ASSIGNMENT

QUESTION 1 - Assume that the production function is given by:

Y = K^αH^φ(AL)^1-α-φ

where Y is output, K is physical capital stock, H is human capital, A is the level of technology, and L is labor. Assume α > 0, φ > 0 and α + φ < 1. L and A grow at constant rates n and g, respectively. Both physical capital and human capital depreciate at the rate δ. Assume the physical capital and human capital accumulation equations are given as: K^· = s_kY - δK and H^· = s_HY - δH. Where S ≡ I = (s_k + s_H)Y where s_k is fraction of output saved or invested in the physical capital and s_H is a fraction of output invested in the human capital, s_k + s_H < 1.

A) Assume k^~ = K/AL and h^~ = H/AL per effective labor values. Derive the law of motion for k^~ and h^~.

B) Derive the steady state values for k^~, h^~ and y^~ i.e. in terms of per effective worker. [show your work to get the steady state values]

C) What is the growth rate of output per worker in the steady stateφ

D) This Solow model with labour augmenting technology can be tested empirically with cross-country data if we assume that all countries are in their steady states. Derive a log-linear regression equation (using discrete time) for output per worker that you could estimate using OLS assuming you have measures for sⁱ_K, sⁱ_H, δⁱ, nⁱ for each country i and that g and A_o are known constants across countries. where A_t = A_oe^gt.

(i) If we exclude human capital from the model i.e. φ = 0 then what is the interpretation of the model.

(ii) if α = φ = 1/3 then what is the elasticity of output per worker in the steady state (y*) with respect to s_K.

(iii) What is the role of S_H in the determination of output per worker in the steady state (y*) as compared to the Solow model with exogenous technology.

(iv) How does population growth rate affects the output per worker in the steady state (y*) as compared to the Solow model with exogenous technology.

QUESTION 2 - Consider a labour augmenting Solow growth model represented by Cobb-Douglas production: Y = K^α(AL)^1-α, savings rate s, depreciation rate δ, population growth rate n, and rate of technological progress equal to g. 

Consider the following empiric al observations for the Canadian economy:

• Capital stock (K) is 2.5 times GDP (Y), population growth (n) is roughly 2%
• Depreciation (δ) accounts for 10% of GDP (Y).
• GDP (Y) grows at a rate of 3% Capital owners' share of output (α) is roughly 30%

(a) Based on these data, is Canadian economy currently at the golden rule level of capitalφ If not then based on these data, what is the golden rule level of capitalφ

(b) Suppose Canadian government implements a policy that achieves the savings rate needed to achieve the golden rule level of capital. Using impulse responses, illustrate how the following variables would change as the Canadian transitions to its new balanced growth path: capital, output, and consumption (all in per effective worker form). Illustrate in level form and log level form i.e.  y^~ and ln(y^~).

QUESTION 3 - Consider the below model:

Y = K^α(AL)^1-α

Where A is the stock of ideas, L_Y is the amount of labour used in the production of the final good, K is the stock of capital and, α is the constant between 0 and 1. the law of month of A is: A^· = δA^φL^λ_A, δ is constant between 0 and 1, L_A is the labour devoted in the production new ideas. We assume φ > 0 and λ  ≤ 1.

The resource constraint of the economy is: L_A + L_Y = L and proportion devoted to both activities is constant: L_A/L = S_R and L_Y/L = S_Y = 1 - S_R. The population grows at the rate of n i.e.:  L^{^}_A = L^{^}_Y = L^{^} = n.

The capital accumulation for the physical capital stock is:  K^· = S_KY. Depreciation is assumed to be zero.

(c) Show that, at any time, the growth of technology, g_A, is:  g_A = δA^φ-1(S_RL)^λ. Explain for each of the two cases: φ = 1 and φ < 1 how the output of the research sector A^·, the technological level A, and the technological growth rate g_A  evolve over time when the labour input into the research sector  L_A = S_RL is constant.

(d) Derive the equation for the growth rate of capital per worker as:

k^{^} = S_KS^1-α(k/A)^α-1 - n

(e) Give an intuitive explanation of what the parameters λ and φ between zero and one imply.

(f) Assume now that λ = 1 and φ =0. Write down the equation for the growth rate of A at the aggregate level, the growth rate of k =K/L and the growth rate of y=Y/L.

(g) Define a balanced growth path (BGP) in this model. Assume that along a BGP the growth rates of LA and LY equal the rate of population growth (n). Derive the growth rate of A and y along the BGP.

QUETSION 4 -

Assume a production function a competitive firms using labor L_Y and collection of intermediate capital goods xj. The amount of human capital per person in the firm determines the range of intermediate capital goods that firm can use. , human capital used in the model is interpreted as skill or experience in using advanced intermediate goods. The production function for a firm employing workers of average skills h is

Y = L^1-α₀∫^hx_j^αd_j                     (1)

Intermediate goods (x_j) are treated symmetrically throughout the model, so that x_j = s for all j. This symmetry also describe the demand for raw capital (K) is equal to ₀∫^h x_j^αd_j i.e. units of an intermediate capital good x_j are created one-for-one with units of raw capital. Market clearing condition ensures K = hx. The above relationships imply that the aggregate production function can be written as:

Y = K^α(hL)^1-α                          (2)

Assume ­L^{^} = n, K^· = S_KY - δK and human capital accumulation equation: h^· = μ^ψμA^γh^1-γ. μ > 0, 0 > γ > 1 and ψ > 0. Where μ is number of years of schooling on measure skills, and A represents the technological frontier, i.e. the total measure of intermediate goods that have been invented to date. Using the above relationships to derive output per worker at BGP (y_BGP) as:

y_BGP = (S_K/(n+δ+g))^α/1-α(μe^ψμ/g)^1/γ A