1 Proving there is a BGP in the Solow Model with technology and population growth Consider the Solow Model with technology and population growth in continuous time. The production function is given by: Y, = A (K )" (L)' and the law of motion of capital is given by K, = sY,6K,. The growth rate of 4 technology is given by g4 = %'L =~ and the growth rate of population is given L by gr = + =n. 1. 2, What are the endogenous and exogenous variables in the model? Define k, = %+ Find the growth rate of k, as a function of k. '_,1:1\" L . Draw the growth rate of k; as a function k; and use the graph to prove that k; converges to a steady state (you can follow similar steps as slides 12-13 of the static Solow model lecture) . Compute the steady state level of k. . Compute income per capita y, = LL: as a funetion of k. Yy . Is there a steady state level of output per capita y;7 Define 3, = t. A" L, Does i have a steady state? Does it imply that the economy converges to a Balanced Growth Path (BGP)? . Solve for the growth rate of income per capita on the BGP based on this question. . Solve for the level of income per capita on the BGP as a function of parameters and exogenous variables, 1. Endogenous and Exogenous Variables: . Endogenous variables: K (capital stock), Y (output), Et (capital per effective worker), and yt (output per capita). . Exogenous variables: At (technology), Lt (labor), s (saving rate), o (depreciation rate), a (output elasticity of capital), n (population growth rate), and y (technology growth rate). 2. Growth Rate of Kt: We first need to find Kt. Rewrite the production function as Yt = At K L;- = At ( K. A ) / (1-@ ) It ) " ( Lt ) 1-a = Al/ (1-@ ) ke Lt . Then, we have: K . ki = Kt A!/ (1-@) Lt (A;/ (1-@) [,) 2 (At 1 - a sYt Skit At 1 A!/(1-@) Lt At 1 - a 3 Graph and Convergence of Kt: Similar to the static Solow model, you can plot , against ; to show convergence towards the steady state. The steady state occurs where k, = 0. 4 Steady State of Kt: In the steady state, kt = 0, so: sY A 1 AL - Sk Al-a .= 0 Solve this equation for & to find the steady-state level. 5 Income per Capita yt: Yt At K. L,- yt = It = Al/(1-@) ka Lt 6. Steady State Level of Output per Capita yt: yt is defined as 1/(1-2) L ). There is a steady state if y. = 0. If y. = 0, it implies convergence to a BGP. 7. Growth Rate of Income per Capita on the BGP: If there's a BGP, then yt = 0. Solve the differential equation for y, and find its value when it equals zero. 8. Level of Income per Capita on the BGP: If there's a BGP, you can solve for the steady-state level of income per capita using the steady- state value of k and the parameters of the model. For detailed calculations and solutions, you would need to substitute the expressions derived above into the relevant equations and solve accordingly. This involves algebraic manipulations and possibly numerical methods depending on the complexity of the model and its parameters