ECON 100C Equilibrium of The Slow Growth Model Question

Instructions

The aim of this exercise is for you to simulate the equilibrium of the Solow growth model on the computer. By programming it on a simple spreadsheet you will obtain a better understanding of the model.

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Follow the prompt below. You will need to create a spreadsheet. To do so, you may use any spreadsheet software, for example, Excel (available to UCI students here
Links to an external site.
). You may also use Python, Matlab, R, or Julia. If you prefer to use another programming language please email Bryan for approval first.

To receive full credit, you will need to answer all the questions and upload your spreadsheet (click “file upload”)

If you have difficulties completing this assignment, make sure to ask questions during office hours, we will be happy to help.

Questions

Suppose that you have a standard Solow model with a Cobb-Douglas production function,
F(
K
t
)=A
K
α
t
L
1−α
t
F(Kt)=AKtαLt1−α. Employment,
L
t
Lt, and TFP (A) are constant. As in lecture, denote capital per worker as
k
t
=
K
t
/
L
t
kt=Kt/Lt. The change in capital per worker can be written as
k
t+1

k
t
=sf(
k
t
)−d
k
t
kt+1−kt=sf(kt)−dkt. Equivalently, it can be written as
k
t+1
=sf(
k
t
)+(1−d)
k
t
kt+1=sf(kt)+(1−d)kt (easier to program).

(a) Derive the closed-form solution for steady-state capital stock,
k

k∗, by hand. (5 points points)

(b) Suppose that A = 1, s = 0.2, d=0.1, and
α
α= 1/3. Create a new spreadsheet. In the top-left cell, using your answer from (a), numerically solve for
k

k∗(you should get a number). (5 points)

(c) Create a column in your spreadsheet corresponding to periods. Let these periods run from period 0 to period 100. Create additional columns for capital stock, output, consumption, investment, depreciation, and change in capital stock. (2 points)

(d) Suppose that in period 0, the capital stock per worker equals its steady state (i.e.
k
0
=
k

k0=k∗), computed in (b). Compute and fill in for output, consumption, investment, depreciation, and change in capital stock in period 0. (3 points)

(e) Using the central equation of the Solow model, compute capital stock per worker in period 1. Then repeat the process and fill in for output per worker, consumption per worker, investment per worker, depreciation, and change in the per worker capital stock in period 1. (2 points)

(f) Iterate this process until you have filled all of the columns up to period 9. What is true about the per worker capital stock in periods 0 through 9 when the per worker capital stock starts in the steady-state in period 0? (3 points)

(g) Suppose that in period 10, the saving rate increases to 0.25 and is expected to forever remain there. What will happen to the capital stock in period 10? (2 points)

(h) Compute the capital stock in period 11, given the capital stock in period 10 and the new, higher saving rate. Compute and fill in for output, consumption, investment, depreciation, and change in capital stock in period 11. (1 point)

(i) Iterate and fill in all of the columns all the way to period 100. (2 points)

(j) Approximately how many periods does it take the capital stock to get halfway to its new, higher steady-state value when s increases from 0.2 to 0.25? (5 points)

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