investment M. R. Grasselli EPOG Seminar Paris Nord, November 18, 2016

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1 Mathematics and Statistics, McMaster University Joint A. Nguyen Huu (Montpellier) EPOG Seminar Paris Nord, November 18, 2016

2 Cycles Small fraction of output (about 1% in the U.S.) but major fraction of changes in output (about 60% for postwar recession in the U.S.) Figure: Blinder and Mancini (1991)

3 Stylized facts is more volatile than output. is strongly countercyclical at very high frequencies (e.g., 2-3 quarters per cycle) but procyclical at business-cycle frequencies (e.g., 8-40 quarters per cycle). Production is less volatile than sales around the high frequencies; it is more volatile than sales only around business-cycle or lower frequencies. Most of the variance of inventory is concentrated around high frequencies rather than around business-cycle frequencies (unlike capital and GDP).

4 Theoretical models Micro theories view inventories primarily as a stabilizing factor (e.g production-smoothing). Incorporating inventories into fully micro-founded DSGE models is akin to incorporating money and finance. Earlier Keynesian model by Metzler (1941), further developed by Franke (1996) provides a more promising starting point. Heterodox (e.g stock-flow consistent) models emphasize the role of inventories, but fully developed models are rare and tend to be overcomplicated.

5 Contributions of this paper Combines the Franke (1996) model for inventory fluctuations Goodwin (1967) model for labor market dynamics. Provides the first stock-flow consistent extension of the Keen (1996) model where both consumption and () are independently specified. Identifies and analyses two important sub-models: (i) the long-run model is a version of the Keen model non-trivial effective demand, whereas (ii) the short-run model gives rise to Kitchin cycles (1923).

6 Notation Potential output: Y p = K/ν Actual output: Y = Y e + I p Capacity utilization: u = Y /Y p Capital accumulation: K = I k δ(u)k Demand: Y d = C + I k Change in inventories: V = I p + I u = Y Y d Unplanned changes: I u = Y Y d I p = Y e Y d. Gross : I = Y C = Y Y d + I k = I p + I u + I k

7 Cost, prices, and financial balances Productivity: a = Y /l (assume ȧ a = α) Employment rate: λ = l/n = Y /(an) (assume Ṅ N = β) Wage rate: w = W /l Unit labour cost: c = W /Y = w/a. Nominal output: Y n = pc + pi k + c V. Profits: Π = Y n W rd pδk Change in debt for firms: Ḋ = p(i k δk) + c V Π = pi k + c V Π p, where Π p = Y n W rd.

8 SFC Table Households Firms Banks Sum Balance Sheet Capital stock +pk +pk +cv +cv Deposits +M M 0 Loans D +D 0 Sum (net worth) X h X f X b X Transactions current capital Consumption pc h +pc pc b 0 Capital Investment +pi k pi k 0 Change in +c V c V 0 Accounting memo [GDP] [Y n] Wages +W W 0 Depreciation pδk +pδk 0 Interest on deposits +r mm r mm 0 Interest on loans rd +rd 0 Profits Π +Π 0 Financial Balances S h 0 S f p(i k δk) c V S b 0 Flow of Funds Change in Capital Stock +p(i k δk) +p(i k δk) Change in +c V +c V Change in Deposits +Ṁ Ṁ 0 Change in Loans Ḋ +Ḋ 0 Column sum S h S f S b p(i k δk) + c V Change in net worth Ẋ h = S h Ẋ f = S f + ṗk + ċv Ẋ b = S b Ẋ

9 Behavioural rules - firms Define π e = Y ne W rd py = y e (1 ω) rd, where y e = Y e /Y, ω = W /(py ) and d = D/(pY ). We assume that sales expectations evolve as Ẏ e = g e (u, π e )Y e + η e (Y d Y e ) Let V d = f d Y e for a constant f d and assume that I p = g e (u, π e )V d + η d (V d V ). Moreover, take I k = κ(u, π e) K. ν

10 Behavioural rules - banks and households We assume that C = θ(ω, d)y. This includes the case pc h = c ih [W + r m M] + c wh M, pc b = c ib [rd r m M] + c wb (D M). In particular, we can have pc = c 1 W + c 2 D θ(ω, d) = c 1 ω + c 2 d. Total demand is then given by py d = pc + pi k = pθ(ω, d)y + p κ(u, π e) K, ν so that y d = Y d Y = θ(ω, d) + κ(u, π e). u

11 Behavioural rules - wages and prices We assume that prices follow ṗ p = η p (m cp 1 ) η q Y e Y d Y = η p (mω 1) + η q (y d y e ) := i(ω, y d, y e ). The dynamics for nominal wages is (1) ẇ w = Φ(λ) + γ ṗ p, (2)

