An overlapping generations model with two types of agents and shift in behaviour Pasquale Commendatore (University of Naples “Federico II”, Italy) Ingrid Kubin (Vienna University of Economics and BA, Austria) Iryna Sushko (NASU and Kyiv School of Economics, Ukraine) NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 1
Contents • Introduction • Sketch of the full model (two types of agents) • Reduced model (only one type of agents) • Preliminary results of the full model • Final remarks • References NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 2
Introduction NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 3
• We present an overlapping generations model describing an economy in which two types of agents may co- exist: ‘workers’ and capitalists’ (Pasinetti, 1962). • Workers and capitalists save on the basis of rational choices (Foley & Michl, 1999; Michl, 2009; Commendatore & Palmisani, 2009). • Workers face a finite time horizon and base their consumption choices on a life-cycle motive. • Capitalists behave like an infinitely-lived dynasty. NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 4
• Capitalists only source of income is profit. • Workers main source of income is wages. However, depending on their income workers may have a switch in behaviour: • above a certain threshold they decide to leave bequests to the offspring according to a ‘warm glow’ motive (see Andreoni, 1989, 1990). • Empirical literature confirms that households with higher levels of (lifetime) income typically leave very large bequests (De Nardi, 2004). NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 5
• The resulting model is in three dimensions with a discontinuity. • We consider a special case in which a distinct class of capitalists does no exist. The resulting model is 2-dimensional with a discontinuity. • Notice that when workers’ income is sufficiently high, so that they never change their behaviour altruism never sets in, the 2-D model reduces to that proposed by Michel and de La Croix (2000) and Chen et al. (2008) with myopic expectations. NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 6
• We will study the dynamics properties of the 2-D model with switch in behaviour both in relation to the local stability properties and to the global dynamics verifying how the threshold impinges on those properties. • We will also verify how changes in workers behaviour affects capital accumulation. This will be crucial, when capitalists are reintroduced, to study the overall effects on the distribution of capital between the two groups. NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 7
Full model (sketch) NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 8
Capitalists • Are represented as a single dynasty with and infinite time horizon (Barro, 1974). • The only source of income is profit. • Given initial wealth 𝐿 𝑑,0 , at the beginning of period 0 they choose consumption quantities ∞ in order to solve: 𝐷 𝑑,0 , 𝐷 𝑑,1 , 𝐷 𝑑,2 , … , 𝐷 𝑑,𝑢, … = (𝐷 𝑑,𝑢 ) 0 ∞ 𝛾 𝑑 𝑢 U(𝐷 𝑑,𝑢 ) max σ 𝑢=0 subject to 𝐷 𝑑,𝑢 + 𝐿 𝑑,𝑢+1 ≤ (1 + 𝑠 𝑢 − 𝜀)𝐿 𝑑,𝑢 • where 0 ≤ 𝜀 ≤ 1 is the depreciation rate and 0 < 𝛾 𝑑 < 1 is capitalists time discount factor. • 𝐿 𝑑,𝑢 is the capital of capitalist at point of time t. • For future reference 𝑙 𝑑,𝑢 is the capitalists’ capital per worker. • N.B. Capitalists have perfect foresight. NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 9
Workers • The population of workers has an overlapping generations structure • Each generation faces a finite time horizon composed of two periods. • Each individual is active, working and earning a wage, when “young” (i.e. in the first period of life). • And she is in retirement when “old” (i.e. in the second period of life). • In each period t a young worker inelastically supplies one unit of labour at the wage rate 𝑥 𝑢 . • 𝑀 𝑢 is the overall number of workers, n is the rate of growth, where 𝑜 > −1 . Thus 𝑀 𝑢+1 = 1 + 𝑜 𝑀 𝑢 . • N.B. Workers have myopic foresight. NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 10
