There are two islands, Avalon and Baltia, which have identical economies satisfying the assumptions of the Solow model. In particular, output in both economies is produced with the neoclassical production function F(K, AL) using inputs capital K, technology A, and labor L. In addition to identical production functions, the depreciation rate δ, savings rate s, population growth rate n, and technology growth rate g are the same in each economy. Variables in the Avalonian economy are denoted with a superscript A and variables in the Baltian economy with a superscript B. For example, Avalonian output is Y A t = F(KA t , AA t L A t ), Baltian output is Y B t = F(KB t , AB t L B t ), and so on... In period t = 0 both economies are at their Solow steady state levels of capital per efficiency unit of labor, and they have identical initial conditions AA 0 = AB 0 , L A 0 = L B 0 , and KA 0 = KB 0 .
(a) In the long run, continuing at the Solow steady state, what is the growth rate of the combined output Y A t + Y B t of the Avalonian and Baltian economies?
(b) From this point onwards, assume that the common production function takes a CobbDouglas form, with F(K, AL) = Kα(AL) 1−α for 0 < α < 1. Write an explicit formula for the capital per efficiency unit of labor in each economy in period t = 0, i.e. k A 0 and k B 0 as a function of parameters only. This means that your formulas should be in terms of constants like s, δ, α, etc... but not variables like K, L, or A.
(c) From this point onwards, assume that a horrible disaster occurred! At the beginning of period t = 1, after investment from period t = 0 has realized and population growth has occurred, a tsunami hit both islands. The Avalonian capital stock was all located on the beach, and half of it was destroyed. But all Avalonian citizens, living in the hills, survived unharmed. The Baltian labor force all lived on the beach, and half of them were carried off to sea. But the Baltian capital stock, located in the hills, survived unharmed. The capital stocks and populations at the beginning of period t = 1 after the tsunami therefore satisfy the equations KA 1 = 1 2 I A 0 + (1 − δ)KA 0 , KB 1 = I B 0 + (1 − δ)KB 0 L A 1 = (1 + n)L A 0 , LB 1 = 1 2 (1 + n)L B 0 . Write an explicit formula for output per capita in Avalon and Baltia in period t = 1 as a function of parameters and initial conditions AA 0 , AB 0 , L A 0 , and L B 0 . Did living standards decrease, increase, or stay the same in Avalon in period t = 1 after the disaster? How about in Baltia? Assume that 1 + g < 2 α.
(d) In the periods t > 1 after the disaster, assuming no other changes to parameters or the structure of the economy, will Avalonian capital per efficiency unit of labor k A t increase, decrease, or stay the same? How about the Baltian level k B t ? Explain your answers.