12 The main dynamical system The full model is described by ω = ω [Φ(λ) α (1 γ)i(ω, y d, y e )] λ = λ [g(u, π e, y d, y e ) α β] ḋ = d [r g(u, π e, y d, y e ) i(ω, y d, y e )] + ω θ(ω, d) ẏ e = y e [ [g e (u, π e ) g(u, π e, y d, y e )] + η ] e (y d y e ) u = u g(u, π e, y d, y e ) κ(u,πe) ν + δ(u) where and i(ω, y d, y e ) = η p (mω 1) + η q (y d y e ) g(u, π e, y d, y e ) = [ f d (g e (u, π e ) + η d ) + 1 ]( y e g e (u, π e ) + η e (y d y e ) ) + η d (y d 1)

13 Interior equilibrium It follows from the second equation that g(u, π e, y d, y e ) = α + β. Inserting this in the fourth equation gives y d = y e and g e (u, π e ) = α + β. Substitution in the definition of g then gives y d = y e = (α + β)f d. Moreover, it follows that v = f d y e, so that the equilibrium level of inventory is the desired level V d = f d y e Y. Furthermore, we see from the definition of i that i(ω, y d, y e ) = i(ω) = η p (mω 1).

14 Interior equilibrium - continued From the third equation, we see that: d = From the last equation, we obtain: ω θ(ω, d) α + β + i(ω) r. (3) κ(π e, u) = ν[α + β + δ(u)], (4) which can be inserted in the demand function to give u = ν[α + β + δ(u)](1 + (α + β)f d) 1 θ(ω, d)(1 + (α + β)f d ) We can then obtain the values of (ω, d) by solving (3)-(4). Finally, the first equation gives Φ(λ) = α + (1 γ)i(ω)..

15 Goodwin model Model in real terms: η p = η q = γ = 0, p = 1. No inventories: f d = η d = V d = I p = 0 Output equals demand: η e, Y e = Y d = Y Constant capital-to-output ratio: u = 1. Constant depreciation: δ(u) = δ > 0. Investment equals profits: κ(u, π e ) = π e = 1 ω rd. No banks: Ḋ = 0, take d = D 0 = 0. All wages are consumed: c ih = c 1 = 1 (and c 2 = r). This leads to { ω = ω [Φ(λ) α] λ = λ [ 1 ω ν α β δ ], (5)

16 Franke model Model in real terms: η p = η q = γ = 0 and p = 1 Variables normalized by K instead of Y, resulting in the intensive variables: u F := Y /K = u/ν, z F := Y e /K = y e u F, v F =: V /K = vu F. Constant wage share ω: ω = 0. Second equation in our system decouples. No banks: ḋ = 0. Constant long-run expected growth: g e (u, π e ) = α + β. Investment as function of utilization: κ(u, π e ) = νh(u F ). Excess demand as a function of u F : y d = e(u F ) + 1. We then obtain the same system as in Franke (1996) from our fourth and fifth equations, leading to v F = f d u F 1 + (α + β)f d = v u F, z F = u F 1 + (α + β)f d = y e u F.

17 Keen model Model in real terms: η p = η q = γ = 0 and p = 1 Same as Goodwin for production and inventories: f d = η d = V d = I p = 0, η e, Y e = Y d = Y, u = 1, δ(u) = δ. Investment as function of profits: is now given by κ(u, π e ) = κ(π e ) = κ(1 ω rd). Accommodating consumption: C = Y d I k = (1 κ(π e ))Y, θ(ω, d) = 1 κ(1 ω rd). With these parameter choices, the system reduces to ω = ω [ [Φ(λ) α] ] λ = λ κ(πe) ν α β δ [ ] ḋ = d r κ(πe) ν δ + ω 1 + κ(π e )

18 Monetary Keen model As shown in Grasselli Nguyen-Huu (2015), it is easy to incorporate inflation in the original Keen model. Adopting all the parameter choices and functional forms of the previous section (including η q = 0) the exception of arbitrary constants η p and γ, we find ω = ω [ [Φ(λ) α (1 γ)i(ω)] ] λ = λ κ(πe) ν α β δ [ ] (6) ḋ = d r κ(πe) ν δ i(ω) + ω 1 + κ(π e ) where π e = 1 ω rd and i(ω) = η p (mω 1). Apart from the usual good and bad equilibrium, this system also admits a new class equilibria of the form (ω 3, 0, d 3 ) or (ω 3, 0, + ) where ω 3 = 1 m + Φ(0) α mη p (1 γ), i(ω 3) < 0.

19 dynamics Take η e = η d = f d = 0 so that Y = Y e. We then have g(u, π e, y d, y e ) = g e (u, π e ). The system then becomes ω = ω [Φ(λ) α (1 γ)i(ω, y d )] λ = λ [g e (u, π e ) α β] ḋ = d [ [r g e (u, π e ) i(ω, y d )] + ] ω θ(ω, d) u = u g e (u, π e ) κ(u,πe) ν + δ(u), where π e = 1 ω rd and i(ω, y d ) = η p (mω 1) + η q (y d 1). In the special case g e (u, π e ) = α + β (as in the Franke model), we have λ = 0, so the interior equilibrium can only be achieved if λ 0 = Φ 1 (α + (1 γ)ω).

