Homework 4
Group 6: James Ottaway, Brianna Norman, Henry Yueng, dfnewell
August 22, 2015
\[Y_{t} = K_{t}^{\alpha}(A_{t}L_{t})^{1-\alpha}\] where \(Y\) is output, \(K\) is the capital stock, L is the labour force, and \(A_{t}\) is a Hicks-neutral technology. The labour force grows according to \(L_{t+1} = L_{t}(1+n)\), and technology grows at \(A_{t+1} = A_{t}(1+g)\). The capital stock evolves according to \(K_{t+1} = (1-\delta)K_{t} + I_{t}\), where \(\delta\) is the depreciation rate and \(I=sY\) is investment, with s being an exogenously determined savings rate. Firms in the economy maximise profits \[\Pi = K_{t}^{\alpha}(A_{t}L_{t})^{1-\alpha} - wL-rK\] with \(w\) being the prevailing wage rate and \(r\) being the cost of capital.
Question 1
When firms are using the amount of capital that maximises profits, how much capital is used? What is the return on capital?
\(y=k^\alpha(AL)^1-\alpha\)
\(\frac{\delta(\Pi)}{\delta(k)}=\alpha(k^\alpha-1)(AL)^(1-\alpha)-r\)
\(\frac{\alpha(k^\alpha)(AL)^(1-\alpha)}{k}-r\)
\(r={\alpha(y)}{k}\)
This is the return on capital
\(k={\alpha(y)}{r}\)
This is the profit maximimising level of k
Question 2
Question 3
Another common production function is the CES production function, which takes the form
\[Y = \left(\alpha K^{\rho} +(1 - \alpha)L^{\rho} \right)^{\frac{1}{\rho}}\]
Derive the marginal products of capital and labour for this form.
\(Y = \left(\alpha K^{\rho} +(1 - \alpha)L^{\rho} \right)^{\frac{1}{\rho}}\)
\(\pi'K = Y'K - r = 0\)
\(\frac{1}{\rho}\left(\alpha \rho K^{\rho-1}\right)\left(\alpha K^\rho+(1-\alpha)L^\rho\right)^{\frac{1}{\rho}-1} -r\)
\(\left(\alpha \rho K^{\rho-1}\right)\left(\alpha K^\rho +(1-\alpha)L^\rho \right)^{\frac{1}{\rho}-1} = r\rho\)
\(\left(\alpha \rho K^{\rho-1}\right)\left(\alpha K^\rho +(1-\alpha)L^\rho \right)^{\frac{1}{\rho}}\left(\alpha K^\rho +(1-\alpha)L^\rho \right)^{-1} = r\rho\) Note that Y = \(\left(\alpha K^\rho +(1-\alpha)L^\rho \right)^{\frac{1}{\rho}}\)
\(\frac{\left(\alpha \rho K^{\rho-1}\right)Y}{\left(\alpha K^\rho +(1-\alpha)L^\rho \right)} = r\rho\)
\(\frac{\left(\alpha \rho K^{\rho-1}\right)Y}{Y^\rho} = r\rho\)
\(\left(\alpha \rho K^{\rho-1}\right)Y^{1-\rho} = r\rho\)
\(\alpha\left(\frac{Y}{K}\right)^{1-\rho} = r\) = MPK
\(Y = \left(\alpha K^{\rho} +(1 - \alpha)L^{\rho} \right)^{\frac{1}{\rho}}\)
\(Y'L = (1-\alpha) L^{\rho-1}(\alpha K^ \rho (1-\alpha) L^\rho)^{\frac{1}{1-\rho}-1}\)
\(MPL = (1-\alpha) \left ( \frac {Y}{L}\right)^{1-\rho}\)
Part 2: Growth
Question 4
Express the Cobb-Douglas version of the model in per-effective worker terms (both 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 \(l\).
\(y=\frac{Y}{LA}=\frac{k^\alpha(AL)^{1-\alpha}}{AL}\)
\(y=k^\alpha(AL)^{1-\alpha}(AL)^{-1}\)
\(y=k^\alpha(AL)^{-\alpha}\)
\(y=k^\alpha\)
Question 5
\(k_{t+1}=k^*\) BGP
\(k^*=\frac {(1-\delta)k^*+sk^{*\alpha}}{1+n+g}\)
\(k^*(1+n+g) = (1-\delta)k^*+sk^{*\alpha}\)
\(k^*(n+g+\delta)=sk^{*\alpha}\)
\(s=k^{*1-\alpha}(n+g+\delta)\)
\(k^* = \left(\frac{s}{n+g+\delta}\right)^{\frac{1}{1-\alpha}}\)
\(y=k^\alpha\)
\(y^*=\left(\frac{s}{n+g+\delta}\right)^{\frac{\alpha}{1-\alpha}}\)
Question 6
As Germany’s capital stock was destroyed, the steady state of capital decreases whilst the technology available was not noticably affected. Capital wss destroyed by the war or diverted to the Nazi war effort. Resulting in a shift downwards of the function of capital and savings curve. Providing us with a new equilibrium point, where the new steady state of capital now intercepts with the capital per unit on population (nk).
Question 7
The Golden Rule savings rate is the rate that maximises consumption over time.
\(y - sy = f(k) - s f(k) = f(k) - k(n+g+d)\)
\(f'(k) = -(n+g+d) = 0\)
As \(sf(k) = (n + g + d)k\)
\(s* = f'(k)k\)