\(\)\(\)\(\)a) At the time of the jump, output per unit of effective labor is reduced. This is implied by the equation \(k\ =\ \frac{K}{AL}\). An increase in L without a jump in K or A causes k to fall. Since \(f'\left(k\right)\ >0\), that is the derivative of the production function with respect to capital per unit of effective labor is positive, then a reduction of \(k\) must lead to a reduction in output per unit of effective labor.
b) In the short run, a jump in the amount of labor means a reduction of capital per unit of effective labor from \(k^\star_1\) to \(k_2\). Once it reaches \(k_2\) however, savings are superior than break-even investment, i.e. \(sf(k) > (g+\delta)k\). The economy is now saving and investing more than enough to offset
depreciation and technological
progress and \(k\) increases towards \(k^\star_1\) which is the initial balanced growth path amount of capital.
c) Once the economy has again reached a balanced growth path, output per unit of effective labor is the same as it was before the new workers appeared. This is so because there is no population growth on the break-even investment path. Thus \(k\) tends toward \(k^\star_1\) after any change in the number of workers and hence \(f(k)\) tends back toward \(f(k^\star_1)\) which is the original balanced growth path output.
1.5
a)
The equation describing the evolution of the capital stock per unit of effective labor is given by:
\(\dot{k}_t = sf(k_t) -(n+g+\delta)k_t\)
With \(f(k) = k^\alpha\) which means that
\(\dot{k}_t = sk^\alpha_t -(n+g+\delta)k_t\)
We have to resolve the differential equation -i.e. what is the path of \(\dot{k}\).
We define \(v = k^{1-\alpha}\) which can be translated in terms of k as:
\(k = v^{\frac{1}{1-\alpha}}\)
\(\dot{k}_t = (\frac{1}{1-\alpha})v^{\frac{\alpha}{1-\alpha}} \frac{dv}{dt}\)
We substitute \(k\) and \(\dot{k}\) in the initial equation
\(\frac{1}{1-\alpha} v^{\frac{\alpha}{1-\alpha}} \frac{dv}{dt} = sv^{\frac{\alpha}{1-\alpha}} -(n+g+\alpha)v^{\frac{1}{1-\alpha}}\)
\(\frac{dv}{dt} = \frac{s-(n+g+\alpha)v}{\frac{1}{1-\alpha}}\)
\(\frac{dv}{dt} = (1-\alpha)(s-(n+g+\alpha)v)\)
\(\frac{dv}{dt} + (1-\alpha)(n+g+\alpha)v = (1-\alpha)s\)
Now we can set to zero:
\(\frac{dv}{dt} + (1-\alpha)(n+g+\alpha)v = 0\)
\(\int \frac{1}{v} dv = -(1-\alpha)(n+y+\delta) \int dt\)
\(ln(v) + c_1 = -(1-\alpha)(n+y+\delta)t +c_2\) With \(c = c_2 - c_1\)
\(e^{ln(v)} = e^{-(1-\alpha)(n+y+\delta)t}e^{c}\) With \(A = e^c\)
\(v = Ae^{-(1-\alpha)(n+y+\delta)t}\)
\(\dot{v} = -(1-\alpha)(n+y+\delta)Ae^{-(1-\alpha)(n+y+\delta)t}\)
We substitute \(\dot{v}\) in: \(\dot{v} + (1-\alpha)(n+g+\alpha)v = (1-\alpha)s\) for \(t=0\)
\(-(1-\alpha)(n+y+\delta)Ae^{-(1-\alpha)(n+y+\delta)0} + (1-\alpha)(n+g+\alpha)v_0 = (1-\alpha)s\)
\(-(1-\alpha)(n+y+\delta)A = (1-\alpha)s - (1-\alpha)(n+g+\alpha)v_0 \)
\(A = v_0 -\frac{s}{n+g+\delta}\)
Hence we have:
\(\dot{v} = -(1-\alpha)(n+y+\delta)[v_0 -\frac{s}{n+g+\delta}]e^{-(1-\alpha)(n+y+\delta)t}\)
We first rearrange the following equation to express it in terms of \(v\) before substituting in \(\dot{v}\).