20 Keen model inventories - real version Take η e = η d = f d = 0 so that Y = Y e as in the long-run dynamics above, so that g(u, π e, y d, y e ) = g e (u, π e ). In addition, consider the model in real terms, that is η p = η q = γ = 0 and p = 1. Setting g e (u, π e ) = κ(u,πe) ν δ(u) leads to ω = ω [ [Φ(λ) α] ] λ = λ κ(u0,π e) ν δ(u 0 ) α β [ ] ḋ = d r κ(u 0,π e) ν + δ(u 0 ) + (1 c 1 )ω c 2 d, where we took θ(ω, d) = c 1 ω + c 2 d. This is the closest model to the original Keen model but y d = c 1 ω + c 2 d + κ(u 0,π e) u 0 and ( v = 1 c 1 ω c 2 d κ(u ) ( ) 0, π e ) κ(u0, π e ) δ(u 0 ) v. u 0 ν

21 Keen model inventories - monetary version Using (1)-(2) as the price-wage dynamics leads to the following monetary version of the model of the previous section ω = ω [Φ(λ) α (1 γ)i] ] [ λ = λ κ(u0,π e) ν δ(u 0 ) α β [ ḋ = d r κ(u 0,π e) ν + δ(u 0 ) i ] + (1 c 1 )ω c 2 d where i(ω, d) = η p (mω 1) + η q (y d 1) As before, we regard this as the closest model to the monetary Keen model in (6), but a non-trivial effective demand and fluctuating inventory levels.

22 dynamics Suppose now that α + β = 0, g e (u, π e ) = 0 (no growth) Assume further that κ(u, π e ) = νδ(u). This leads to v = [1 + f dη d ]y e 1 η d, g(y e, y d ) = η e (1 + f d η d )(y d y e ) + η d (y d 1), and the main system reduces to ω = ω[φ(λ) (1 γ)i(ω, y d, y e )] λ = λg(y e, y d ) ḋ = d[r g(y e, y d ) i(ω, y d, y e )] + ω θ(ω, d) ẏ e = y e g(y e, y d ) + η e (y d y e ) u = ug(y e, y d )

23 Planar dynamics Assume now that η p = 0 and Φ( ) 0, so that i(ω, y d, y e ) = i(y d, y e ) = η q (y d y e ). Moreover, let δ(u) = δu for δ > 0 and θ(ω, d) = c 1 ω + c 2 d = c 1 ω (i.e. c 2 = 0) (7) This gives y d = c 1 ω + νδ so the system decouples and we can focus on { ẏd = (1 γ)y d η q (y d y e ) (8) ẏ e = η e (y d y e ) y e g(y e, y d ) (ω, λ, d) satisfying a subordinated system that can be solved after.

24 Equilibrium analysis The previous system admits the equilibria (1, 1), (0, 0) and (+, + ). The equilibrium (1, 1) is locally stable provided γ < γ 0 := 1 η e η d f d /η q. At γ = γ 0 there is a sub-critical Andronov-Hopf bifurcation and for γ γ 0 the equilibrium is unstable. The equilibrium (0, 0) is unstable provided η d > η e and fails to be asymptotically stable, even if η d < η e. The equilibrium (+, + ) is characterized by a finite-time blow-up y d /y e 0 for a large set of initial conditions.

25 Figure: dynamics i(ω, y d, y e ) = i(y d, y e ) = η q (y d y e )

26 Kitchin cycles Consider now ṗ p = η p (m cp 1 ) η q V d V Y which, along previous assumptions, provides the inflation rate i(y e ) = η q (1 y e )/η v. This now leads to { ẏd η = (1 γ)y q d η v (1 y e ) ẏ e = η e (y d y e ) y e g(y e, y d ), The slight difference the latter concerns the first equation, for which the isocline is given by {y e = 1} instead of {y d = y e }. (9)

27 Equilibrium analysis The new system also admits the equilibria (1, 1), (0, 0) and (+, + ). The equilibrium (1, 1) is now locally unstable for all parameters. On the other hand, the equilibrium (0, 0) is locally stable provided η d < η e. The finite-time blow-up is similar to the previous case.

28 Figure: dynamics i(ω, y d, y e ) = i(y d, y e ) = η q (1 y e )/η v

29 Concluding remarks We have introduced a stock-flow consistent model for inventory. The model unifies features of several simpler models previously proposed in the heterodox economics literature (Goodwin, Franke, Keen). We identified the interior equilibrium of the full model and analyzed in detail the stability of two classes of sub-models. The long-run dynamics arising from ignoring short-run fluctuations can be regarded as a Keen model inventories. The short-run dynamics arising solely from tracking inventory fluctuations in an imperfect information setting can be regarded as a formalization of Kitchin cycles.

30 Thank you!