Macroeconomics Problem Set -

I. Malthusian model

In the Malthusian model, suppose that the quantity of land increases. Using the diagrammatic analysis presented in class, determine what effects this development has on the steady state of the model. Explain your results.

II. Solow-Swan growth model

1. Describe how, if at all, each of the following developments affects the actual and replacement investment lines in our diagram depicting the solution to the Solow model. Also discuss the effects on output per capita growth rate resulting from each development.

(a) The rate of technological progress rises.

(b) The production function is Cobb-Douglas, f(k) = k^α, and α rises.

(c) The rate of depreciation falls.

(d) Workers exert more effort, so that output per unit of effective labor for a given value of capital per unit of effective labor is higher than before.

2. In class we considered labour-augmenting technological progress. Another view of technological progress is that the productivity of capital goods built at t depends on the state of technology at time t and is unaffected by subsequent technological progress. This is known as embodied technological progress (technological progress must be "embodied" in new capital before it can raise output). Basically, the source of technological progress here is improvements in the quality of capital (computers being the best example). This problem asks you to investigate its effects. We will develop the analysis in two steps - first by considering a version of the Solow-Swan model with capital-augmenting technological progress, and then switching to embodied technological progress.

(a) Let's start by modifying the basic Solow-Swan model studied in class to make technological progress capital-augmenting rather than labor-augmenting. For the balanced growth path to exist, assume that the production function is Cobb-Douglas: Y(t) = [z(t)K(t)]^αL(t)^1-α. Assume that z(t) grows at rate μ: z^·(t) = μz(t). Assume that evolution of capital is given by K^·_t = sY_t - δK_t, as in the original Solow-Swan model.

Show that the economy converges to a balanced growth path, and find the growth rate of Y and K (levels) on the balanced growth path. (Hint: Show that we can re-write the model in stationary terms as ratios of z^α/(1-α)L. Then analyze the dynamics of this stationary model using diagrammatic approach used in class.)

(b) Now consider embodied technological progress. SpeciÖcally, let the production function be Y (t) = J(t)^αL(t)^1-α, where J(t) is the effective capital stock. The dynamics of J(t) are given by J^·(t) = sz(t)Y(t) - δJ(t).

The presence of the z(t) term in this expression means that the productivity of investment at t depends on the technology at t.

Show that the economy converges to a balanced growth path. What are the growth rates of Y and J (levels) on the balanced growth path? (Hint: Let J^{^}(t) = J(t)/z(t). Then use the same approach as in part (a), focusing on J^{^}/z^α/(1-α)L instead of K/z^α(1-α)L.)

(c) What is the elasticity of output on the balanced growth path with respect to savings/investment rate s?

(d) Compare your result in (c) with the corresponding result in the basic Solow-Swan model presented in class.