In this section, we will combine the Solow and Romer models together and derive a model that will exhibit both transition dynamics (catching up of poor countries) as well as persis- tent growth. The derivation of the model closely follows the textbook and the slides. Time is discrete and indexed by t = 0, 1, 2, .... The economy starts at time 0 with an initial stock of ideas An and stock of capital Ko, and is described by the following five equations, with variables that we defined in class. 1 2 3 4 5 Y = AKL AAt+1 = ZA Lat : AKt+1=8Y₁ - 8Kt L=Lyt + Lat You should obtain : Lat= IL Question 4.1 Give names to each of the five equations, representing their economic mean- ing. Question 4.2 Solve for the growth rate of ideas 94,1 = +¹ We now introduce a method how to solve separately for the transition dynamics. The method will consist of rescaling output Y, and capital K, by the level of technology. Define so-called scaled variables Y₁ Y₂₁ = (4₂) We can interpret Y, and K, as output and capital levels relative to the level of technology. You may wonder why A, has the exponent in this scaling, but for now taking it as a 'good guess' which will become clearer very soon. and K Question 4.3 Express Y, and K, from equations (1) and plug them into the output production function (first equation of Solow-Romer model above). Show that you can simplify the equation so that the technology A, drops out of the equation. K₂ (4₂) = Question 4.4 In the same way, plug in Y, and K, from equations (1) into the capital ac- cumulation equation (third equation of Solow-Romer model above). Simplify the equations and show that the production function will now not depend on A directly, and the capital accumulation equation will only contain A, as a ratio , which is equal to 1 + 9A,t- Hint: This is very similar to the scaling procedure in the Solow growth model with population growth. Before you start, it is advantageous to write the capital accumulation equation as 4 Kt+18Yt +(18) Kt. = S Kt+1= (1 + 9A,t) a -Ÿ + -Ÿ₁ 1-8 (1+9₁) (1) -K₁. Question 4.5 Show that we can write this capital accumulation equation as Akt+1 = 8Ỹt - 8K Y₁ = KL Kt+1 - Kt = sỸ, - ŠK₁ (2) where and are parameters that can be called 'adjusted saving rate' and 'adjusted de- preciation rate'. Express these parameters as a function of the original parameters of the model and the growth rate ga,t- Observe that we can now write the production function and the capital accumulation equation as Question 4.6 Express Lut in terms of the parameters of the model. Now determine the 'steady state' in variables Y₁, K₁. Question 4.7 When K, and Y, are in steady state, at what rate do capital K, and output Y grow? Notice that you will obtain that K, and Y, grow at constant rates such a situation is called the balanced growth path. ■ Question 4.8 Verify that gy> 9A. Why does output and capital grow at a faster rate than technology? ■