model[END_REF], hold at the golden rule steady state:

Theorem 1.1. At the golden rule allocation:

l C = f k C -k C f k C f (k C ) (12)
l I = k C f k C f (k C ) ( 13
)
l C k C k = g k I -k I g k I g (k I ) ( 14
)
l I k I k = k I g k I g (k I ) ( 15
)
where all variables are at their golden rule level -

k = k * , ω = ω * , k I = k I (ω * ) , k C = k C (ω * ) , l I = l I (k * , ω * ) and l C = l C (k * , ω * ) .
Proof. Proof is straightforward by combining equations ( 3), ( 6) and [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF].

That is, the fraction of total labor supply employed in the consumption sector is equal to the labor share in the consumption sector; the fraction of total labor supply employed in the investment sector is equal to the capital share in the consumption sector; the fraction of total capital supply employed in the consumption sector is equal to the labor share in the investment sector; and the fraction of total capital supply employed in the investment sector is equal to the capital share in the investment sector.

Further, it is immediate from first-order conditions (see equations (2) in footnote 8) and from equations ( 12) to ( 15) that consumption is equal to the macroeconomic labor share of income and that investment is equal to the macroeconomic capital share of income at the golden rule steady state 7 :

c = l C f k C = w p C (16) i = l I g k I = rk p I (17)
And so:

(y ≡) p C c + p I i = w + rk (18)
In extensive form, we can write [START_REF] Pakko | Investment-Specific Technology Growth: Concepts and Recent Estimates[END_REF] as:

(Y ≡) p C C + p I I = wL + rK,
which reflects the fact that labor plus capital income equals product at the macroeconomic level.

Consequently, the golden rule property that the saving rate is equal to the macroeconomic relative share of capital 8 also holds:

s ≡ p I i p C c+p I i = rk w+rk (19) s = k/ω 1 + k/ω . ( 20
)
2 The neutrality of technological shocks in the consumption goods sector

Imagine that the economy is initially on its steady state characterized by [START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF] if θ = 0 or by [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF] if θ = 0. What happens when the economy experiences a permanent technological shock in the consumption-sector? If this shock is Hicks-neutral -i.e. if it turns the technology F K C , L C into BF K C , L C -then basically nothing. It is apparent in [START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF] and in [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF], that quasi-golden rule and pure golden rule steady-state values of ω, k C , k I , l C , l I , and k are not affected by a Hicksneutral, consumption-specific technological shock. Therefore, steady-state consumption increases (if the shock is positive) but the distribution of inputs, as well as the capital-labor ratio, remain unaltered. In particular, the economy does not react to consumption-specific technical progress by capital accumulation 9 .

Therefore, we have the following neutrality theorem: 7 This proposition notably makes the model that suppose that wages are consumed and profits are invested not so absurd, even if it is unpleasant to take the shortcut to make arbitriraly this kind of assumption. It is not really clear why this double duality (wages-consumption, capital-investment) prevails, but it is a tenacious conclusion of the golden rule. [START_REF] Srinivasan | Optimal Savings in a Two-Sector Model of Growth[END_REF] shows that this property is no longer true when the social planner has a rate of preference for the present strictly less than unity; in that case, savings are less than non-wage income. 8 See Phelps (1961) and (1965) and Robinson (1962) for the original one-sector formulation, and [START_REF] Srinivasan | Optimal Savings in a Two-Sector Model of Growth[END_REF] on the two-sector version. 9 As steady-state ω is not altered by a shock on B, w and r increase proportionally.

Theorem 2.1. Let B denote total factor productivity in the consumption sector and c denote quasi-golden rule (or pure golden rule) steady-state consumption per capita level. Then:

B c ∂c ∂B = 1. (21)
This important property was missed by the models focusing on economy-wide technical change versus investment-specific technical change, like [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF][START_REF] Greenwood | The Role of Investment-Specific Technological Change in the Business Cycle[END_REF] or Pakko (2000).

To our knowledge, [START_REF] Kimball | Proof of Consumption Technology Neutrality[END_REF] was the first to prove, in a dynamic framework with specific utility functions (i.e. logarithmic and power), the result of the neutrality of consumption-specific technical change 10 . The result presented here supplements Kimball's theorems in a static framework and in this case, the result does not hinge on any specific assumption about preferences. Coupling both approaches naturally yields the following interpretation: consumption-specific productivity only influences the optimal growth path of the economy through the choices of intertemporal substitution of consumption, and not via any 'first-order' effect.

More intriguing is the fact that the shape of the production function of the consumption good does not influence the wage-rental ratio 11 . A direct consequence is that capital-augmenting or labor-augmenting technical change can happen in the consumption industry, without any influence on the wage-rental ratio 12 .

The relative price of investment and consumption goods p ≡ Bf k C (ω) /g k I (ω) also notably increase one-for-one with B, as shocks on consumption-specific productivity do not influence the steady-state wage-rental ratio.

Technological shocks in the investment goods sector

What happens to steady-state consumption after a (permanent) positive technological shock on the investment technology? Such a shock makes the maintenance of the existing stock of capital less resource-demanding, and thus expands steady-state production possibilities and steady-state consumption.

So, after an investment-specific technical shock, final consumption increases through capital accumulation in the consumption sector. The quantitative response depends on the planner's subjective discount rate: the more patient the planner is, the more we can expect him to take advantage of investment-specific technical progress. We escape from this difficulty by deriving comparative statics properties only for the pure golden rule steady state. This makes algebra easy and gives insight into the effect of investment-specific technical progress on the production possibilities of the economy. 10 See also Fisher (1997, footnote 8) and [START_REF] Basu | Sector-Specific Technical Change[END_REF]. 11 Still, the shape of function f matters for equilibrium capital-labor ratio in the consumption sector k C as can be seen in [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF]. Consequently, steady-state values of l I , l C , and k depend on the exact shape of the consumption production function. 12 As is apparent in equations ( 8) and [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF], the distribution of inputs across sectors as well as the overall steadystate capital-labor ratio will adjust to a non-Hicks neutral technological shock within the consumption industry.

Some general properties

When the TFP of the investment sector is A > 0, the golden rule steady-state values of ω, k I , k C , and k are determined through the following system 13 :

       Ag k I = δ f (k C ) f (k C ) -k C = g(k I ) g (k I ) -k I = ω k C -k k C -k I Ag k I = δk (22)
Notice that Ag/Ag = g/g , which reflects the invariance of the distribution of income in one particular industry to a Hicks-neutral technological shock. The wage-rental ratio, here, increases because the overall capital-labor ratio increases, and the two sectoral capital intensity ratios increase as well.

