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ECON 2203Assignment #4: Suggested Answers e)See Figure 1. Figure 1:f) We see that those countries which had the lowest GDP per capita in 1981 tended to grow at a faster rate than those that had higher GDP per capita in 1981. This downward sloping trendline is a good indication that convergence was o...

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ECON 2203 Assignment #4: Suggested Answers 1. e) See Figure 1. Figure 1:

f ) We see that those countries which had the lowest GDP per capita in 1981 tended to grow at a faster rate than those that had higher GDP per capita in 1981. This downward sloping trendline is a good indication that convergence was occurring between Canadian provinces during this period. i) We see that Alberta, Saskatchewan, and Newfoundland and Labrador are all oil producing provinces. This new figure is show in Figure 2. j) We see that the slopes of the convergence lines are quite similar, indicating that convergence is happening at the same rate in all provinces, and that this rate is faster than suggested by Figure 1. Oil producing provinces simply have larger growth rates than non oil producing provinces by around 0.75% per year. This causes their convergence line to be shifted upwards. 2. Suppose Canada’s aggregate production function is given by the following: 1

Figure 2:

1

2

Y = K3N 3

Variables are defined as they were in class. Suppose the savings rate in Canada is 33.33% (s = 13 ) and the depreciation rate is 15% (δ = 0.15). Assume that Canada is not currently experiencing any technological change. a) Using our results from the example in class, we know the steady state level of capital per worker is given by the following:  s 1 K∗ 1−α = δ N  1  K∗ 0.3333 1− 13 = 0.15 N  23 20 K∗ = N 9 K∗ = 3.331269 N 2

Output per worker is given by the following: s α Y∗ 1−α = N δ  13 0.333 1− 13 0.15   12 Y∗ 20 = 9 N

Y∗ = N



Y∗ = 1.4906 N So equilibrium level of capital stock per worker is 3.331 and the equilibrium level of output per worker is 1.491. b) In this version of the model, there is no growth, so the annual growth rate is 0%. c) Output per worker is given by the following: α  s  1−α Y∗ = δ N

 13 0.4 1− 13 0.05   12 Y∗ 8 = 3 N

Y∗ = N



Y∗ = 1.633 N To determine the growth rate, we simply use our equation for how capital stock evolves over time:  α Kt+1 Kt Kt Kt − −δ =s N N N N   13 Kt Kt Kt Kt+1 − 0.15 + 0.4 = N N N N To determine how output grows over time, we plug this into our output per capita equation:   13 2 1 K Y K3 N 3 = = N N N To calculate the growth rate we use the standard equation:   Y − NY t N t+1 Y  Growth Rate = N t

3

See Figure 3 for the graphed growth rate. Savings has not led to a large growth in the economy. A relatively large increase in the savings rate only increased GDP per worker by 9.6% over a large time span. Figure 3:

d) Begin by finding an expression for consumption per capita: Y =C+S C S Y = + N N N C S Y − = N N N C sY Y − = N N N From our steady state condition, we sub out the last term: Y C K = −δ N N N Then we plug in the equations we derived earlier:  s α 1  s  1−α C 1−α = −δ N δ δ Then we plug in for the known values of δ and α:  s 1  s 23 C 2 = − 0.15 N 0.15 0.15 4

We can simplify this to the following: 1

3

C s2 s2 − = 1 1 N 0.15 2 0.15 2 1

3

C s2 − s2 = 1 N 0.15 2 Our goal is to find the value of s which maximizes this expression. You could conduct a simple search using excel, or you could use calculus, take the first derivative and set it equal to zero to get the following:   ∂ NC 1 1 − 12 3 1 2 − = =0 s s 1 ∂s 2 0.15 2 2 1 −1 3 1 s 2 − s2 = 0 2 2 3 1 1 −1 s 2 = s2 2 2 1

1

s− 2 = 3s 2 1 = 3s 1 s= 3

e) Consumption per capita is maximized in this economy when the savings rate is equal to 31 . The increase in the savings rate from 31 to 0.4 will indeed produce more capital, and thus more output. However all of this additional output will go into covering the increased depreciation associated with the increased capital stock. There will not be any extra output left to consume. Therefore, this is a bad policy. 3. a) Begin by converting to output per effective worker: Y = AN



K AN

 13

Plug this into our steady state condition: s



K AN

31

= (δ + gN + gA )

K AN

Plug in the known values and solve: 

K 0.2 AN

 13

= (0.05 + 0.02 + 0.04)

5

K AN

K = 2.452 AN Plug this into the production function: 1 Y = (2.452)3 AN

Y = 1.348 AN For the growth rates, we know that output and capital per effective worker are not growing at all, since they are in a steady state. The growth of technology is given at 4% and growth of workforce is given at 2%. These growth rates imply that capital and output both must be growing at 6% to ensure that the ratios are unchanging. Output per worker and capital per worker both grow at the rate of technological advancement, 4%. A N K Y

4% 2% 6% 6% 4% 4% 0% 0%

Y N K N Y AN K AN

b) Following the same steps, we plug in the known values and solve using our equilibrium condition. 1  K 3 K 0.25 = (0.05 + 0.02 + 0.04) AN AN K = 3.426 AN Plug this into the production function: 1 Y = (3.426)3 AN

Y = 1.508 AN For the growth rates, we know that output and capital per effective worker are not growing at all, since they are in a steady state. The growth of technology is given at 4% and growth of workforce is given at 2%. These growth rates imply that capital and output both must be growing at 6% to ensure that the ratios are unchanging. Output per worker and capital per worker both grow at the rate of technological advancement, 4%. c) As we can see, the answer is no. Although output per effective worker does grow, 6

A N K Y

4% 2% 6% 6% 4% 4% 0% 0%

Y N K N Y AN K AN

the growth rate of output per worker does not change in the long-run. It still grows at the rate of technological progress, 2%.

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