\(\dot{v} + (1-\alpha)(n+g+\alpha)v = (1-\alpha)s\)
\(v = \frac{s}{n+g+\delta} - \frac{\dot{v}}{(1-\alpha)(n+g+\delta)}\)
\(v = \frac{s}{n+g+\delta} + [v_0 -\frac{s}{n+g+\delta}]e^{-(1-\alpha)(n+y+\delta)t}\)
Since \(v = k^{1-\alpha}_t\)
\(k^{1-\alpha}_t = \frac{s}{n+g+\delta} + [k^{1-\alpha}_0 -\frac{s}{n+g+\delta}]e^{-(1-\alpha)(n+y+\delta)t}\)
With time the amount of capital will converge toward its balanced growth path:
\(\lim_{t\to\infty} k^{1-\alpha}_t = \frac{s}{n+g+\delta} \)
Hence:
\(k^{\star} = [\frac{s}{n+g+\delta}]^{\frac{1}{1-\alpha}}\)
Since \(y = k^\alpha\) the output at the balanced growth path is:
\(y^{\star} = [\frac{s}{n+g+\delta}]^{\frac{\alpha}{1-\alpha}}\)
And since \(c = y - sy = (1-s)y\) consumption at the balanced growth path is:
\(c^{\star} = (1-s)[\frac{s}{n+g+\delta}]^{\frac{\alpha}{1-\alpha}}\)
b)
The golden-rule level of the capital stock is that balanced growth path level at which consumption per unit of effective labor is maximized.
We rearrange the following equation to find savings:
\(k^{\star} = [\frac{s}{n+g+\delta}]^{\frac{1}{1-\alpha}}\)
\(s = (n+g+\delta)k^{\star (1-\alpha)}\)
As we know that consumption at the balanced growth path is equal to:
\(c^{\star} = (1-s)[\frac{s}{n+g+\delta}]^{\frac{\alpha}{1-\alpha}}\)
We use substitution to find:
\(c^{\star} = [1-((n+g+\delta)k^{\star (1-\alpha)})][\frac{(n+g+\delta)k^{\star (1-\alpha)}}{n+g+\delta}]^{\frac{\alpha}{1-\alpha}}\)
\(c^{\star} = k^{\star \alpha} - (n+g+\delta)k^{\star}\)
\(\frac{dc^{\star}}{dk^{\star}} = \alpha k^{\star (\alpha-1)} - (n+g+\delta)\)
Because \(1 > \alpha >0\), the second derivative his negative, which means that
\(\frac{d^{2}c^{\star}}{(dk^{\star})^2} = (\alpha-1)\alpha k^{\star (\alpha-2)}\)
\(\frac{d^{2}c^{\star}}{(dk^{\star})^2} < 0\) since \((\alpha - 1) < 0\) and \(\alpha k^{\star (\alpha-2)} > 0\)
This means that the function \(c^{\star}(k)\) is concave and therefore there is a maximum when \(\frac{dc^{\star}}{dk^{\star}} = 0\)
\(0 = \alpha k^{\star (\alpha-1)} - (n+g+\delta)\)
\(\alpha k^{\star (\alpha-1)} = n+g+\delta\)
Since \(f(k)^{\star} = k^{\alpha}\) we have \(f'(k)^{\star} = \alpha k^{\star\alpha -1}\)
\(f'(k)^{\star} = n+g+\delta\)
The last equation means that consumption is maximized when the slope of the output function at the balanced growth path is equals the slope of the break-even investment function.
So the golden-rule amount of capital is:
\(\alpha k^{\star (\alpha-1)} = n+g+\delta\)
\(k^{\star}_{GR} = [\frac{\alpha}{n+g+\delta}]^{\frac{1}{1-\alpha}}\)
c)
At the balanced growth path we have:
\(s = (n+g+\delta)k^{\star (1-\alpha)}\)
Hence we substitute \(k^{\star}_{GR}\) in the equation:
\(s^{\star}_{GR} = (n+g+\delta)[(\frac{\alpha}{n+g+\delta})^{\frac{1}{1-\alpha}}]^{(1-\alpha)}\)
\(s^{\star}_{GR} = \alpha\)
With a Cobb-Douglas production function, the saving rate required to reach the golden rule is equal to the capital share in output or to the elasticity of output with respect to capital.