Theorem 3.1. ∂ω ∂A = g k I Ag (k I ) = g k I δ > 0 ( 23
)
and so the golden rule wage-rental ratio ω is a strictly increasing function of A. Also ω

A→0 ---→ 0 and ω A→∞ ----→ ∞.
Proof. The value of the derivative of ω with respect to A is obtained by differentiating the first equation in [START_REF] Srinivasan | Optimal Savings in a Two-Sector Model of Growth[END_REF] with respect to A, and using (5):

g k I = δ A ⇒ ∂ω ∂A ∂k I ∂ω g k I = -δ A 2 ⇒ ∂ω ∂A = g(k I ) δ
Limits of ω(A) directly derive from the Inada conditions on g.

The effect of Hicks-neutral, investment-specific technical shock on golden rule steady-state consumption is summarized by the following theorem: Theorem 3.2. Let c denote the pure golden rule steady-state consumption per capita level. Then:

A c ∂c ∂A = k ω ( 24
)
where k is the macroeconomic golden rule capital-labor ratio and where ω is the golden rule wage-rental ratio.

Proof. In virtue of ( 12), we have that

c = l C f k C = f k C -k C f k C at
golden rule steady state. Therefore:

∂c ∂A = -k C ∂k C ∂ω ∂ω ∂A f k C (25)
Differentiation of the first equation in [START_REF] Srinivasan | Optimal Savings in a Two-Sector Model of Growth[END_REF] with respect to A yields:

- ∂ω ∂A ∂k I ∂ω g k I = δ A 2 (26)
In view of ( 4) and ( 23) and of the golden rule identities ( 13) and ( 14), we can successively re-write [START_REF] Whelan | New Evidence on Balanced Growth, Stochastic Trends, and Economic Fluctuations[END_REF] as:

∂c ∂A = 1 A k C f 2 (k C ) f (k C ) g(k I ) g (k I ) = 1 A l I g(k I ) g (k I ) f k C = 1
A kf k C Finally, remark that, in virtue of (3) and ( 12):

f k C l C f (k C ) = 1 ω ,
which gives the final result:

A c ∂c ∂A = k ω .
From equations ( 16) and ( 17), the ratio k ω is also equal to the ratio between investment and consumption, when expressed in the same unit:

k ω = rK wL = p I i p C c = p I I p C C
= s 1-s . Intuitively, maybe one would tend to think that, because of decreasing marginal returns to capital in the consumption sector, consumption cannot increase asymptotically solely as result of technical progress in the investment sector, and so the ratio k ω should tend to zero as A tends to infinity. Under this scenario, investment-specific technical progress barely makes the existing stock of capital less costly to maintain, freeing resources for the consumption industry, until the consumption sector asymptotically employs the whole of capital and labor inputs. According to this intuition, notably, l I shall tend to zero as investment-specific TFP goes to infinity. But this intuition is not always true. In principle, k/ω, can either increase, decrease, or remain constant when A tends to infinity, and consequently the ratio does not need to tend to zero.

The evolution of the relative price of investment and consumption goods p = f (k C (ω)) Ag (k I (ω)) after an investment-specific productivity shock is such that:

Theorem 3.3. A p ∂p ∂A = - k k C (27)
Proof. The denominator in the expression of p is constant and equal to δ at steady state. Then, using ( 4) and ( 23):

∂ ln(p) ∂A = ∂ ln(f (k C )) ∂A = ∂ω ∂A ∂k C (ω) ∂ω f (k C ) f (k C ) = -g(k I ) Ag (k I ) f (k C ) f (k C )
And so, from golden rule identities ( 13) and (15):

A p ∂p ∂A = 1 l I f (k C ) f (k C ) -Al I g(k I ) Ag (k I ) = -k k C
Thus, the price of investment goods in terms of consumption goods always decreases after an investment-specific technological shock. If the consumption sector is relatively more capitalintensive than the investment sector -that is, if k C (ω) > k I (ω) for all ω > 0 -then the elasticity of p with respect to A is greater than -1. In the contrary, if it is the investment sector that is capital-intensive, then this elasticity is less than -1 and relative price overshoots the size of the shock.

Investment-specific technological shocks and the distribution of inputs

How do investment-specific technical shocks affect the steady-state distribution of inputs across sectors? Let's first define the sectoral elasticities of substitution by:

σ C (ω) = - f f -k C f k C f f (28) σ I (ω) = - g g -k I g k I gg (29)
We can reasonably suppose that σ C (ω) and σ I (ω) are bounded.

Remark that those elasticities are invariant to Hicks-neutral technical shocks. Further, it is immediate by combining (3), ( 4) and ( 5) with ( 28) and (29) that:

∂k C ∂ω = σ C k C ω ( 30
)
∂k I ∂ω = σ I k I ω (31)
So, consequently:

∂ ln k C ω ∂ω = σ C -1 ω (32) ∂ ln k I ω ∂ω = σ I -1 ω (33)
An increase in ω will increase (resp. decrease) the relative share of capital in industry X if the elasticity of substitution between capital and labor in industry X is greater (resp. less) than unity.

By combining (3) and the golden rule identities from ( 12) to [START_REF] Litina | Do Inada conditions imply that production function must be asymptotically Cobb-Douglas? A comment[END_REF], it is clear that the following identities hold at steady state:

l I l C = k C ω (34) l I k I l C k C = k I ω ( 35
)
Consequently we can state the following proposition:

Proposition 3.1. After a positive Hicks-neutral investment-specific technical shock:

-labor is transferred from the investment industry to the consumption industry if σ C < 1;

-labor is transferred from the consumption industry to the investment industry if σ C > 1;

-capital is transferred from the investment industry to the consumption industry if σ I < 1;

-capital is transferred from the consumption industry to the investment industry if σ I > 1.

The sustainability of growth regimes led by investmentspecific technical change

As noted previously, investment-specific technological progress makes final consumption grow solely as a result of capital accumulation within the consumption sector. Intuition suggests that this pattern must stop somewhere: if the Inada conditions are fulfilled, doesn't marginal productivity of capital in the consumption sector fall to zero, and so marginal benefits from investment-specific productivity growth?

We already know from the existing literature that this intuition is wrong. Meade (1962) and [START_REF] Whelan | New Evidence on Balanced Growth, Stochastic Trends, and Economic Fluctuations[END_REF] have shown that, when both production functions F and G are of Cobb-Douglas type, consumption eventually does increase asymptotically with only investment-specific technical change. In this case, real consumption and investment grow at constant, however different, rates; and nominal consumption and investment grow at the same constant rate14 .

The macroeconomic elasticity of substitution

The results presented here suggest that whether investment-specific productivity growth will bring less and less consumption benefits depend on the evolution of the k/ω ratio, that is on the ratio of capital to labor share at the macroeconomic level. When investment-specific productivity growth makes this ratio decrease -that is, when the macroeconomic share of capital decrease relatively to the macroeconomic share of labor -consumption growth will continuously slow down. When this ratio increases, consumption growth will tend to accelerate 15 .

