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

\[Y_t=K_t^\alpha(A_tL_t)^{1-\alpha}\]

where \(Y\) is output, \(K\) is the capital stock, \(L\) is the labour force, and At 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 \(L_{t+1}=(1-\delta) K_t+I_T\), 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

\[Y_t=K_t^\alpha(A_tL_t)^{1-\alpha}-wL-rK\]

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

Part 1: 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?

\(\pi=K_t^a(A_tL_t)^{1-\alpha}-wl-rk\)

\(\frac{\partial \pi}{\partial K}= \alpha K^{\alpha-1}(A_tL_t)^{1-\alpha}-r\)

\(=\alpha K_t^\alpha K_t^{-1} A_t^{1-\alpha} L_t^{1-\alpha} -r\)

\(=\frac{\alpha K_t^\alpha(A_tL_t)^{1-\alpha}}{K_t}-r\)

\(Y_t=K_t^\alpha(A_tL_t)^{1-\alpha}\)

\(\frac{\partial \pi}{\partial t}= \frac{\alpha Y}{K_t}-r\)

Profit maximisation:

\(0=\frac{\alpha Y}{K_t}-r\)

\(K_t=\frac{\alpha Y}{r}\)

return on capital:

\(MP_K = \frac{\partial Y}{\partial K}\)

\(Y_t=K_t^\alpha(A_tL_t)^{1-\alpha}\)

\(\frac{\partial Y}{\partial K} =\alpha K_t^{\alpha-1}(A_tL_t)^{1-\alpha}\)

\(=\frac{\alpha K_t^{\alpha}(A_tL_t)^{1-\alpha}}{K_t}\)

\(MP_K=\frac{\alpha Y_t}{K_t}\)

Question 2

Repeat the above, but for labour. If all income \(Y\) is either labour income \(w\) or capital income \(rK\), what is the relationship between the parameters of the production function and the capital/labour shares?

\(\pi=K_t^a(A_tL_t)^{1-\alpha}-wl-rk\)

\(=K_t^\alpha A_t^{1-\alpha} L_t{1-\alpha}-wl-rk\)

\(\frac{\partial \pi}{\partial L} =K_t^\alpha A_t^{1-\alpha}(1-\alpha)L_t^{1-\alpha-1}-w\)

\(=\frac{K_t^\alpha A_t^{1-\alpha}(1-\alpha)L_t^{1-\alpha}}{L_t}-r\)

\(=\frac{(1-\alpha)K_t^\alpha(A_tL_t)^{1-\alpha}}{L_t}-r\)

\(=\frac{(1-\alpha)Y}{L_t}-r\)

Profit Maximization:

\(0=\frac{(1-\alpha)Y}{L_t}-r\)

\(L_t=\frac{(1-\alpha)Y}{w}\)

Return on Labour:

\(MP_L = \frac{\partial Y}{\partial L}\)

\(Y_t=K_t^\alpha(A_tL_t)^{1-\alpha}\)

\(\frac{\partial Y}{\partial L} =K_t^\alpha A_t^{1-\alpha}(1-\alpha)L_t^{1-\alpha-1}\)

\(=\frac{(1-\alpha)K_t^\alpha A_t^{1-\alpha}L_t^{1-\alpha}}{L_t}\)

\(MP_L=\frac{(1-a)Y_t}{L_t}\)

Question 3

Another common production function is the “CES” production function, which takes the form

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

Derive the marginal products of capital and labour for this form. Hint: apply the chain rule.

