The production function

QUESTION-1: [30] Assume that the production function is given by:      1 Y K H (AL) where Y is output, K is physical capital stock, H is human capital, A is the level of technology, and L is labor. Assume   0 ,   0 and   1. L and A grow at constant rates n and g , respectively. Both physical capital and human capital depreciate at the rate  . Assume the physical capital and human capital accumulation equations are given as: 𝐾̇ = 𝑠௞𝑌 − 𝛿𝐾 𝑎𝑛𝑑 𝐻̇ = 𝑠ு𝑌 − 𝛿𝐻. Where S  I  (sk  sH )Y where Ks is fraction of output saved or invested in the physical capital and H s is a fraction of output invested in the human capital,   1 K H s s A) [10] Assume 𝑘෨ = ௄ ஺௅ and ℎ෨ AL H  per effective labor values. Derive the law of motion for 𝑘෨ 𝑎𝑛𝑑 ℎ෨. B) [15] Derive the steady state values for 𝑘෨ , ℎ෨ 𝑎𝑛𝑑 𝑦෤ i.e. in terms of per effective worker. [show your work to get the steady state values] C) [5] What is the growth rate of output per worker in the steady state? QUESTION-2. [15] Consider a labour augmenting Solow growth model represented by Cobb-Douglas production:    1 Y K (AL) , savings rate s, depreciation rate δ, population growth rate n, and rate of technological progress equal to g. Consider the following empirical observations for the Canadian economy:  Capital stock (K) is 2.5 times GDP (Y), population growth (n) is roughly 2%  Depreciation ( ) accounts for 10% of GDP (Y).  GDP (Y) grows at a rate of 3% Capital owners’ share of output () is roughly 30% (a) [10] Based on these data, is Canadian economy currently at the golden rule level of capital? If not then based on these data, what is the golden rule level of capital? Created in Master PDF Editor (b) [5] 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: [30] Consider the below model:      1 Y K ALY Where A is the stock of ideas, LY is the amount of labour used in the production of the final good, K is the stock of capital and,  is the constant between 0 and 1. the law of month of A is:   A  A LA  ,  is constant between 0 and 1, LA is the labour devoted in the production new ideas. We assume   0 and   1 The resource constraint of the economy is: LA  LY  L and proportion devoted to both activities is constant: R A s L L  and Y R Y s s L L   1 . The population grows at the rate of n i.e.: L L L n ˆ A  ˆ Y  ˆ  The capital accumulation for the physical capital stock is: K  sKY . Depreciation is assumed to be zero. [6 marks each part = 30] (a) Show that, at any time, the growth of technology, gA, is:     g A A sR L 1  . Explain for each of the two cases:   1 and  1 how the output of the research sector A , the technological level A, and the technological growth rate A g evolve over time when the labour input into the research sector LA  sR L is constant. (b) Derive the equation for the growth rate of capital per worker as: n A k k s s K Y           1 ˆ 1   (c) Give an intuitive explanation of what the parameters  and  between zero and one imply. (d) Assume now that  = 1 and  =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. Created in Master PDF Editor (e) 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: [20 marks] Assume a production function of a competitive firm using labor L 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 ଵିఈ ∫ 𝑥௝ ఈ𝑑௝ ௛ ଴ (1) Intermediate goods (xj) are treated symmetrically throughout the model, so that xj = s for all j. This symmetry also describe the demand for raw capital (K) is equal to ∫ 𝑥௝ ఈ𝑑௝ ௛ ଴ i.e. units of an intermediate capital good xj 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 ఈ(ℎ𝐿) ଵିఈ (2) Assume 𝐿෠ = 𝑛,𝐾̇ = 𝑠௄𝑌 − 𝛿𝐾 𝑎𝑛𝑑 ℎ𝑢𝑚𝑎𝑛 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛: ℎ̇ = 𝜇நఓ𝐴 ఊℎ ଵିఊ . 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. Derive the output per worker at BGP (𝒚𝑩𝑮𝑷) as given by: 𝑦஻ீ௉ = ቀ ௦಼ ௡ାఋା௚ቁ ഀ భషഀ ቀ ఓ௘ಠಔ ௚ ቁ భ ം 𝐴. QUESTION-5: [15 marks] Consider a endogenous growth model with human capital. The aggregate technology is Y = Kα [h(1 − u)L] 1−α , where h is the level of human capital and u is the fraction of time the population spends acquiring human capital. Physical capital is accumulated in the usual fashion, 𝐾̇ = sKY − δK. Human capital is accumulated according to the following differential equation: ℎ̇ = ψuh. Assume that both sK and u are constant and exogenously given. A. Define capital and output per unit of “effective labour” by 𝑦෤ ≡ Y /hL and 𝐾෩ ≡ K/hL respectively. Derive expressions for 𝑘෨ and in 𝑦෤ the steady-state.