Like shown in (30) and (31), the evolution of the sectoral ratios k C /ω and k I /ω depend on the sectoral elasticities of substitution between labor and capital. The ratio k/ω is a weighted average of those two quantities.

Whether the ratio k/ω goes up or down as the wage-rental ratio increases depends on the economy-wide elasticity of substitution σ, which governs the distribution of value added between labor and capital at the macroeconomic level. We define σ(ω) by:

∂k ∂ω = σ k ω . ( 36
)
Around the golden rule steady state, we can link σ to the value of the sectoral elasticities of substitution.

Proposition 4.1.

σ = l C (σ C -1) + l I k I k (σ I -1) + 1 (37)
Proof. From the golden rule identities ( 13) and ( 15) we can write the golden rule value of k as:

k = k C f (k C ) f (k C ) g(k I ) g (k I ) (38)
Log-differentiating this expression with respect to ω and using (4), ( 5), ( 6), ( 13), ( 15), (30), (31), (34) and (35) yields:

∂ ln k ∂ω = 1 k C ∂k C ∂ω + ∂k I ∂ω g (k I ) g(k I ) -∂k I ∂ω g (k I ) g (k I ) -∂k C ∂ω f (k C ) f (k C ) + ∂k C ∂ω f (k C ) f (k C ) = σ C ω + σ I k I ω g (k I ) g(k I ) + g (k I ) g(k I ) -σ C k C ω f (k C ) f (k C ) -f (k C ) f (k C ) = σ C ω + σ I ω l I k I k + l I k -σ C ω l I -l I k C = σ C ω + σ I ω l I k I k + 1 ω l C k C k -σ C ω l I -l C ω = 1 ω l C (σ C -1) + l I k I k (σ I -1) + 1 .
Remark that, if σ C and σ I are both greater (resp. less) than 1, then σ is greater (resp. less) than 1 as well.

If σ > 1 then k/ω increases as A increases, and consecutive investment-specific productivity shocks bring increasing consumption rewards. If σ < 1 then k/ω decreases and consumption growth diminishes.

'Long-term' consequences of investment-specific technical change

When both production functions satisfy Inada conditions

It is often believed that the long-run elasticity of substitution is equal to one 16 . Formally, [START_REF] Barelli | Inada conditions imply that production function must be asymptotically Cobb-Douglas[END_REF] and [START_REF] Litina | Do Inada conditions imply that production function must be asymptotically Cobb-Douglas? A comment[END_REF] have shown that, for all production function that fulfills the Inada conditions, the elasticity of substitution must be equal to one in the long run. As an application of this property, we can state the following theorem:

σ C (ω) ω∞ --→ 1 σ I (ω) ω∞ --→ 1
Because l C and l I k I /k are bounded, σ also tends to 1 as A tends to infinity. So, if both production functions satisfy the Inada conditions, then:

ω k ∂k ∂ω A∞ --→ 1,
which means that, asymptotically, ω and k increase proportionally after an investment-specific shock, so that k/ω tends to some constant, like in the Cobb-Douglas case, and consumption grows asymptotically with investment-specific technical change 17 .

When the production function of the consumption does not satisfy Inada conditions

We have supposed throughout the paper that both production functions verified Inada conditions, but the existence of the golden rule only needs Inada condition to hold in the investment sector. In virtue of the Barelli-Pessôa-Litina-Palivos theorem, this implies that the elasticity of substitution of production function G must be asymptotically one 18 .

Let's drop the Inada conditions for the consumption sector.

First, suppose that σ C is less than unity, in the precise sense that there exists some ε > 0 such that, for all ω, σ C < 1 -ε. In this case, function f (k C ) tends to a constant (and no more to infinity) as k C tends to infinity 19 , and so investment-specific technical change alone can never increase consumption asymptotically.

It is immediate from (30) that k C /ω tends to 0 as investment-specific productivity A tends to infinity. That is, the relative share of capital in the consumption sector eventually vanishes. But from (34), we see that l I → 0 and l C → 1, and so the macroeconomic elasticity of substitution in (37) is such that:

σ(ω) -σ C (ω) ω∞ --→ 0
In this regime, consumption does not increase asymptotically as investment-specific productivity tends to infinity. But the economy takes advantage of investment-specific TFP growth by 16 See Jones(2003). 17 See appendix. 18 Litina and Palivos (2008) show that this does not imply that function g must be asymptotically equivalent to some Cobb-Douglas function. 19 See de La Grandville (2009, chapter 4).

freeing up labor resources from the investment to sector to allocate them to the consumption sector 20 . But as the macroeconomic ratio k/ω tends to zero, the relative share of capital tends to nil. When capital and labor are complements in the consumption sector, productivity increases within the investment sector ultimately impoverish those who own capital. In this regime, investmentspecific technical change does not contribute to growth asymptotically but is a powerful engine of equality.

If σ C is greater than unity, then k/ω tends to infinity, and so the macroeconomic relative labor share vanishes as A tends to infinity. Also l I → 1 and l C → 0, and so the asymptotic macroeconomic elasticity of substitution tends to 1. In this case, despite the fact that the elasticity of substitution is always greater than one, the observed macroeconomic elasticity of substitution tends to one exactly.

Conclusion

This article put in regards the consequences of consumption-and investment-specific technical change in a neoclassical, optimal growth framework.

Consumption-specific and investment-specific technical change are largely orthogonal. Consumptionspecific shocks barely affect consumption, and virtually anything else, while the dynamic response of the economy to investment-specific shocks depends on the sectoral elasticities of substitution between labor and capital inputs. Different regimes exist according to whether elasticities are greater or less than one. Unfortunately, to our knowledge there does not exist any estimation of the sectoral elasticities of substitution 21 .

What are the real-world insights that one could gain from this results with regards to modern growth phenomenon? The results we have presented yield some predictions about the links between technology and input and output prices and quantities, which could lead to some interesting empirical view on current and past growth regimes. In the perspective given in the paper, one can speculate that China has grown so fastly for several decades because productivity growth was faster in the investment industry than in the consumption industry -and it seems that it has, at the global level -while concomitantly the Chinese economy has continuously showed a high ratio of domestic investment to domestic consumption, thus accentuating the effect of investment-biased technical change on product and income relatively to what occurred in other countries.

