Week 4 ECO3EGS Group 3
Nick Russell, Declan Gleeson, Gao Yu
20 August 2015
Part 1 : Working with the aggregate production function
\[ Y_{t} = K_{t}^\alpha (A_{t} L_{t})^{1-\alpha} \]
\[ \Pi = K_{t}^\alpha(A_{t} L_{t})^{1-\alpha} - wL - rK \]
Question 1
When firms are using the amount of capital that maximises profits, how much capital is used? What is the return on capital?
\[\frac{\delta \Pi} {\delta K} = \alpha K^{a} (AL)^{1-\alpha}K^{-1}-r = 0\]
\[\frac{ \alpha K^{a} (AL)^{1-\alpha}}{K} = r\]
Substitute in \(Y_{t}\)
How much capital is used?
\[ K = \alpha \frac{Y_{t}} {r}\]
Return on Capital
\[ r = \alpha \frac{Y_{t}} {K}\]
Question 2
Repeat the above, but for labour. If all income Y is either labour income wL or capital income rK, what is the relationship between the parameters of the production function and the capital/labour shares?
\[\frac{\delta \Pi} {\delta L} =(1- \alpha) K^{a} (AL)^{1-\alpha} L^{-1}-w = 0\]
\[\frac{(1- \alpha) K^{a} (AL)^{1-\alpha}}{L} = w \]
Substitute in \(Y_{t}\)
How much labour is used?
\[ L = (1 - \alpha) \frac{Y_{t}} {w}\]
Return on labour
\[ w = (1 - \alpha) \frac{Y_{t}} {L}\]
If all income is from capital, then \(\alpha = 1\). As a result,
\[Y_{t} = K_{t}^1 A_{t}^{1-1} L_{t}^{1-1}\] \[Y_{t} = K_{t}\]
If all income is from labour, then \(\alpha = 0\). As a result,
\[Y_{t} = K_{t}^0 A_{t}^{1-0} L_{t}^{1-0}\] \[Y_{t} = A_{t}L_{t}\]
Question 3
Another common production function is the “CES” production function, which takes the form:
\[ Y = (\alpha K^\rho + (1 - \alpha) L^\rho)^{\frac{1}{\rho}}\]
Derive the marginal products of capital and labour for this form.
\[MPK = \frac{\delta Y}{\delta K}\]
\[\frac{\delta Y}{\delta K} = \frac {1}{\rho} (\alpha K^\rho + (1 - \alpha) L^\rho)^{\frac{1}{\rho}-1}\rho \alpha K^{\rho - 1}\]
Therefore
\[\frac{\delta Y}{\delta K} = \alpha K^{\rho - 1}(\alpha K^\rho + (1 - \alpha) L^\rho)^{\frac{1}{\rho}-1}\]
\[MPL = \frac{\delta Y}{\delta L}\]
\[\frac{\delta Y}{\delta L} = \frac {1}{\rho} (\alpha K^\rho + (1 - \alpha) L^\rho)^{\frac{1}{\rho}-1}\rho (1 - \alpha) L^{\rho - 1}\]
Therefore
\[\frac{\delta Y}{\delta L} = (1 - \alpha) L^{\rho - 1}(\alpha K^\rho + (1 - \alpha) L^\rho)^{\frac{1}{\rho}-1}\]
Part 2
Question 4
Express the Cobb-Douglas version of the model in per-effective worker terms (bothe production function and capital accumulation equation). That is, divide both parts of the model by AL. Call the resulting capital and output per effective units of labour k and l.
The original capital accumulation equation can be written by \(K = s y - ( n + g + d ) k\) and the Cobb-Douglas production function in this model is \(Y_{t} = K_{t}^\alpha ( A_{t} L_{t})^{1-\alpha}\)
First, we let lowercase letters denote variable divided by the stock of unskilled labor, L, and rewrite the production function in terms of output per worker as \(y_{t} = k_{t}^\alpha (A_{t} l_{t})^{1-\alpha}\)
Here, since lt is constant we can define the state variables by dividing by Al, Denote these state variables with a tilde, implies that. (equation 1)
The capital accumulation equation in terms of the state variables as (equation)
The steady-state variable value of k~ and y~ are found by seting k~. =0, which yields (equation)
Substituting this condition into the production function in equation 1 (equation)
Rewriting this in terms of output per worker, we get \(y_{t}^* = \left(\frac{S_{k}}{n+g+s}\right)^{\frac{\alpha}{1-\alpha}}A_{t}l_{t}\)
In equation (equation), since y~ is constant, if k increases, that (equation) will increase as will, and vice versa. However, the output per worker also deponds on \(\frac{\alpha}{1-\alpha}\), if is closer to 1, the output per worker will get higher. (Since, \(A_{t}\) and \(l_{t}\) is positive and not equal to 0) Which the variable l is directly affect \(y_{t}^*\)
\((n + g + s)\) 
Question 5
Under balanced growth, the capital-output ratio is constant. Express the balanced growth path of ystar and kstar in terms of the exogenous variables. How quickly are the capital and output growing during balanced growth?
