Introduction

• Inability of the AK model to produce a convingin model of long term growth and convergence.
• Endogenous growht theory: Innovation-based models.

Similarities

• No demand for leisure so they offer L inelastically.
• Utilitiy function:

\[u(c)= c^{1-\epsilon}/(1-\epsilon)\]

• Total amount of final good used in producing intermediate products:

\[X_{t} = \int_{0}^{M_{t}} X_{i} di\]

(x=xi for all i).

• Free market entry.

Differences

• Alternative assumption of the Romer Model with Labour as R&D Input:
• L can be used in manufacturing the final good (\(L_{1}\));
• L can be used in research (\(L_{2}\)).

\(L= L_{1}+ L_{2}\)

Product-Variety model (I)

• Each intermediate good producer is a monopolist for the product.
• Mazimizing the flow of profit at each date.
• Equilibrium profit:
• \[\pi= ((1-\alpha)/\alpha) L\alpha^{2/(1-\alpha)}\]

Product-Variety model (II)

• Because \(x=xi\) for all i then, \(x=X_{t}/M_{t}\) and using \(M_{t}x\) the final good output and GDP will both be proportional to the degree of product variety:

\[Y_{t} = M_{t}(L^{1-\alpha}x^{\alpha}-x)\] - Growth rate:

\[g=(1/\epsilon) (\lambda (1-\alpha/\alpha L\alpha^{2/1-\alpha}-\rho)\]

Romer model(I)

• Each intermediate good producer is a monopolist for the product.
• Mazimizing the flow of profit at each date.
• Equilibrium profit:
• \(\pi =((1-\alpha)/\alpha) L_{1}\alpha^{2/(1-\alpha)}\)
• If profits increase, the prospect of these rents will motivate research activities aimed at discovering new varieties.

Romer Model (II)

\[\phi<1\]

• the final good output and GDP will both also be proportional to the degree of product variety:

\(g = M = \lambda L_{2}\) so we have \(r=(L-g)\) \[g = (\lambda\alpha L-\rho)/(\alpha+\epsilon)\]

Final results(I)

• We obtain similar conclusions for both models.
• Growth increases with:
• Productivity of research activities and the size of the economy.
• The equilibrium growth rate is always less than the social optimum.

Final results (II)

• Intermediate firmsdo not internalize their contribution to the division of labour;
• Researchers do not internalize research spill overs;
• Ideas are non-rival and exclusive "goods".