The first-best results presented may also serve as a benchmark for further theoretical research. In particular, when it comes to the deep nature of investment in the growth process 22 , it would constitute an interesting extension to analyze the dynamical consequences of investment-specific technical progress when some friction distorts the investment decisions. 20 Remark that, from ( 13) and [START_REF] Litina | Do Inada conditions imply that production function must be asymptotically Cobb-Douglas? A comment[END_REF], the allocation of capital input across sectors stays exactly the same. The investment sector is, then, ultimately more capital-intensive than the consumption sector. 21 ), using a one-sector framework, shows that the elasticity of substitution has tended to during the last decades in most industrialized countries. 22 See the discussion in the present introduction.

Appendix: steady growth and the golden rule in the Cobb-Douglas case

In this appendix we build on the description of the steady growth path by Meade (1962) and [START_REF] Whelan | New Evidence on Balanced Growth, Stochastic Trends, and Economic Fluctuations[END_REF] We suppose that the production technologies are now:

c = Bl C k C b (39) i = Al I k I a (40)
with 0 < a < 1 and 0 < b < 1. The effect of investment-specific technical progress on golden rule steady-state consumption is now characterized through a very simple equation. Proposition 5.1. At the golden rule steady state, with Cobb-Douglas production functions, one has that:

k ω = A c ∂c ∂A = b 1 -a . (41)
Proof. Efficiency equation ( 3) is now an equation of proportionality:

ω = 1 -a a k I (42)
From equation ( 13), we have that l I = b at the golden rule, and so equation ( 14) with Cobb-Douglas production functions yields:

k = l I k I a a (k I ) a-1 = b a k I = b 1 -a ω.
Therefore, when production functions are of Cobb-Douglas type the elasticity of final consumption with regards to investment-specific TFP is constant. Unsurprisingly, this figure is the higher when the production functions F and G are more capital-intensive.

It is beyond the scope of this paper to give a complete characterization of the optimal path of allocation of resources for any initial conditions. Instead, we show a much more pleasant result that supports previous conclusions: if technologies are of Cobb-Douglas type in equations ( 39) and (40), then there exists a rule of thumb for the allocation of resources that, for some initial capital-labor ratio k 0 , makes consumption grow at the gross rate of:

1 + z c = (1 + z B )(1 + z A ) b 1-a , (43)
where 1 + z B = Bt+1 Bt and 1 + z A = At+1 At . This rule of thumb we characterize through the golden rule identities from [START_REF] Greenwood | The Role of Investment-Specific Technological Change in the Business Cycle[END_REF] to [START_REF] Litina | Do Inada conditions imply that production function must be asymptotically Cobb-Douglas? A comment[END_REF]. Suppose that the social planner allocates resources according to the following rule, at all dates 23 :

         l C = 1 -b l I = b l C k C k = 1 -a l I k I k = a (
44) 23 Notice that this allocation is always feasible, as it obviously verifies [START_REF] Fisher | Relative Prices, Complementarities and Co-movement among Components of Aggregate Expenditures[END_REF].

which corresponds to sectoral capital intensities:

k C = 1-a 1-b k k I = a b k (45)
The accumulation equation, when inputs are allocated according to this rule, becomes:

k t+1 = (1 -δ)k t + bA 0 (1 + z A ) t a b k t a (46)
Let's first prove the following lemma:

Lemma 5.1. If the initial capital-labor ratio satisfies the following adjusted golden rule:

A 0 g a b k 0 = δ + (1 + z A ) 1 1-a -1 (47)
and if the allocation of resources follows (44) for all dates, a similar adjusted golden rule is verified at all dates:

∀t, A t g a b k t = δ + (1 + z A ) 1 1-a -1 (48)
Proof. If (48) is satisfied for some t ≥ 0:

A 0 (1 + z A ) t a a b k t a-1 = δ + (1 + z A ) 1 1-a -1.
Then, from (46): Therefore, if the property holds for t = 0, then it holds for all t ≥ 0.

A t+1 g a b k t+1 = A 0 (1 + z A ) t+1
We now prove the following existence theorem: Proof. If initial capital-labor ratio is at the level given by equation (47), and if the golden rule of thumb of allocation of resources is followed, (48) holds and so equation (46) can be reduced to:

k t+1 = (1 -δ)k t + aA 0 (1 + z A ) t a b k t a-1 k t = 1 -δ + δ + (1 + z A ) 1 1-a -1 k t = (1 + z A ) 1 1-a k t .
Consequently, for all t:

k t = k 0 (1 + z A ) 1 1-a t . (49)
Investment per capita, in this case, reduces to:

i t = A t l I t a b k t a = A 0 (1 + z A ) t a b k 0 (1 + z A ) 1 1-a t a = i 0 (1 + z A ) 1 1-a t .
From (39), (44), ( 45) and (49), we can finally write that:

c t = (1 -b)B 0 (1 + z B ) t 1-a 1-b k 0 (1 + z A ) 1 1-a t b = (1 -b)B 0 1-a 1-b k 0 b (1 + z B )(1 + z A ) b 1-a t = c 0 (1 + z c ) t .
Paradoxically, the growth rate of real consumption and investment, but also of the stock of capital, do not depend on the depreciation rate δ.

Along this 'pseudo-balanced' path, with only investment-specific technical progress (z A > 0 and z B = 0), the ratio of real consumption to real investment and the ratio of real consumption to real capital-stock both tend to zero when as time goes on. With both types of technical progress, consumption and investment do not generally grow at the same rate. Indeed, the only balanced path eventually arises when:

(1 + z A ) 1-b = (1 + z B ) 1-a . ( 50
)
Perhaps surprisingly, if we let p denote the relative price of the investment good in terms of the consumption good, the quantities pi and pk grow at the same rate than c. Equality of technical marginal rates of substitution across sectors implies that the relative price can be written as 24 :

p = Bf k C Ag (k I ) . (51)
Along the growth path of theorem 5.1, the gross rate of growth of p is constant and equal to:

1 + z p = (1 + z B ) (1 + z A ) -1-b 1-a , (52)
which can be more or less than unity, and is exactly one if and only if technical progress in both sectors satisfy condition (50) 25 . One important conclusion is that the relative price of the investment good may rise without technical progress in the investment industry being steadier, and conversely. 24 See (2). 25 See equation (3.3).

Figure 1 :

 1 Figure 1: Macroeconomic ratio of real investment to real private consumption, 1947-I to 2007-IV (source: Bureau of Economic Analysis, NIPA database). Grey areas represent recessions according to NBER.

Figure 2 :

 2 Figure 2: Ratio of investment deflator to private consumption deflator, 1947-I to 2007-IV (source: Bureau of Economic Analysis, NIPA database). Grey areas represent recessions according to NBER.

Theorem 4 . 1 .

