1(a) In the Solow Growth Model the output function for the economy is given by: Y = 1.02 K 1/3 (L x E)2/3 Y = output; K= capital stock; L = labour force; E = efficiency of labour so that L x E = labour force measured in efficiency units) The depreciation rate () is 0.1, the growth rate of the population (n) is 0.03 and the rate of advance in technology (g) is 0.07. Use mathematical analysis to calculate the golden rule steady state capital stock per
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- Consider an economy described by the Solow model with the following production function: Y = F(K, L) = K“ (L)'-« L grows at the rate n, the depreciation rate is 8, and the country saves a constant fraction s of its income. The change in capital per-worker is given by Ak = sy – (n+ 8)k. (a) Derive the per-worker production function. (b) Assuming population growth equals n and the depreciation rate equals 8, find the steady state level of capital per worker. It will depend on a, s, n and 8. Imagine the economy begins at the steady state you found in part b. Then there is a war that destroys a substantial amount of the economy's capital. The war does not affect the size of the labor force, population growth, the depreciation rate, or the saving rate. c) What is the immediate effect of the war on output per worker? Explain. d) After the war, is the growth rate of output per worker higher or lower than it was in steady state? Explain. e) How does the war affect steady state output per…Consider a standard Solow growth model. Denote capital stock as K, population as N, capital depreciation rate as d, saving rate as s, output as Y. Output is produced by a representative firm according to the production function Y=zF(K, N), where z is current total factor productivity. The law motion for capital is K' = (1 - d)K + I, where K' is the future capital stock. Population grows at a constant rate n, that is N' = (1 + n)N, and household supply labor inelastically, so population equals labor force. (a) In a graph, show the steady state level of capital per worker. Use lower case letters to denote per-capita terms and use * to denote steady state.(b) Suppose a country is initially at a steady state, then a war destroyed some of its capital stock. Determine the long run effects on the quantity of capital per worker and on output per worker in the steady state. Show by a graph. (C)Define golden rule saving rate. What does it maximize? Determine the effects of a decrease in the…Problem 2: Growth Models Solow Growth Model: Assume there is an imaginary country where output is produced according to the following constant returns to scale production function that lies at the heart of the standard Solow growth model: Y = 10K0.5 L0.5 Also assume that the savings rate is 13% (s = 0.13), the population grows at a rate of 2% per year (n 0.02), and capital depreciates at a rate of 2% per year (d = .02). (a) Which of the following is an expression of the production function in per-capita terms y expressed as a function of capital per worker k: i. Y = k0.5 ii. Y = 10k0.5| iii. y = k0.5 iv. y = 10k0.5 (b) What is the steady state level of capital per worker? (c) What is the steady state level of output per worker? (d) Suppose that the populist leader of our imaginary country increases the savings rate from 13% to 15%, i.e., (s = 0.15), what is the new steady state level of capital per worker? [Note: Assume that the other parameters n and d remain unchanged.] (e) [TRUE or…
- In the Solow Growth Model, a country's production function is defined by the following: Y = F (K/L) = kºs Where K is capital and I. labour; labour grows at a rate (n), and the economy faces depreciation at a rate (8) The initial Capital Stock per worker: k, = 9 units The savings rate: s = 0.20 and the rate of depreciation: a = 0.1 Using equations and identities from the Solow Growth Model, calculate the level of capital k, Output y, Consumption c, depreciation and change in capital from the equation of motion for periods 1, 2 and 3. Comment on the accumulation of capital and output over the three periods.Consider the continuous-time Solow growth model as discussed in the lecture. The economy is on its balanced growth path with labor augmenting technological progress at rate g and population growth at rate n. The depreciation rate is 8. (a) Derive the dynamic equation of aggregate capital K. (b) Normalize the aggregate capital K by technology A and population L and denote the capital per unit of effective labor as k = K/(AL). Derive the dynamic equation of k .