In an economy, aggregate production is produced according to the Cobb Douglas production function:

\[Y_{t}=K^{α}_{t}(A_{t}L_{t})^{1−α}\]

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 \(K_{t+1}=(1−δ)K_{t}+I_{t}\), where \(δ\) is the depreciation rate and \(I=sY\) is investment, with s being an exogenously determined savings rate. Firms in the economy maximise profits

\[\pi=K^{α}_{t}(A_{t}L_{t})^{1−α}−wL−rK\]

with \(w\) being the prevailing wage rate and \(r\) being the cost of capital.

Part I - Working with the aggregate production function

Question 1 - When firms are using the amount of capital that maximises profits, how much capital is used? What is the return on capital?

Firms maximise profit for a given level of K when:

\[\frac{\partial\pi}{\partial K} = 0\]

Marginal Product of Capital

1 \[\frac{\partial\pi}{\partial K} = \alpha K^{\alpha - 1}_{t}(A_{t}L_{t})^{1 - \alpha} - r = 0\]
2 \[\alpha K^{-1}_{t}K^{\alpha}(A_{t}L_{t})^{1-|alpha} = r\]

\[\alpha\frac{Y_{t}}{K_{t}} = r\]

Profit Maximising Level of K

Above is the \(MP_K\). We can rearrange to get the profit maximising level of K:

3 \[K^{*}_{t} = \alpha\frac{Y_{t}}{r}\]

Return on Capital

4 \[\frac{Y_{t}}{K_{t}} = \frac{r}{\alpha}\]

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?

Firms maximise profit for a given level of L when:

1 \[\frac{\partial\pi}{\partial L} = (1 - \alpha)K^{\alpha}_{t}(A_{t}L_{t})^{1 - \alpha - 1} - w = 0\] 2 \[(1 - \alpha)K^{\alpha}_{t}(A_{t}L_{t})^{1 - \alpha}L^{-1}_{t} = w\]

Marginal Product of Labour

3 \[(1-\alpha)\frac{Y_{t}}{L_{t}} = w\]

Profit Maximising Level of Labour

4 \[L^{*}_{t} = (1 - \alpha)\frac{Y_{t}}{w}\]

Return on Labour

5 \[\frac{Y_{t}}{L_{t}} = \frac{w}{(1 - \alpha)}\]

Question 3 - Another common production function is the ‘CES’ production function, see below. Derive the marginal products of capital and labour for this form.

\[Y=(αK^{ρ}+(1−α)L^{ρ})^{\frac{1}{ρ}}\]

Marginal Product of Capital

Set \(\frac{\partial\pi}{\partial K} = 0\), then, using the chain rule take:

1 \[\frac{\partial Y}{\partial K} = \frac{\partial Y}{\partial u} * \frac{\partial u}{\partial K}\]

2 \[\frac{\partial Y}{\partial u} = \frac{1}{\rho}u^{\frac{1}{\rho}-1}\]

3 \[\frac{\partial u}{\partial K} = \alpha\rho K^{\rho-1}\]

4 \[\frac{\partial Y}{\partial K} = \frac{1}{\rho}(\alpha\rho K^{\rho-1})(\alpha K^{\rho} + (1 - \alpha)L^{\rho})^{\frac{1}{\rho} - 1}\]

5 \[\frac{\partial\pi}{\partial K} = \frac{1}{\rho}(\alpha\rho K^{\rho-1})(\alpha K^{\rho} + (1 - \alpha)L^{\rho})^{\frac{1}{\rho} - 1} - r = 0\]

6 \[(\alpha\rho K^{\rho-1})Y^{1-\rho} = r\rho\]

7 \[\alpha(\frac{Y}{K})^{1-\rho} = r\]

Marginal Product of Labour

1 \[\frac{\partial Y}{\partial L} = \frac{\partial Y}{\partial u} * \frac{\partial u}{\partial L}\]

2 \[\frac{\partial Y}{\partial u} = \frac{1}{\rho}u^{\frac{1}{\rho} - 1}\]

3 \[\frac{\partial u}{\partial L} = (1 - \alpha)\rho L^{\rho - 1}\]

4 \[\frac{\partial Y}{\partial L} = \frac{1}{\rho}(1 - \alpha)\rho L^{\rho - 1}(\alpha K^{\rho} + (1 - \alpha)L^{\rho})^{\frac{1}{\rho} - 1}\]

5 \[\frac{\partial\pi}{\partial L} = \frac{1}{\rho}(1 - \alpha)\rho L^{\rho - 1}(\alpha K^{\rho} + (1 - \alpha)L^{\rho})^{\frac{1}{\rho} - 1} - w = 0\]

6 \[(1 - \alpha)\rho L^{\rho - 1}(\alpha K^{\rho} + (1 - \alpha)L^{\rho})^{\frac{1}{\rho} - 1} = w\rho\]

7 \[(1 - \alpha)\rho L^{\rho - 1} Y^{1-\rho} = w\rho\]

8 \[(1 - \alpha)(\frac{Y}{L})^{1 - \rho} = w\]

Part II - 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 \(l\).

Note, the Cobb-Douglas Production Function is given as: \[Y_{t}=K^{α}_{t}(A_{t}L_{t})^{1−α}\] \[\frac{Y_{t}}{A_{t}L_{t}}=\frac{K^{α}_{t}(A_{t}L_{t})^{1−α}}{A_{t}L_{t}}\] \[\frac{Y_{t}}{A_{t}L_{t}}=\frac{K^{α}_{t}}{(A_{t}L_{t})^{α}}\] \[y_{t}=k_{t}^{α}\] Where \(y_{t}\) denotes output per effective unit of labour and \(k_{t}\) represents capital per effective unit of labour.

Question 5 - Under balanced growth, the capital-output ratio is constant. Express the balanced growth path of \(y^{∗}\) and \(k^{∗}\) in terms of the exogenous variables. How quickly are capital and output growing during balanced growth?

\[k=\frac{(1−δ)k+sk^{α}}{(1+n+g)}\] \[(n+g+δ)k = sk^{α}\] \[k^{1-α} = \left( \frac{s}{n+g+δ} \right)\] \[k^{*} = \left( \frac{s}{n+g+δ} \right)^{\frac{1}{1−α}}\] \[y =k^{α}\] \[y^{*} = \left( \frac{s}{n+g+δ} \right)^{\frac{α}{1−α}}\]

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.

Assumptions:

Question 7 - Finally, we’re not interested in just finding the balanced growth path; we want to find the one that maximises some measure of wellbeing. One measure of wellbeing is the dollar value of material goods that we purchase.

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?

The golden rule of savings is determined at the point at which consumption is maximised, that is the first derivative is equal to 0.

Below is the mathematical derivation of the lden rule savings rate:

\[C = y - sy\] \[C = k^{\alpha} - (\delta + n + g)k\] \[\frac{\partial C}{\partial k} = \alpha k^{\alpha - 1} - (\delta + n + g) = 0\] \[\alpha k^\alpha = (\delta + n + g)k\] \[\alpha k^\alpha = sk^\alpha\]

Bibliography

Solow, R 1957, ‘Change and the Aggregate Production Function’, The Review of Economics and Statistics, vol. 39, no. 3, pp. 312-320.

“A Reduction in the Capital Stock - War! - Solow Model Application Part 1 of 4” https://www.youtube.com/watch?v=K3XmPnV_j-I

“Is China’s Growth Rate Healthy Now? Golden Rule of Capital”" http://snbchf.com/economic-theory/golden-rule-of-capital/