 41 (Unit long-run elasticity of substitution in both sectors) If functions F and G satisfy the Inada conditions, then:

  Basu et al. (2010) provide estimates of the capital intensities of the consumption sector and of the investment sector in the USA for the period from 1961 to 2005. De La Grandville (2009, chapter 5

a a b a- 1 ( 1 -k t a a- 1 =a a b k t a-1 k t a- 1 =- 1 1 1 1-a -1 k t a- 1 = 1 = δ + ( 1 + z A ) 1 1

 1111111111 δ)k t + bA 0 (1 + z A ) t a b A 0 (1 + z A ) t+1 a a b a-1 (1 -δ)k t + A 0 (1 + z A ) t A 0 (1 + z A ) t+1 a a b a-δ + δ + (1 + z A ) A 0 (1 + z A ) t a a b a-1 (k t )a--a -1.

Theorem 5 . 1 .

 51 If the production functions of the two sectors are of Cobb-Douglas type like in (39) and (40) and if technical progress in each sector is Hicks-neutral and increases at steady rate, with Bt+1 Bt = 1 + z B and At+1At = 1 + z A for all t ≥ 0, then there exists a growth path where investment and the capital stock grow at the same constant gross rate of (1 + z A ) 1 1-a , and where consumption grows at the constant gross rate of (1 + z B )(1 + z A ) b 1-a .

[START_REF] Kimball | Proof of Consumption Technology Neutrality[END_REF] shows that those effects cancel out in the dynamic framework if utility is logarithmic.

Notice that because production function G satisfies the Inada conditions, there exists a maximum sustainable capital stock defined by G K, L = δK.

[START_REF] Senouci | Optimal Growth and the Golden Rule in a Two-Sector Model of Capital Accumulation[END_REF].

See Phelps (1961, 1965) and[START_REF] Cass | Optimum Growth in an Aggregative Model of Capital Accumulation[END_REF] for the original one-sector formulation of the golden rules.[START_REF] Srinivasan | Optimal Savings in a Two-Sector Model of Growth[END_REF],Uzawa (1965), Shell (1967), Haque (1970),[START_REF] Galor | A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System[END_REF] and[START_REF] Cremers | Dynamic Efficiency in the Two-Sector Overlapping Generations Model[END_REF] among others present the golden rule in two-sector frameworks.

Since the rest of this section only deals with the golden rule value of all variables, we omit the star subscript.

In the appendix, we present a golden rule adjusted for steady investment-specific technical change in the case the production functions are both of Cobb-Douglas type, through which we derive the same growth rates the ones found byMeade and Whelan.

Notice that this property has something of a paradox, as consumption at steady state is exactly equal to the macroeconomic share of labor.


Ning Ge
email: ning.ge@enseeiht.fr

Marc Pantel
email: marc.pantel@enseeiht.fr

Xavier Crégut
email: xavier.cregut@enseeiht.fr

Time Properties Dedicated Transformation from UML-MARTE Activity to Time Petri Net

Critical Real-Time Embedded Systems (RTES) have strong requirement regarding system's reliability. UML and its profile MARTE are standardized modeling language that are getting widely accepted by industrial designers to cope with the development of complex RTES. Relying on Model-Driven Engineering (MDE), critical time properties' verification in UML-MARTE model at early phases of the system lifecycle becomes possible. However, many challenges still exist. A key challenge is to eliminate the gap between UML semiformal semantics and fully formal executable semantics using model transformation. The model transformation must ensure on the one hand the consistency between high-level user dedicated models and lower-level verification dedicated ones, and on the other hand that the subsequent verification is not too expensive and can be applied to real size industrial models. This paper presents an approach to translate UML-MARTE Activity Diagrams to Time Petri Net (TPN) with the aim of verifying efficiently time properties. This work is under the framework of the UML-MARTE Model Checker which is dedicated to verifying time properties (synchronization, schedulability, boundedness, WCET, etc.) in RTES. This contribution focuses on how to define the TPN formal semantics to avoid the core problem of state space explosion in model checking. The proposed method is validated using a representative case study. Experimental results are given that demonstrate the method's performance.

Introduction

As the complexity of RTES increases, requirements for system's reliability become more and more stringent. The challenges are to guarantee the system reliability while ensuring that the functional requirements are completely met with less trade-off for system design's efficiency. Model-driven Engineering (MDE) allows an early integration of feasibility analysis in the conception phase, and enables a rapid iterative design-prototype cycle. The core issue of MDE for RTES is how to rapidly validate and verify that the system's behavior matches the specification, especially for the temporal aspect.

UML (Unified Modeling Language) and its profile MARTE (Modeling and Analysis of Real Time and Embedded systems) are standardized modeling languages that are getting widely accepted by industrial designers to cope with the development of complex RTES. But, properties cannot be verified directly using the UML semi-formal semantic. A trans-lation from UML into a formal domain must be performed where the key challenge consists in expressing the temporal semantics between the user high level models and the verification low level formal ones while keeping the verification scalable enough to be applied to real industrial models.

This paper presents an approach to translate UML-MARTE Activity Diagrams to Time Petri Net (TPN), which is driven by the requirement that the time properties' verification needs to be conducted in a practical time scale. This work is under the framework of UML-MARTE Model Checker that is dedicated to verifying time properties (synchronization, schedulability, boundedness, WCET, etc) in RTES. This contribution focuses on how to define TPN based formal semantics for UML-MARTE Activity Diagrams to avoid the core problem of state space explosion in model checking. TPN is selected as verification model, not only because of the maturity of both its theory and the associated TINA toolset [START_REF] Berthomieu | The tool tina -construction of abstract state spaces for petri nets and time petri nets[END_REF], but also because of its powerful capacity to express temporal semantics.

This paper is organized as follows: Section 1 introduces some preliminaries. Section 2 compares this work with related ones. Section 3 presents a brief overview of UML-MARTE Model Checker. Section 4 describes the methodology. Section 5 presents the transformation approach. Section 6 evaluates the proposed approach by illustrating experimental results and discussing the method's performance based on a case study. Section 7 gives some concluding remarks.

Preliminaries

UML Activity Diagram

UML Activity [START_REF]OMG Unified Modeling Language TM (OMG UML), Superstructure[END_REF] modeling emphasizes the sequence and conditions for coordinating lower-level behaviors, rather than which classifiers own those behaviors. These are commonly called control flow and object flow models. The actions coordinated by activity models can be initiated because other actions finish executing, because objects and data become available, or because events occur external to the flow.

MARTE

MARTE [START_REF]UML Profile for MARTE: Modeling and Analysis of Real-Time Embedded Systems Version 1[END_REF] is a UML profile standardized by the OMG. It adds capabilities to UML for model-driven development of RTES. MARTE provides foundations for modelbased description of RTES. These core concepts are then refined for both modeling and analysis concerns.

