Macroeconomic Theory
Econ 385 – AS01
Winter 2016
Homework Assignment 1
William Easterly in his book ‘The Elusive Quest for Growth’ described the Solow’s growth theory as a ‘shocker’. Discuss why the Solow’s growth model was surprising. Which policy implication of the Solow model was surprising and how did Solow arrive at this policy implication?
Up until the Solow model, economists mainly believed that capital accumulation (machinery and buildings) is the primary determinant of growth (capital fundamentalism). Solow’s growth model concludes that investment in machinery does not lead to long-term growth. The only possible source of growth is technological progress. According to Solow’s model, the more machinery and men an economy had, the higher its production. Growth translates into improved standards of living (through increased labor productivity). Labor productivity increases due to increased machines per worker. However, by increasing the number of machines per worker (if it is indeed possible) does not increase productivity due to diminishing returns to capital. The impact of additional capital diminishes with each additional unit per worker. Even most importantly, buildings and machines comprise a small proportion of GDP, compared to labor income (about two thirds). Effectively, the diminishing returns to capital investments are likely to be severe.
Further, Solow found that saving (which increase investments) is equally impotent in sustaining growth, because it has the effect of diverting resources from consumption to purchase capital in the future, which is futile in creating long term growth. Effectively, countries with high rates of savings are unlikely to experience growth in the long-term, because growth will drop to zero with diminished returns to capital. However, countries such as the US had experienced sustained productivity per worker for over two centuries because of sustained technological progress (p. 51-52). Labor saving technical changes is responsible for long-term economic growth. However, countries with little capital stock must save and invest more in order to maintain their capital-output ratio, and ultimately transition from short-term to long-term growth when technical progress occurs.
Using the Solow growth model, discuss the likely impact of the following changes on the level of Canadian output per worker in the long run (that is steady state):
The government of Canada has introduced a Tax Free Saving Account legislation that allows Canadians to open up a savings account that is sheltered from income tax.
The tax incentive will lead to an increased accumulation of savings, which will be available for investment borrowing in the subsequent years. The rate of consumption or living standards fall. In a steady state, at which the capital per effective worker is given by K*, increasing savings rate leads to increased investments in capital, which pushes productivity up. This increases the steady state capital per effective worker from K* to K1 and increases output/GDP. However, increased capital also leads to increased depreciation rate (δ). There will be an increase in GDP up until when the new level of savings is just equal to the level of depreciation, δ. At this level, a new steady state will be attained, at which the growth level will be equal to the initial growth rate (at K*).
Where output and capital grow at the same rate, gy, then:
Kt+1 = (1+gy) Kt
Kt+1 = (1−δ) Kt +sYt
(1 + gy)(Kt/Yt) = (1 − δ)(Kt/Yt) + s
Obtaining K/Y gives the steady state capital-output ratio as:
KY=s∂+gy
Canadian female participation (but constant population) is expected to continuously increase in the coming years.
The capital per effective worker i.e. k*=KAL will reduce due to an increase in labor supply, there should be an increase the productivity per worker, because the available capital is spread to many workers, which increases the returns to scale for capital. In a steady state, the capital-labor ratio is constant and output per effective worker is constant, but total output and capital would increase proportionally to the increased rate of labor supply.
Suppose that the production function of a country is given by
Y=AKL1-, where A is the technology parameter, L is labor, and K is capital. (50 percent)
Show that the production function exhibits constant returns to scale.
Multiply all inputs by an arbitrary parameter λ >0 where +(1-) = 1
Y=λ(AKαL1-α)
Y=λ(AKαL1-α)
Y=λαAKαL1-αλ ^(1-α)
Y=λ1AKαL1-α
Hence the constant returns to scale since if the inputs are increased by λ, the output will increase by the same multiple.
Are there decreasing returns to capital? (hint: MPK= AK-1L1-)
The slope of the marginal product curve i.e. the derivative of the MPK or the second derivative with respect to capital should be negative for there to be decreasing returns to scale.
∂Y/∂K = αAKα−1L1−α = αY/K
∂2Y/∂K2 =α (α − 1) AKα−2L1−α
Since α is equal to 1, then α (α − 1) <1
Similarly, labor exhibits decreasing returns to scale i.e.
∂Y/∂L = (1 − α) AKαL−α = (1 − α) Y/L
∂2Y/∂L2 = −α (1 − α) AKαL−α−1
Derive the equation for steady state output per worker and steady state consumption per worker in terms of the saving rate (s), the technology parameter (A), and depreciation rate ().
Output per worker (Y/L)
Y=AKαLb
Capital per worker=KL
Investment per worker=IL
Output per worker
Output per worker=YL
YL=AKLa*1L
YL=(AKa)1L*L1-a*L-1 .
YL=(AKa)1L*La-1 =AKaL-a.
Consumption per worker
Consumption per worker=CL
C=1-sY
CL=1-sYL
c=1-sy
I=sY=sYL
i=sy
Steady state level of capital per worker in terms of depreciation (∂) and savings rate (s)
Steady state output per worker
Kt+1L-KtL=sfKL-∂KL
Since the capital stock does not change in a steady state, then
Kt+1L-KtL=0=sfKL=∂KL
=sAKaL-a = ∂KL
s∂=K/(LAKaL-a)
K=s∂A11-aL.
Does the steady state capital stock is equal to K=s∂A11-aL.
Since YL=(AKa)1L*La-1 =AKaL-a.=AKLa
However,
KL=s∂A11-a
Thus
YL=As∂A11-aa
Consumption per worker = C/L
CL=1-sYL=A (1-s) s∂A11-aa
Suppose that = 0.05, = 0.5, and A = 1. With your favorite spreadsheet software, compute steady state output per worker and consumption per worker for s = 0.1, 0.2, 1.
Based on (d) above graph the steady state consumption per worker and output per worker against saving (saving on the horizontal axis). Is there a saving rate that maximizes steady state consumption per worker?
The consumption per worker is optimized at interest rate =0.5
Bibliography
Easterly, William. The Elusive Quest for Growth: Economists' Adventures and Misadventures in the Tropics. Boston: The MIT Press; Revised ed. edition, 2002.
