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Consider a full general equilibrium model of the economy with no uncertainty, perfectly competitive markets, inelastic labor supply.

The representative household’s preferences are given by ∑∞t = 0 βtu(ct), with β ≡ 1⁄(1 + ρ) and utility function u(ct) = log ct.

The production function (with lt = 1 at all t) has the form. yt = f(kt) = Akt, with A constant.

Assume 5 > δ + ρ.

[In class we discussed how this could be the reduced form. of a production function with either human capital accumulation or learning-by-doing. Here we ignore the reasons behind the linearity in capital and focus on characterizing the dynamics implied by this production function]

Capital accumulation depends on investment It (net of depreciation): kt+1 = It + (1 − δ)kt.

The resource constraint for the economy is ct + It = yt.

1. Write the equilibrium dynamical system (the two dynamic equations) for c and k.

[You don’t need to show all the maximization problems and the derivations; just write the two equations first for general functions f(k) and u(c), and then for these specific production function and utility function]

2. Construct the phase diagram:

- Take the first equation and ask where c is stationary, growing or declining. What does the equation tell you about the growth rate of consumption?

- Take the second equation and ask where k is stationary, growing or declining.

- Is there a steady state for c and k (apart from the origin)?

3. We have seen in class that if the levels of the variables do not reach a steady state, in the long run the economy may follow a balanced growth path (BGP), defined as a path along which c and k (as well as output y) grow at constant rates.

Try to characterize a balanced growth path for this economy:

- First, show that a BGP exists: i.e. show that there is a constant growth rate for k and for c such that the two equilibrium conditions are satisfied, and y also grows at a constant rate.

[Hint: you know already what c is doing from the first equation; now use the dynamic equation for capital to look at kt+1/kt and argue about what the growth rate of k must be to have a BGP; then use the production function to argue about the growth rate of y.]

- What is the BGP growth rate of y, k and c? What is the ratio c/k along the BGP? Show in the phase diagram what the BGP equilibrium path looks like.

[If you know what the ratio c/k is along the BGP, you can draw the BGP path in the (c, k) graph]

4. Finally, comment very briefly on the implications of this model for long-run growth.

- Is there persistent long-run growth in per capita income? Why? What is the crucial difference with the standard neoclassical growth model that explains the different implication for long-run growth?

- Given an arbitrary initial condition k0, does the economy have to go through a transitional dynamics before it settles on a balanced growth path or is it possible for the economy to be from the initial period on the BGP?

[You don’t need to solve the system or prove that the economy will start immediately on the BGP;just think why in the standard model we have a transition and why we may not need it here.]

- Optional: try to prove that the economy will start immediately on the BGP

[Use the expression for the growth rate of k from the dynamic equation for k; check what happens next if you start with % above or below the BGP; would you converge to the BGP or not?]