Workers • If workers’ income is low, they save out of income only to be able to consume during retirement according to a life cycle motive. • There is not intergenerational transfer of wealth (Diamond, 1965). • A young worker chooses the quantities of current 𝑑 𝑥,𝑢 and future consumption 𝑑 𝑥,𝑢+1 solving the following constrained utility maximization problem: max[𝑉(𝑑 𝑥,𝑢 ) + 𝛾 𝑥1 𝑉(𝑑 𝑥,𝑢+1 )] 𝑑 𝑥,𝑢+1 subject to 𝑑 𝑥,𝑢 + 1+𝑠 𝑢 −𝜀 ≤ 𝑥 𝑢 + 𝑐 𝑢 , • where 0 ≤ 𝛾 𝑥1 < 1 is the workers’ consumption discount factor. NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 11
Workers • If workers’ income is high, workers decide to behave altruistically towards their offspring • Parents value the bequest per se following the “warm glow” (joy of giving) approach (Andreoni 1989, 1990) • The amount of the bequest is one of the arguments of the workers’ intertemporal utility function. A single worker solves: max 𝑉(𝑑 𝑥,𝑢 ) + 𝛾 𝑥1 𝑉(𝑑 𝑥,𝑢+1 ) + 𝛾 𝑥2 𝑉((1 + 𝑜)𝑐 𝑢+1 ) • Subject to: 𝑑 𝑥,𝑢+1 1+𝑠 𝑢 −𝜀 + 𝑐 𝑢+1 (1+𝑜) • 𝑑 𝑥,𝑢 + ≤ 𝑥 𝑢 + 𝑐 𝑢 , 1+𝑠 𝑢 −𝜀 • where 0 ≤ 𝛾 𝑥2 < 1 is the discount factor workers apply to bequests, with 𝛾 𝑥2 ≤ 𝛾 𝑥1 . NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 12
Reduced version (no capitalists) NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 13
• In the full version, the model is three-dimensional (the state variables are: capitalists’ capital, workers’ capital and workers’ bequests) with a discontinuity. • We simplify by assuming 𝑙 𝑑,𝑢 = 0 which holds for all t (no capitalists), the resulting model is two-dimensional (the state variables are reduced to capital and bequests). • We drop the subscript w . NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 14
• We assume a CIES Utility function ( 𝜏 intertemporal elasticity of substitution): −1 1 − 1 𝑑 1−1 𝑔𝑝𝑠 𝜏 > 0 𝑏𝑜𝑒 𝜏 ≠ 1 𝜏 𝑉 𝑑 = 𝑔 𝑦 = 𝜏 ln(𝑑) 𝑔𝑝𝑠 𝜏 = 1 • and a Cobb-Douglas production function: 𝑔 𝑙 = 𝛽𝐵𝑙 𝛽 with 0 < 𝛽 < 1 and 𝐵 > 0 NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 15
• If 𝑥 𝑢 + 𝑐 𝑢 < ത 𝑧 (i.e. agent’s income is low, with ത 𝑧 ≥ 0 ) solutions satisfy the condition: 𝜍 𝑢 (𝑥 𝑢 +𝑐 𝑢 ) • 𝑑 𝑢 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 𝜍 𝑢 (𝜍 𝑢 𝛾 1 ) 𝜏 (𝑥 𝑢 +𝑐 𝑢 ) • 𝑑 𝑢+1 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 • A single agent saving corresponds to: (𝜍 𝑢 𝛾 1 ) 𝜏 (𝑥 𝑢 +𝑐 𝑢 ) • 𝑡 𝑢 = 𝑥 𝑢 + 𝑐 𝑢 − 𝑑 𝑢 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 where 𝜍 𝑢 = 1 + 𝑠 𝑢 − 𝜀 NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 16
• If 𝑥 𝑢 + 𝑐 𝑢 ≥ ത 𝑧 (i.e. agent’s income is high, with ത 𝑧 ≥ 0 ) solutions satisfy the condition: 𝜍 𝑢 (𝑥 𝑢 +𝑐 𝑢 ) • 𝑑 𝑢 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 +(𝜍 𝑢 𝛾 2 ) 𝜏 𝜍 𝑢 (𝜍 𝑢 𝛾 1 ) 𝜏 (𝑥 𝑢 +𝑐 𝑢 ) • 𝑑 𝑢+1 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 +(𝜍 𝑢 𝛾 2 ) 𝜏 𝜍 𝑢 (𝜍 𝑢 𝛾 2 ) 𝜏 (𝑥 𝑢 +𝑐 𝑢 ) • 𝑐 𝑢+1 1 + 𝑜 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 +(𝜍 𝑢 𝛾 2 ) 𝜏 • A single agent saving corresponds to: [(𝜍 𝑢 𝛾 1 ) 𝜏 +(𝜍 𝑢 𝛾 2 ) 𝜏 ](𝑥 𝑢 +𝑐 𝑢 ) • 𝑡 𝑢 = 𝑥 𝑢 + 𝑐 𝑢 − 𝑑 𝑢 = 𝜍 𝑢 +(𝜍 𝑢 𝛾 1 ) 𝜏 +(𝜍 𝑢 𝛾 2 ) 𝜏 where 𝜍 𝑢 = 1 + 𝑠 𝑢 − 𝜀 NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 17
Equlibrium conditions in the labour and capital markets Given the production function 𝛽 𝑔 𝑙 𝑢 = 𝐵𝑙 𝑢 • Equilibrium in the capital market involves: 𝛽−1 𝑠 𝑢 = 𝑔′ 𝑙 𝑢 = 𝛽𝐵𝑙 𝑢 • Equilibrium in the labour market involves: 𝛽 𝑥 𝑢 = 𝑔 𝑙 𝑢 − 𝑔′ 𝑙 𝑢 𝑙 𝑢 = 1 − 𝛽 𝐵𝑙 𝑢 NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 18
Dynamic equations • Considering that 𝑀 𝑢+1 = 1 + 𝑜 𝑀 𝑢 and using the equilibrium condition savings = investments, • for 𝑥 𝑢 + 𝑐 𝑢 < ത 𝑧 we can write: (𝜍 𝑢 𝛾 1 ) 𝜏 (𝑥 𝑢 + 𝑐 𝑢 ) 1 1 𝑙 𝑢+1 = 1 + 𝑜 𝑡 𝑢 = 𝜍 𝑢 + (𝜍 𝑢 𝛾 1 ) 𝜏 1 + 𝑜 𝑐 𝑢+1 = 0 • for 𝑥 𝑢 + 𝑐 𝑢 ≥ ത 𝑧 we can write: [(𝜍 𝑢 𝛾 1 ) 𝜏 +(𝜍 𝑢 𝛾 2 ) 𝜏 ](𝑥 𝑢 + 𝑐 𝑢 ) 1 1 𝑙 𝑢+1 = 1 + 𝑜 𝑡 𝑢 = 𝜍 𝑢 + (𝜍 𝑢 𝛾 1 ) 𝜏 + (𝜍 𝑢 𝛾 2 ) 𝜏 1 + 𝑜 𝜍 𝑢 (𝜍 𝑢 𝛾 2 ) 𝜏 (𝑥 𝑢 + 𝑐 𝑢 ) 1 𝑐 𝑢+1 = 𝜍 𝑢 + (𝜍 𝑢 𝛾 1 ) 𝜏 + (𝜍 𝑢 𝛾 2 ) 𝜏 1 + 𝑜 NED 2019 - Kyiv School of Economics, Kyiv, Ukraine 19
Recommend
More recommend