(c) Use the phase diagram to illustrate the steady state level of k and the dynamics of k if k starts from a level lower than the steady state level. (d) Suppose that the economy is on its balanced growth path at time to. Unex- pectedly, the population growth rate drops down to n' <n at to, stays at n' until t1, and resumes to n after t1. How does k respond, between to and t1, and afterwards?Question 1 1(a) In the Solow Growth Model the output function for the economy is given by: Y = K 1/3 (L x E) 2/3 (Y = output; K = capital stock; L = labour force; E = efficiency of labour so that L x E = labour force measured in efficiency units) The depreciation rate (8) is 0.05, the growth rate of the population (n) is 0.03 and the rate of advance in technology (g) is 0.07. Use mathematical analysis to calculate the golden rule steady state capital stock per efficiency unit of labour (k*gold). 1(b) Create a schedule that confirms that the result in part (a) is correct. The schedule should include the following seven columns: the savings rate, the steady state capital stock per efficiency unit of labour, the steady state output per efficiency unit of labour, the sum of the depreciation rate, population growth rate and the rate of advance in technology all with respect to the steady state capital stock per efficiency unit of labour, the steady state consumption per efficiency unit of…
- 1(b) Assume that the production function for an economy in the Solow Growth Model is given by: Y = K 1/3 (L x E) 2/3 (where Y = Total output; K = Capital stock; L = Labour force; E = advances in technology, which are measured as improvements in the efficiency of labour). The savings rate in the economy is 0.4, the depreciation rate is 0.1, the growth in the labour force is 0.05 and technology advances at a rate of 0.05 (each of these changes relate to a given time period). Calculate the steady state capital stock per efficiency unit of labour (k*).Consider the Solow model with a production function Y(t) = A*K(t)αL(t)1-α, Where A is a fixed technological parameter. Explicitly solve for the steady-state value of the per capita capital stock and per capita income. How do these values change in response to a rise in (a) the technological parameter A, (b) the rate of saving s, (c) α , (d) δ, the depreciation rate, and the population growth rate n?Consider the basic Solow growth model. Let the aggregate production function be defined as Y = F(K, L) = K0.5 L0.5 where Y is output, K is capital, and L is labor. Furthermore, let the saving rate be 48%, population growth be 2%, and depreciation rate be 10%. a. Find the steady-state levels of capital per worker k and output per worker y.. %. b. Now assume that, because of the proliferation of financial technologies, the saving rate increases suddenly to 54% after one year. The steady-state level of capital will increase by ____%, and the steady-state level of output will increase by_
- Consider the Solow growth model. Suppose that F(K, N) = zK^a N^1-a,where a = 0.3. Also, assume that capital depreciation rate is 10% (that is d = 0.1), savings rateis 25% (that is s = 0.25), populations growth rate is 2% (that is n = 0.02), and z = 2.• First, determine capital per worker, income per capita, and consumption per capita in thesteady state.• Second, assume that the savings rate has increased to 40% but the total factor productivitydecreases to 1. Discuss the effect of savings and productivity on the steady state level ofconsumption per worker. PLEASE SHOW ALL HAND WRITEN STEPS!!Problem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 8, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (i) Does this production function exhibit constant returns to scale? Explain why. (ii) Does it exhibit diminishing returns to physical capital? Explain why. • (ii) Express output per person, y =- -, as a function of capital per person, k =. • (iv) Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) (v) Use the production function in per capita terms to write the fundamental equation of the Solow-Swan model. • (vi) Suppose first that sA 8 + n. Draw the savings and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these…Problem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 8, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (i) Does this production function exhibit constant returns to scale? Explain why. (ii) Does it exhibit diminishing returns to physical capital? Explain why. • (ii) Express output per person, y =- as a function of capital per person, k =. • (iv) Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) (v) Use the production function in per capita terms to write the fundamental equation of the Solow-Swan model. • (vi) Suppose first that sA 8 + n. Draw the savings and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these circumstances,…