Please note that if any software is utilized in the answer such as Matlab, Maplesoft, Mathematica or others, I expect to receive ALL of the proper documentation including related software files

Instructions: This homework assignment is focused on the papers we reviewed in class: "Leapfrogging in International Competition: A Theory of Cycles in National Technology Leadership" , by Brezis, Krugman, and Tsiddon (AER 1993). There are 5 related questions with total worth of 100 points. Please try to provide a detailed explanation of each question you attempt to answer. I encourage you to use the Mathematica/Matlab or any other programming software to plot draw figures or to find solutions numerically.

Consider a world economy consisting of two countries home country (denoted by h) and foreign country (denoted by f). There are 2 sectors in each country: agriculture and manufacturing. Labor is the only factor of production and the size of labor force, in both countries, is normalized to unity. Production functions in agricultural and manufacturing sectors in each country j = h, f , are Q^j_F = L^j and Q_i^j (t) = a_i^j(t)(1 - L^j) respectively, where L^j and 1- L^j represent the employment in each sector.

Q_i^j(t) denotes the manufacturing commodity produced at time t, by employing production technology of ith generation. Suppose a_i^j(t) = A_i [K_i^j(t)] = 0.025i(1 + K_i^j(t)) denotes the Total factor Productivity (TFP) of manufacturing sector in period t, if the sector operates with production technology of ith generation and possesses K_i(t) units of stock of knowledge of operating the production technology of i^th generation. Stock of knowledge is subject to a learning-by-doing externality and change in stock of knowledge in country j = h, f is given by K_i^j(t) = Q_i^j(t). Both countries are inhabited by a continuum of infinitely lived homogeneous agents of unit mass. Agents supply one unit of labor inelastically and consume both commodities in each period. Agents' maximize the utility

U = c^μ_M c^1-μ_F, where μ ∈ (0.5, 1).

Use the fact that the solution of the differential equation

x(t) = ax(t) - be^c(t-T) with a given x_T, = x(T)

is

x(t) = (x_T + b/c-a).e^a(t-T) - (b/c-a).e^c(t-T).

Question 1: Manufacturing technology of 1^st generation becomes available in 1940 and country-f adapts it. What is a₁^f (t) for t ∈ G [1940,19601 if both countries operate in autarky?

Question 2: Manufacturing technology of 2^st generation becomes available in 1960. For what values of parameter μ would country-f adapt it if countries operate in autarky.

Question 3: Suppose countries open their borders for trade in 1950. That time both countries operate manufacturing technology of 1^st generation. TFP in these countries are a₁^h(1950) = 0.045 and a₁^f(1950) = 0.055. In which year the world economy would achieve a complete specialization if μ = 0.60 and manufacturing technology of 2nd generation becomes available in 1960?

Question 4: Suppose μ = 0.60. If a manufacturing technology of 2^nd generation becomes available in 1960 would country-h adapt it? would country-f adapt it?

Question 5: Suppose μ = 0.60. If country-h adapts a manufacturing technology of 2^nd generation while country-f continues operation of a manufacturing technology of 1st generation would there be a leapfrogging in income leadership? if yes when?