What is the average growth rates of the economy along the

Question

Questions from Matsuyama:

Consider a model which combines two views of growth, one based on factor accumulation and the other based on innovation of new products, motivated by monopoly profits.

Let's assume that labor supply and productivity parameters are set to unity, L =1 and ^{^}A = 1. There is a fixed cost for innovation which is set to unity as well, F = 1.

Parameters Ψ ∈ (0, ∞) and σ ∈ (1, ∞) describe marginal cost of production and the elasticity of substitution between any two varieties.

Let K_t-1 is the amount of final goods left unconsumed in period t - 1 and let N_t-1 is the size of varieties available in period t-1.

Let k_t-1 = K_t-1/θσN_t-1 denotes the relative size of resource base in terms of existing products.

Question 1: Establish the relationship between profit of a monopolist and the borrowing cost. For which values of rt, would monopolist earn a positive profit? zero profit? negative profit?

Question 2: Let

^{^}r := (σ - 1/σ)²(Ψ^1-σ/σ-1)^1/σ and ^{^}x :=  σ - 1/Ψ

What is the amount of innovation when r_t < ^{^}r? r_t> ^{^}r if and r_t = ^{^}r?

Question 3 Let k_t-1 = K_t-1/θσN_t-1. Establish the relationship between r_t and K_t-1.

Question 4

where Dynamics of k_t is given by the following time one map

where

f(k_t) = Gk_t^σ-1/σ  and g(k_t) = Gk_t/1 + θ(k_t-1)

Suppose σ > 2 and G ∈ (1, θ - 1) then kg fluctuates endogenously. What are the growth rates of investment, output, and innovations along the balanced growth path? Suppose the economy converges to a unique cycle of order 2, (k_L, k_H), where

(a) 1 < k_L < k_H;

(b) 0 < k_L, < 1 < k_H.

What is the average growth rates of the economy along the path (k_H, K_L, k_H, k_L, k_H, ...) and how it compares with the average growth rate along the the balanced growth path?

Question 5 Steady state k** = 1 + θ - 1/G becomes unstable through a Flip bifurcation when g'(k**) = -1 ⇔ G = θ - 1. At G = θ - 1 there exists a unique and globally stable cycle of order 2, (k_L; k_H) where 1 < k_L < k_H < ∞. As G decreases further, (k_L, k_H) where 1 < k_L < k_H < ∞, becomes unstable and a unique and globally stable cycle (k_L, k_H) where 0 < k_L, < 1 < k_H < ∞ emerges. For which value of G would (k_L, k_H), where 0 < k_L, <1 < k_H < ∞, becomes unstable?

Can you guess what happens after (k_L, k_H) becomes unstable?