Question

Q2)

Consider two imaginary countries, indexed A and B. Each economy can be characterised by the model above, but the population is constant in both economies. In the steady state, GDP per worker in country A is 1.44 times that of country B and the ratio of physical investment to output is 0.3 in country A and 0.25 in country B. The rate of depreciation is the same in both countries. What must α be in order for the model to fit these facts?

Q3)

Consider the two countries A and B above, but modify the model along the lines of Mankiw, Romer and Weil (1992) so that human capital, H, is included as a factor input. For simplicity, labour efficiency is assumed to be 1 in both countries. Output in country i is thus given by

Y_it = K_it^αH_it^βL_i^{(1−α−β)},

and capital is accumulated according to

∆K_it+1 = s_ikY_it − δK_it,

∆H_it+1 =s_hY_it − δHit,

where we note that s_h is the same in both countries.

A. Let y ≡ Y/L denote GDP per worker. Derive an expression for the steady-state value of the ratio y_A/y_B in terms of s_Ak, s_Bk, α and β.

B. Suppose that β = 0.5.  What must α be in order for this model to fit the facts stated in Question 2?

Answer 1

Q2

answer is , alpha = 1/3

Let Y_t = K_t alpha L_t (1 - alpha), y^alpha_A = 1.44 y^alpha B, n = 0 in both Nation (I/Y)^A = 0.3 (I/Y)^B = .25, delta_A = delta_B, alpha = ? at Steady state: delta y = delta k y_t = k_t^alpha then s = delta k^1 - alpha k^alpha = (8/delta)^1/1 - alpha then y^alpha = k^alpha = (s/delta)^alpha/1 - alpha thus (s_A/delta)^alpha/1 - alpha = (1.44) (s beta/delta)^alpha/1 - alpha (S_A/S_B)^alpha/1 - alpha = 1.44 as I = S Y, then S^A = .3 & S^beta = .25 I/Y = S thus (.3/.25)^alpha/1 - alpha = 1.44 .30/.25 = (1.44)^(1 - alpha)/alpha 1.2 = (1.44)^(1 - alpha/alpha) ln (1.2) = (1 - alpha/alpha) ln (1.44) .182 (1 - alpha/alpha) [.364] .182 alpha =.364 - .364 alpha .564 alpha = .364 thus alpha = .364/.546 = .67, alpha = .67 = 1/3