[growth] AK model

Two approaches to deal with endogenous growth theory:

1. Alternative specification of aggregate production function: (e.g.) AK model with y(t)=Ak(t).
2. Endogenize the technology as a choice variable (e.g.) AKh model.

 It turns out that AK model serves as a reduced form model for lots of interesting models.

[1] AK model - Basic

 With CRRA UF, common growth rate is given as g=[beta(A+1-delta)]^{1/sigma}. Two restrictions on g are

• g>1 for sustained growth
• Transversality condition: boundedness of lifetime utility on BGP: (beta*g^(1-sigma))<1 

 These are restrictions on model parameters: A should be neither too big nor too small.

Question: Is feasible BGP the unique optimal path? (In other words, could we have an optimal path -with a constant growth rate of consumption -but non-constant growth rates of output, capital, and investment?)

• Yes. From RC, if g(c)>g(k) -> consumption exhausts the budget. If g(c)<g(k), violation of TVC.

Properties of AK model

1. No transitional dynamics: a balanced growth path is immediately attained since optimal consumption growth is always constant.
2. Growth rate is a function of underlying (not exogenously given) technology and preference parameters: Policy implications on growth rate effect vs. level effect. 
3. No convergence of income levels across countries: again, underlying preferences are different.

[2] Example: Romer's Externality Model

 

 Production externality in capital accumulation: (e.g.) Arrow (1962), "Learning by doing".^[각주:1] 

• Arrow: learning from doing is embodied in the capital goods accumulated, used in production, enhances production efficiency.
• Romer: spillovers result from investments in knowledge acquired during capital accumulation, knowledge enhances production efficiency.

 - Individual firm's production function is  where \bar{k}: total amount of capital in the economy. Since the firm is actually representative, kbar is also average capital level too.

 - Externality acts as labor-augmenting technological change.

 - Leisure is not valued: AK production function as a reduced form.

Recall: Market Failures (i) Imperfect competition, (ii) Informational imperfections (MH and Adverse Selection), (iii) Missing Markets (PG and Externality)

 [Externality in the framework of GE]

 - Production externality: SMC (e.g. measured in labor hours) of producing one more unit of output of externality producing firm =/ SMB (also measured in labor hours) of producing one more unit of output.

 - Consumption externality: (e.g.) relative consumption level matters (agents care status etc.): 

: In this case, over-consumption and over-working than socially optimal levels.

A. SP level of growth rate

 Aggregate PF: Y=AK. Therefore g=[beta*(1-delta+A)]^(1/sigma) as before.

B. CE level of growth rate

 Easily shown that equilibrium rental rate r(t)=alpha*A. Making g(CE)<g(SP): under-accumulation of capital.

 

Problems

• Little evidence that externalities from capital accumulation are large.

[3] Data 

 - In AK model, there's no labor: Empirically not an interesting model.

 - No transitional dynamics: We tend to see transitional dynamics in the data.

 - When simulated, AK model produces too much divergence in output levels across countries.

[4] Issues

Distortionary Taxes and Growth

 - Capital income tax, equivalent lump-sum transfer

 - Since there's tax, SP problem is no longer the same as CE problem.

 - HH's BC: c(t)+k(t+1)=k(t)(r(t)+1-delta)*(1-tau(t)) + t(t)

 - Government's BC requires that t(t)=tau(t)*(r(t)+1-delta)k(t).

 - Easy to show that tax negatively affects LR growth rate.

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Reference

 [1] Growth II: 

2. Endogenous Growth 47-106 (AK Model, Externality).pdf

 [2] Krusell lecture note

1. My understanding on this capital accumulation production externality is more like network effect. (e.g.) Prevalent use of same software making positive production externality. [본문으로]