Time Petri Net

Time Petri Net [START_REF] Merlin | Recoverability of communication protocols-implications of a theoretical study[END_REF] model is an extension of Petri Net which allows the symbolic expression of execution time. A Time Petri Net T is a tuple (P, T, E, W, M 0 , (α, β)), where

• P represents a finite set of places,

• T represents a finite set of transitions,

• E represents the edges by E ⊆ (P × T) ∪ (T × P),

• W represents the weight of the edges,

• M 0 represents the initial marking,

• α ∈ (Q ≥ 0) and β ∈ (Q ≥ 0 ∪ ∞) respectively represent
the earliest and latest firing time constraints.

Related Works

According to the OMG, UML 2 Activity models are based on Petri Net semantics making them the best choice as tools for system behavior description and for property verification. Many transformation approaches from UML Activity Diagrams (AD) to Petri Nets exist. They rely on Colored Petri Net (CPN) [START_REF] Spiteri | Intuitive mapping of uml 2 activity diagrams into fundamental modeling concept petri net diagrams and colored petri nets[END_REF], Stochastic Petri Nets (SPN) [START_REF] Pablo López-Grao | From uml activity diagrams to stochastic petri nets: application to software performance engineering[END_REF], Instantiable Petri Net (IPN) [START_REF] Thierry-Mieg | Uml behavioral consistency checking using instantiable petri nets[END_REF], Timed Color Petri Net with Inhibitor Arc (TCPNIA) [START_REF] Yang | Mapping uml activity diagrams to analyzable petri net models[END_REF], or Time Petri Net (TPN) [START_REF] Andrade | A methodology for mapping sysml activity diagram to time petri net for requirement validation of embedded real-time systems with energy constraints[END_REF].

[Sta08] presented an intuitive conversion from UML-AD into CPN. It focused on how to ease the understanding of models whilst the fundamental features are retained.

[LGMC04] aimed to translate UML-AD into SPN and applied it in Software Performance Engineering (SPE) process to measure system's performance.

[TMH08] defined IPN and translated UML model into IPN models with the purpose of verifying the correctness and inter-diagram consistency of UML diagrams.

[YYSQ10] maps UML-MARTE AD into TCPNIA to provide a possible foundation for analyzing non-functional properties of system. But it does not propose the corresponding analysis approach, which remains open according to our understanding. Thus, we cannot evaluate the performance of this mapping method in verification aspect.

[AMCN09] is an approach relying on TPN as verification model. It proposed a methodology aiming to map SysML Activity Diagrams to TPN with energy constraints (ETPN) to estimate the energy consumption and execution time of the system. Compared with the other works, it dealt with the temporal semantics in UML, although it only estimated execution time of system. However, its relative static transformation method leads to a limit for its extension capacity to cover more properties.

To summarize these translation approaches, none of them aimed to verify time properties for safety critical RTES, which means that, to our knowledge and understanding, time property verification dedicated transformation from UML-MARTE Activity to Petri Net has not been dealt with so far.

Overview of UML-MARTE Model Checker

The goal of the UML-MARTE Model Checker1 (Figure 1) is to verify whether a UML-MARTE RTES Model satisfies the designed Time Specifications or respects the required Time Constraints. It takes the System Model and Time Specification/Constraint as input. The System Model consists of two concerns: Behaviour Model and Architecture Model. The former defines how the system will act and respond to its outside world while the latter describes the interconnection relation between sub-components of the system. The Time Specification/Constraint consists of two categories: functional and non-functional. The former concerns whether system's output value is consistent with what it is designed for and whether the output is generated at desired time, while the latter focus on the performance requirements, etc.

System Model and Time Specification/Constraint are respectively transformed to associated semantic components at TPN level through Behavior/Structure Mapping and Time Specification/Constraint Mapping approaches. All the transformations are performed automatically and the executable formal model is hidden to the end user.

After TPN Optimization, the formal verification is performed using the generated TPN Model and Generated Quantitative Timing Property, in which a dedicated iterative observer-based model checking method and the TINA toolset are applied. Finally, Computation is performed with Quantitative Results and Formal Time Specification/Constraint to get the target Time Property Verification Result used to refine the original design.

We have proposed efficient approach for each core issue in the whole framework. In this paper, we focus on the approach Behavior/Structure Mapping of UML Activity.

Methodology Description

General Transformation Pattern

In order to automate the assembly of the TPN elements transformed from UML, a general pattern (Figure 2) of target TPN is predefined. Only the TPN elements (transition, place, arc) in real line are belonging to the transformed result of given AD elements; those in dotted line are the mapping result of other related AD elements.

Transformation Principle

Our translation approach from UML-MARTE-AD is driven by the assessed time properties. The approach should respect the following 6 principles.

1. This transformation approach corresponds to the TPN abstraction without data, which means the models are data independent.

2. The transformation of one UML-AD element may be different according to the time property to be assessed.

3. For some intuitive elements not influencing time properties, the translated TPN semantic can be standardized and homogeneous for all the property.

4. The transformation should guarantee the consistency of the semantics between high-level user model and lowerlever verification model. However, a correct transformation here does not imply a 100% semantic preservation, but rather to make sure that the semantics necessary for the properties to be assessed are preserved during the model transformation.

5. The generated TPN models should be able to perform a highly efficient verification of time properties, especially in large scale asynchronous applications.

6. The transformed elements should facilitate the assembly, which may cause the performance a little degraded than manual modelled one.

Mono-clock/Multi-clock Based System

For RTES components, execution can be driven by the same clock or by different clocks. The main difference between these two scenarios for time properties' verification is that clock drifts must be taken into account in multi-clock environment. In mono-clock context, it is not mandatory to distinguish the notion of tick and tick cycle, because the difference between tick duration and physical time is of the same proportion at any given time. If a clock drift occurs, it is also effective for every component in the system. In a multi-clock based system, however, the model transformation need to be compatible to present the correct polychronic semantics of clock-driven system as well as the clock drift. The main idea is to assume a global physical clock and project each time consumption and drift on this precise time reference. In our study, we use strictly the physical time notion as the exact reference for both mono and multi-clock based system. On the other hand, as the physical time is dense and the verification tools rely on dense symbolic time, our approach can handle both dense time problem together with discrete time.

Resource Scheduling

The transformation of some well-known scheduling algorithm to TPN models will often introduce some semantic ambiguities. Moreover, the exact behaviour of some dynamic scheduling algorithm could not be modelled by TPN in trivial ways. To cope with the above problems, our transformation method for resource proposes a universal scheduling algorithm for both non-preemptive scheduling and pre-emptive scheduling rather than a specific one.

Transformation Rules

We present transformation rules for control nodes, action, resource, object, and connections in UML Activity.

