Question

AutoSave OFF OF UA HW2P3 - Compatibility Mode a Home Insert Draw Design Layout References Mailings Review View Tell me Share Comments Times New... v v 12 A A Aa A EEE 24 AalbCelld! AnBIX Dde Anklede ABCcDde AaBbCcDc Aabotee Normal No Spacing Hercing 2 Title Subtitle Pagic B I x? == Heading Dictate Sensitivity Sty es Pare 1. Assume the following production function: Y = AK L -- where: A-1 and <2-0.5 (arx Y-real GDP, K - capital, 1. - labur) Let's also assure that: (where s= savings rate) 8-0.2 (where d-depreciation rate) kr 16 (wherek-KL) a) Are we at the steady state level of capital (*)? b) What is the steady state level of capital (A*? c) Are we at the Golden Rule Stendy State vnlue of K? c) What value of allow this entry to operate at the Colden Rule Steady State value olk? What is the value of the Golden Rule Steady State value of ? e) Describe how consumption and investment will change over time as this ecxininy adjusts and We couverge on the Golden Rule Steady State value of . 2. Assurne that we have an econciny where the quantity of latxır is no longer fixed and that technological change is possible, and that we have the following production fauction: Y-AK" L x 5 where: A = 1 and a=0.5 and Y = rcal GDP, K = capital, L = labor, E = efficiency of Labor and Lx-elective number of laborers) a. Let's assume Country X has the following values for depreciation rate (8) and technological progress (6): 8-0.04 nnd g=0.02. Explain how rapid increases in population growth affect this country's steady state level of capital versus an incrense in the country's sovinus rate. b. Assume that there is a second country, Country Y, and that Country Y las the same values for their depreciation rate mod technological progress as Country X Let's assume that Country X has a slow-growing population and high savings rate, vs Country Y which has a fast-growing population and low savings rate. Their respective values for thesc growth rates is as follows: Country X -0.28 11-0.01 Country Y s-0.10 I=0.04 is-saving rate) (n = population growth rate) Cliven these values fors and n in each country, how much vulput per effective laborer does each country produce when operating at the steady state level of capital? Page 1 of 1 392 words DE Focus --+ 100%

Please answer question 1, parts A,B,C, and D+E if possible. Use formulas to express your solutions! Be sure to show all work used in the problems including equations, and how you got to the solution. I need to understand the math behind the solution and how you got to the answer. Thank you!

Answer 1

Production function, Y = K^0.5 * L^0.5

=> Y/L = (K/L)^0.5

=> y = k^0.5

Here, y = output per worker

k = capital per worker

Consider the change in capital per worker,

dk = s*y - c*k

Here, s = saving rate

D = depriciation

At steady state, dk = 0

=> y/k = D/s

Substituting y = k^0.5 in the above equation, we get,

k^0.5 / k = D/s

=> k = (s/D)^2 ---> Eq1

Thus, steady state level of capital per worker is (s/D)^2

Now, consumption, C = y - sy

=> At steady state, sy = Dk and also, y = k^0.5, so,

C = k^0.5 - Dk

At golden rule steady state, consumption level is maximized, so,

dC/dk = 0 = 0.5/k^0.5 - D

=> k = (1/2D)^2 ---> Eq2

Thus, at steady state capital per worker = (1/2D)^2

For s = 0.4 and D = 0.2

the steady value of capital per worker, k = (0.4/0.2)^2 = 4 units (From Eq1)

and Golden rule steady state of capital per worker, k* = (1/2*0.2)^2 = 6.25 units (From Eq2)

a) Thus, capital per worker of 16 units is not the steady state level of capital per worker

b) Steady state level of capital per worker = 4 units

c) No, capital per worker of 16 units is not the golden rule steady state of capital per worker

d) The golden rule steady state level of capital per worker, k* = 6.25 units

So, at steady state capital per worker,

k* = (s*/D)^2 (From Eq1)

Here, s* is the new saving rate which allows capital per worker to be equal to golden rule steady state,

Thus, s* = k^0.5 * D = 6.25^0.5 * 0.2 = 0.5

Thus, new saving rate which maximizes consumption is 0.5.

e) Currently capital per worker is 16 units which is way more than the golden rule steady state level of 6.25 units. So, the depriciation at this level is more than the investment being made which will decrease the stock of capital per worker over time till it comes to the level of 6.25 units where the economy is at steady state and investment being made at new savings rate (=0.5) equals the depriciation of capital per worker.

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