1.8)
a)
We know that the elasticity of the balanced growth path output with respect to the saving rate is equal to:
\(\frac{dy^\star}{ds}\frac{s}{y^\star} = \frac{\alpha(k^\star)}{1-\alpha(k^\star)}\)
Since on the balanced growth path the share of income going to capital is equal to \(\alpha(k^\star)\) and since we assume that capital’s share is equal to \(\frac{1}{3}\), the elasticity of the balanced growth path output with respect to the saving rate is equal to:
\(\frac{dy^\star}{ds}\frac{s}{y^\star} = \frac{\frac{1}{3}}{1-\frac{1}{3}}=0.5\)
An increase of \(s\) from 0.15 to 0.18 is a 20% increase. Thus, since \(\frac{dy^\star}{ds}\frac{s}{y^\star} =0.5\), this 20% increase in \(s\) implies an increase of 10% in output.
b)
Since consumption equals:
\(c=(1-s)f(k)\)
\(\frac{dc}{ds}\frac{s}{c} = \frac{s}{c}[(1-s)f'(k) - f(k)]\)
\(\frac{dc}{ds}\frac{s}{c} = \frac{sf'(k)}{f(k)} - \frac{s}{1-s}\)
We have \(\frac{sf'(k)}{f(k)}\) which is the elasticity of output with respect to the saving rate and which equals 0.5.
For \(s\), we can take the midpoint between 0.15 and 0.18.
\(\frac{dc}{ds}\frac{s}{c} = 0.5 - \frac{0.165}{1-0.165} \approx 0.30\)
The elasticity of consumption with respect to the saving rate is approximately 0.3. Our 20% increase in \(s\) implies an increase of 6% in consumption. .
c)
Since \(c=(1-s)y\), consumption goes from \(0.85y\) to \(0.82y\) in the short run. In other words, consumption drops by about 3.5% in the very short run. In the long run, consumption increases by 6% -i.e. \(c=0.901\).
From equation (1.30), we know that:
\(y(t) \approx y^\star + e^{-\lambda t} [y(0)-y^\star]\)
With \(\lambda \approx [1-\alpha_k](n+g+\delta)\)
Since \(\)\(\)\(c = (1-s)y\), thenthenth c and y grow at the same rate and we have:
\(c(t) \approx c^\star + e^{-[1-\alpha_k](n+g+\delta)t} [c(0)-c^\star]\)\(\)\(\)
According to David Romer, \((n+g+\delta)\) is approximately equal to 6%. We know that \(\alpha\) is equal to 1/3. This means that we have:
\(c(t) \approx c^\star + e^{-0.04t} [c(0)-c^\star]\)
\(c(t)\approx 0.901y+e^{-0.04t}[0.82y\ -0.901y]\)
\(0.85y\approx 0.901y+e^{-0.04t}[0.82y\ -0.901y]\)
\(-0.051y \approx e^{-0.04t}[-0.081y]\)
\(ln(\frac{-0.051y}{-0.081y}) \approx -0.04t\)
\(t \approx 11.5\)
It would take 11.5 years for consumption to reach its original level.
How does the Bureau of Labor Statistics measures labor productivity?
"Labor productivity is the ratio of the output of goods and services to the labor hours devoted to the production of that output."
What is the latest productivity figure?
Productivity increased 1.5 percent in the nonfarm business sector in the second quarter of 2017 .
What has happened to productivity over time in the US since the 1980s?
According to the BLS, the following figures describe prodctivity change in the non-farm business sector from 1979-2015.
1979-1990: 1.5%
1990-2000: 2.2%
2000-2007: 2.6%
2007-2015: 1.3%
The rate of productivity growth grew in the 80's, 90's and 2000's until 2007, after which productivity growth shrank.
What are some of the explanations that people offer to explain changes in productivity trends?
Some people argue that productivity isn't actually slowing down but that we're just measuring growth incorrectly, in particular because of the gowth of new technologies. This “mismeasurement hypothesis” holds that many of the fastest-diffusing technologies since 2004—like smartphones, online social networks, and downloadable media—involve consumption of products that are time-intensive but do not impose a large direct monetary cost on consumers. Hence, productivity is said to decrease only because many productions are "escaping the market." Chad Syverson (2016) gives some evidences that the productivity slowdown is too large to be explained by such an explanation. Some other authors such as Cowen (2011) have argued that the productivity slowdown can be explained by a reversal of the productivity accelerations in the manufacturing and utilization of information and communication technologies.
Some economists have also pointed out the the possible existence of hysterisis effects. The Great Depression lead people who remain unemployed for a long time lose their skills and because slowing investment prevents the latest technologies embedded in capital goods from being used.
Why is the productivity slowdown debate important?
It is importnat because seemingly small changes in growh rates have huge impacts on GDP per capita in the long run. A country with a GDP per capita growing at 1% per year would double its production every 70 years against 35 years if the rate of growth was of 2%.