\(MP_K\)

Let \(U=\alpha K^{\rho}+(1-\alpha)L^\rho\)

\(\frac{\partial Y}{\partial U}=1/\rho U^{1/\rho-1}\)

\(\frac{\partial U}{\partial K} = \rho \alpha K^{\rho-1}\)

\(\frac{\partial Y}{\partial U}*\frac{\partial U}{\partial K} = 1/\rho(\alpha K^{\rho}+(1-\alpha)L^\rho)^{1/\rho-1}*\rho \alpha K^{\rho-1}\)

\(\frac{\alpha K^{\rho-1}(\alpha K^{\rho}+(1-\alpha)L^\rho)^{1/\rho}}{\alpha K+(1-\alpha)L^\rho}\)

\(\frac{\alpha K^{\rho-1}Y}{\alpha K+(1-\alpha)L^\rho}\)

\(MP_L\)

Let \(U=\alpha K^{\rho}+(1-\alpha)L^\rho\)

\(\frac{\partial Y}{\partial U}=1/\rho U^{1/\rho-1}\)

\(\frac{\partial U}{\partial L} = \rho\alpha K^{\rho-1}\)

\(\frac{\partial U}{\partial U}*\frac{\partial U}{\partial L}=1/\rho(\alpha K^\rho+(1+\alpha)L^\rho)^{1\rho-1}*\rho \alpha K^{\rho-1}\)

\(=(1-\alpha)L^{\rho-1}(\alpha K^\rho+(1-\alpha)L^\rho)^{1/\rho}\)

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

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 \(l\).

Output Per effective Unit of Labour.

\(Y=K^\alpha(AL)^{1-\alpha}\)

\(y=\frac{k^\alpha(AL)^{1-\alpha}}{AL}\)

\(y=k^\alpha\)

Capital Accumulation per worker.

\(K_{t+1}=K_t+s_t+Y_t-\delta K_t\)

\(K_{t+1}/AL = K_t/AL+sY_t/AL-\delta K_t/AL\)

\(k_{t+1}=sk_t^\alpha+(1-\delta)k_t\)

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?

Assuming balanced growth and a constant capital-output ratio, the Balanced Growth Path (BGP) of \(y*\) and \(k\) can be expressed as:

\[y*=\left(\frac{sK}{n+g+\delta}\right)^{1/1-\alpha}\]

And:

\[k*=\left(\frac{s}{n+g+\delta}\right)^{1/1-\alpha}\]

The BGP is a situation where output per worker and capital per worker are growing at a constant rate.

Capital accumulation equation: \(K=sY-\delta K\) Dividing both sides by K yields the growth of capital: \(gk=\frac{\dot{K}}{K}=\frac{sY}{K}-\delta\) substituting the fact that: \(gk=\frac{\dot{k}}{k}=s\frac{y}{K}-n\)

Then we can summarise that \(gk=\frac{\dot{k}}{k}=s\frac{y}{K}-(n+\delta)\)

Along the BGP \(gk\) is constant, therefore \(Y/K\) is constant and it follows that \(gy=gk\). Therefore, \(gy=gk=g\)

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. Tell a story!

During the 1950’s and 1960’s growth may have been artificially and temporarily high in the years following World War II because of the application to the private sector of new technologies created for the war. However, the destruction of the country’s capital stock generated a gap between current income and steady-state income. This gap will change growth rates until the economy returns to it steady-state path. Based on Solow’s model, the massive destruction of Germany’s capital stock meant that now Germany had a much higher marginal product to capital. This would explain the huge levels of growth Germany experienced as it sought to reach its steady state level of capital. The rate of which is based on its exogenous factors which also would have been affected by the war such as population growth. Thus, the economic growth is not permanent (unsustainable in the long-run), but growth resulting from the capital stock approaching the steady-state value, k*.

Today, Germany has replenished its capital stock and is one of the largest economies in the world. The Romer model helps explain why countries which have invested highly in human capital have a higher steady state and therefore higher income per capital along BGP. Since Germany’s stock of ideas were not lost during the war, it is now producing a greater output with its capital than other countries comparably.

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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 well-being. One measure of well-being 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?

Golden Rule saving rate is \(s=a\) the marginal product of capital \(MP_K\) is given by \(y=k^\alpha\) which means \(\alphak^{\alpha-1}\). Therefore, the rate that would be recommended is where \(K*=MP_K=\alpha(k*)^{\alpha-1}=\alpha(n+d/s)\) and when saving is set to maximise the consumption per person at \(s*=\alpha\) and therefore \(MP_K*=n+d\) when consumption is maximised.

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Source: Hill, J http://www.unc.edu/~jbhill/Solow-Growth-Model.pdf