Compare two both production function in this model and Balanced grow model by Kaldor.
\[Y_{t} = K_{t}^\alpha (A_{t} L_{t})^{1-\alpha}\]
\[Y_{t} = A_{0} K_{t}^\alpha (\gamma^t L_{t})^{1-\alpha}\]
Both divided by Labor force L,
\[Y_{t} = k_{t}^\alpha (A_{t}l_{t})^{1-\alpha}\]
\[Y_{t} = A_{0} k_{t}^\alpha (\gamma^t l_{t})^{1-\alpha}\]
In the Balance growth model, settle the variable A to the initial data \(A_{0}\), and non- related to coefficient \(1-/alpha\). And add a new coefficient \(\gamma\) to measure grow rate. Here is few explanation can answer the Q5 in some extent:
\(\frac{Y_{t}}{N_{t}} = (\gamma^{1-\alpha})^t A_{0}(\frac{K_{t}}{N_{t}})^\alpha (\frac{L_{t}}{N_{t}})^{1-\alpha} = \gamma^t A_{0} k^\alpha l^{t\alpha}\) grows at rate \(\gamma-1\)
\(\frac{K_{t}}{N_{t}} = \gamma^t k\) grows at rate \(\gamma-1\)
\(\gamma_{t} - \delta = \alpha(\gamma^{1-\alpha})^t A_{0} K_{t}^{\alpha - 1} L^{1-\alpha} - \delta = \alpha A_{0} k^{\alpha - 1} - \delta = \frac{\gamma \eta}{\beta - 1}\) is constant
\(\frac{K_{t}}{Y_{t}} = k(A_{0}k^{1-\alpha}l^{1-\alpha})\) is constant
\(\frac{\gamma K_{t}}{Y_{t}} = \alpha\), \(\frac {ne_{t} L_{t}}{T_{\alpha}}=1-\alpha\) are constant.
Rate of growth of \(\frac{Y_{t}}{N_{t}}\) is determined solely by \(\gamma\).
\(\frac{Y_{t}}{N_{t}} = (\gamma^{1-\alpha})^t A_{0}(\frac{K_{t}}{N_{t}})^\alpha (\frac{L_{t}}{N_{t}})^{1-\alpha} = \gamma^t A_{0} k^\alpha l^{t\alpha}\) grows at rate \(\gamma-1\)
\(\frac{K_{t}}{N_{t}} = \gamma^t k\) grows at rate \(\gamma-1\)
\(\gamma_{t} - \delta = \alpha(\gamma^{1-\alpha})^t A_{0} K_{t}^{\alpha - 1} L^{1-\alpha} - \delta = \alpha A_{0} k^{\alpha - 1} - \delta = \frac{\gamma \eta}{\beta - 1}\) is constant
\(\frac{K_{t}}{Y_{t}} = k(A_{0}k^{1-\alpha}l^{1-\alpha})\) is constant
\(\frac{\gamma K_{t}}{Y_{t}} = \alpha\), \(\frac {ne_{t} L_{t}}{T_{\alpha}}=1-\alpha\) are constant.
Rate of growth of \(\frac{Y_{t}}{N_{t}}\) is determined solely by \(\gamma\).
Question 6
We don’t always assume that economies are at the equilibrium point; more, it is an attracting point. During WWII, much of Germany’s capital stock was destroyed, though technology was not. Illustrate what happened to Germany during the post-war years on a Solow-Swan diagram.
Due to the destruction of much of its capital stock, Post-war Germany had a very low capital to labour ratio (at k*). Whilst capital was heavily impacted by the war, labour was relatively unaffected. In accordance with the Swan-Solow model, Germany’s capital stock quickly replenished as they converged to the steady state at a fast rate (at k).
Question 7
Plot the savings rate (x axis) against the amount of consumption in the economy (y axis). What is the shape? If a country wanted to maximise its consumption, what rate would you recommend?
To maximise consumption, savings and consumption levels should be equal. This can be seen at the point k where in graph 2 consumption is maximised. The shape of this graph is an inverted parabola