Control Nodes

The mapping of some control elements is intuitive, because TPN has the semantic equivalence of these control characteristics (branch, concurrence, sequence, etc...). In addition, the semantics of some control nodes are dual. So the TPN modeling is also dual for these elements.

Initial Node & Flow Final Node

Initial node and flow final node (Figure 3) are dual elements for flow token control. The dual characteristic is that one emits the control token, and one destroys the control token.

As in UML-AD, an initial node could not have predecessor and a flow final node could not have successor, a derivation to the general pattern of the TPN model transformation occurs: initial node does not have C_IN and flow final node does not have C_OUT.

Activity Final Node

Activity final node (see Figure 4) requires terminating all the activity flow immediately and destroying all control tokens, while in TPN the later one could not be modelled perfectly.

The compromise we make in this paper is only considering to terminate all the activity flow once the activity final node receives the control token. In TPN, this "sudden exit" is implemented by using inhibitor arc to link to all the transitions in this mapping TPN. As the mass usage of inhibitor will potentially degrade the computation performance in TPN model checker, we strongly recommend using flow final node in UML-AD to model those normally ended flow.

[0,0] C_OUT

C_IN

Activity Final Node

For each transitions in this TPN, an inhibitor arc is linked Fork node and join node (Figure 5) are dual nodes for concurrence control. The common characteristic is that each branch has an "eventually and" relation, which means their execution are eventually concurrent. Decision node and merge node (Figure 6) are dual nodes for branch control. The common characteristic is that each branch has a "or" relation, which means their execution are mutually exclusive.

Decision Node (with Finite Loop)

If the decision node (Figure 7) has loop branches which execute finitely, the bound of loop should be added into the transformation and the reset function of loop bound should be executed for preparing next loop activation once the decision node chooses a non-loop branch (the loop is over). One intuitive doubt with this semantic is that the state space may grow up rapidly along with the increase of loop bound

Action

Action is the fundamental unit of executable functionality. It takes a set of inputs and converts them into a set of outputs. Depending on its abstraction scales, an action could represent a complex processing flow or a primitive one which either carries out a computation or assess object memory. UML-AD defines 55 types of actions. In order to focus on the core semantics concerned by this paper, we generalize the concept and harmonize it with the UML-AD usage. Then we introduce the transformation semantics separately for the mono-time based systems and the multi-time based systems.

Action Semantic Pattern

An action is an n-tuple of (I, C, T , R, D), in which:

• I refers to identification, which is derived from its behavior semantics. Only two actions with exactly the same behavior can have the same identification.

• C refers to context. An action's behavior model could be the same but if the behavioral context is different, then it should be labelled with the same identification but different context.

• T refers to time measure. In this paper, only the minimum and maximum execution time are considered.

• R refers to resource usages. The execution of an action will go on only when its required resources are ready and allocated to it. More precisely, the resource usages is a set of < R, N >, which indicates that for a given resource type R, the action requires N of its available instances.

• D refers to data section. It contains both inputs and outputs. It should be clear that the data modeling in AD is value-independent; it only makes sense for data dependency and data type definition.

The transformation approach is illustrated by Figure 8.

• All input resource should be linked to A; All output resource should be linked to D.

• All input data-related flows should be linked to B, if in the action there is an <InputPin>, or connected by an <ObjectLink>; All output data-related flows should be linked to C, if in the action there is an <OutputPin>, or connected by an <ObjectLink>.

• The execution time is modelled by transition C.

Mono-Clock Scenario Action

For mono-clock actions, the measured execution time is directly used after a global normalization of the time units. For example, if action A takes [3.4 ms, 4.7 ms] and actions B [78.9 us, 463.5 us], the correspondent min time and max time on the TPN transition is [34000, 47000] and [789, 4635] respectively, with the common unit of 0.1 us to keep all the results to be integers.

Multi-Clock Scenario Action

For multi-clock actions, the measured execution time need to be translated first into tick numbers from the global physical clock, and then its physical model time is deduced by associating each clock's drift. We use the same example but give respectively its correspondent clock property: let clock A and B tick theoretically every 1 us, and their backward drift and forward drift are both 1%, therefore action A's tick number is [3400, 4700] and [78.9, 463.5] for action B. As tick number must be integer, a rounding strategy must be designed to handle this problem without introducing unreasonable conversion error. In our study, we use the floor function for t min and ceiling function for t max . Therefore, we have A for [3400, 4700] and B for [78, 464] as tick numbers after the rounding.

As the corresponding tick time range is [0.99 us, 1.01 us] due to the mentioned clock property, the drift-based action physical time duration is calculated by multiplying this range and action's tick number range. Following the same principle of unit normalization, the final min time and max time on the TPN transition is [336600, 4747000] and [7821, 46763] respectively, with the common unit of 0.01 us. Comparing to the same purpose in mono-clock context, it is noticed that under the principle of physical time reference, the main difference is that the model precision is increased.

Resource

Resource Semantic Pattern

In UML-MARTE, more than 10 types of resource are proposed to be used in the modeling. According to the same paradigm of concept generalization:

A resource is a 3-tuple of (I, S, Q), in which:

• I refers to identification, which designates the type of the resource.

• S is the scheduler used to respond the requirement of the resource. A scheduler has two main characteristics: scheduling algorithm and whether it allows pre-emption.

• Q is the instance quantity of the given resource.

For example, a 4-core CPU could be modelled as (CPU-CORE, pre-emptive / Round Robin, 4 ).

Non-preemptive Resource Scheduling

The non-preemptive resource scheduling can be transformed to TPN as shown in Figure 9.

Pre-emptive Resource Scheduling

The TPN with stopwatch is used to cope with the preemptive modeling. In our study, however, it is found that it is very expensive in terms of reachability graph generation when performing the model checking. Therefore, an alternative solution is proposed to avoid this problem. The idea is to use the time slice of preemptive scheduler as the time unit to segregate the action's execution. The transition of execution in pre-emptive scenario has been divided into the structure presented in Figure 10. The place representing the resource will be connected to each transition to model the preemptive scheduling.

Input Ready ...

[A,ts] (0,ts] [ts,ts] (0,

Object

The most important factor in object element transformation is to keep the data dependency information in the TPN. Each data dependency is represented by a TPN place. This transformation approach (Figure 11) has a little drawback however to present the non-transient semantics. The expected behaviour of a non-transient object element (like <DataS-tore>) is once the data is feed, it will never lose. In contrast, the model transformed in TPN will only guarantee that data persists only in this current execution of activity.

Connection

Object flow and control flow are translated directly into TPN transition (Figure 13).

Case Study

The real-time embedded applications of aircraft control system are extremely strict in temporal aspect. A simplified Flight Warning System (FWS) Application (Figure 14) of a well-known aircraft program is introduced as case study to evaluate the proposed approach.

