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".