The Sensor (Figure 15) periodically captures at a given rate two categories of data: airborne system data like operational status and availability, and air data like attack angle and relevant air speed. Both of them are sent through AFDX to the redundant calculators CPIOM, where FWS Applications (Figure 16) are running. Once a previewed warning is detected, the Multi-Function Display (MFD) (Figure 17) receives the redundant alerts via AFDX and displays the information to the pilot after comparing the redundant alerts to make sure they are the same.

Assessed Time Properties

The system integrator may want to know by verification:

• Whether the pilot will be informed in time when some abnormal event occurs. This can be answered by knowing the min/max time interval between Data Acquisition of Sensor and Update Display of MFD.

• The designer may want to optimize the sampling rate of sensor, because if sensor updates too frequently the data, the FWS may not be able to handle them correctly due to its computation capacity; if too slowly, the other system like flight control may lose air-plane real-time status.

Transformation & Optimization Result

The transformation approach with property dedicated optimization approach produces the TPN of Figure 18.

Verification Result

We apply an observer based verification approach and use TINA toolset to compute the time properties mentioned above. The verification results and computation time are shown in Table 3.

Performance Analysis

The whole framework will optimise the generated TPN in accordance with time property. The optimization will remove the irrelevant parallelism of the TPN as much as possible because in each verification, we need to only focus on the time properties between two actions. Thus the key factor influencing the computation performance in terms of both time and space is the transformed components in a relatively sequential TPN. In this section, we choose to prove, using the case study, that the transformation method for Decision Node with finite loop branch has a complexity of O(n 2 ), where n is the loop bound. The performance result for other transformed components are ignored because they are all linearly scalable along with the TPN's dimension. The computation time and state class number grows respectively along with the loop bound (Figure 19 state and computation time, we can deduced that it satisfies the O(n 2 ) relation. Theoretically this conclusion is also valid because at restore phase for the loop bound, the restore transition will fire the same times than loop bound to get back the initial state of the decision node.

Conclusion and Future Work

Time property verification issue becomes a stringent requirement in RTES. In the existing translation approaches from UML-AD into Petri Nets, none of them aimed to verify time properties for safety critical RTES so far. To solve the key challenge in it, we present in this paper an efficient and property driven transformation approach for mapping UML-MARTE Activity Diagram into Time Petri Net. By applying this transformation approach, our UML-MARTE Model Checker covers a large range of time properties' verification, including synchronization, schedulability, boundedness, WCET, etc, in both mono-clock and multiclock system, for discrete and dense time concept, and for finite and infinite time scope.

With the result of the case study illustrated, we demonstrate that this verification-driven approach is able to guarantee the correctness of system's temporal properties, perform efficient verification and assist in the optimization of system's design at early phase of RTES's lifecycle.

In the future, we will focus on improving the transformation rules for each functional action and extending this framework by covering a wider range of properties. On the technical side, we will try to optimise UML to TPN transformation by finding some structural patterns which can be reduced without influencing the time properties.

Figure 1 :

 1 Figure 1: UML-MARTE Model Checker

Figure 2 :

 2 Figure 2: General Pattern

Figure 3 :

 3 Figure 3: Initial Node & Flow Final Node

Figure 4 :

 4 Figure 4: Activity Final Node

Figure 5 :

 5 Figure 5: Fork Node & Join Node

Figure 6 :Figure 7 :

 67 Figure 6: Decision Node & Merge Node

Figure 8 :

 8 Figure 8: UML Action Transformation to TPN

Figure 9 :

 9 Figure 9: Non-Preemptive Resource Scheduling

Figure 10 :

 10 Figure 10: Pre-emptive Resource Scheduling

Figure 11 Figure 12 :

 1112 Figure 11: Object Transformation A more common pattern to model the data dependency is

Figure 13 :

 13 Figure 13: Connection

Figure 14

 14 Figure14: FWS Architecture Due to the modeling rules, the MARTE profiles are used to represent the time specification, including the execution time for each action (Table1) and the time configuration for communication (Table2).

Figure 18 :

 18 Figure 18: FWS Transformation & Optimization TPN

  (a), 19(b)). In order to illustrate a O(n 2 ) relation, we add in Figure20(a)and 20(b) their deviation ratio with the increment of the loop bound. As the deviation ratio is nearly a constant for both

Figure

  Figure 19: Computation Time & State Class Number

Figure 20 :

 20 Figure 20: Derivation of (Computation Time/Loop Bound) & (State Class Number/Loop Bound)

Table 1

 1 ) and the time configuration for communication (Table2).

	<<Allocated>>sys data
	<<ResourceUsage>>
	System Data Acquisition
		<<ResourceUsage>>
			Wait
	<<ResourceUsage>>
	Air Data Acquistion
	<<Allocated>>air data
	Figure 15: FWS Sensor Behavior
	<<Allocated>>sys data
	<<ResourceUsage>>	<<ResourceUsage>>
	Alerts Dectection	Send Alerts
	<<Allocated>>air data	<<Allocated>>alert
	Figure 16: FWS Application Behavior
	<<ResourceUsage>>
	Receive Alert 1 <<Allocated>>alert 1	100
	<<ResourceUsage>>	<<ResourceUsage>>
	Compare	Update Display
	<<ResourceUsage>>
	Receive Alert 2
	<<Allocated>>alert 2
	Figure 17: FWS Display Behavior

Table 1 :

 1 Action Execution Time

	Behaviour	Action	T M in	T M ax
		System Data Acquisition	14	34
	Sensor	Air Data Acquisition	28	46
		Wait	3000	3000
	FWS	Alerts Detection Send Alert	128 3	189 10
		Recv Alert 1	3	10
	MFD	Recv Alert 2 Compare	3 11	10 19
		Update Display	110	133

Table 2 :

 2 Communication Configuration

	Component	Component	Pin	Comm. Delay	ADFX
	From	To		T M in / T M ax	Switch
	A/C System	CPIOM-1	sys data	3 / 8	S1
	A/C System	CPIOM-1	air data	3 / 9	S1
	A/C System	CPIOM-2	sys data	3 / 8	S1
	A/C System	CPIOM-2	air data	3 / 9	S1
	CPIOM-1	Display	alert	3 / 8	S2
	CPIOM-2	Display	alert	3 / 9	S2

Table 3 :

 3 Verification Result for Time Properties

	Time Property	Result(T U nit )	Computation Time(ms)
	Min Time Interval	154	4276
	Max Time Interval	2334	5711
	Optimized Wait Time	2034	6437

This work was funded by the French ministries of Industry and Research and the Midi-Pyrénées regional authorities through the ITEA2 OPEES and FUI Projet P projects