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ECON-GA 1005 Macroeconomics (MA)
PROBLEM SET n. 7
Deadline: Monday, Nov 18 - 4:00pm
Submit a clear legible scan of your solution through Brightspace’s Assignments before 4pm next Monday
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 ∑"
!#$ "!#(%!), with " ≡ 1⁄(1 + *) and utility function #(%!) = log %!.
The production function (with 0! = 1 at all 1) has the form 2! = 3(4!) = 54!, with 5 constant.
Assume 5 > 7 + *.
[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 8! (net of depreciation): 4!%& = 8! + (1 − 7)4!. The resource constraint for the economy is %! + 8! = 2!.
1. Write the equilibrium dynamical system (the two dynamic equations) for % and 4.
[You don’t need to show all the maximization problems and the derivations; just write the two equations first for general functions 3(4) and #(%), and then for these specific production function and utility function]
2. Construct the phase diagram:
- Take the first equation and ask where % is stationary, growing or declining. What does the equation tell you about the growth rate of consumption?
- Take the second equation and ask where 4 is stationary, growing or declining.
- Is there a steady state for % and 4 (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 % and 4 (as well as output 2) 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 4 and for % such that the two equilibrium conditions are satisfied, and 2 also grows at a constant rate.
[Hint: you know already what % is doing from the first equation; now use the dynamic equation for capital to look at 4!%&⁄4! and argue about what the growth rate of 4 must be to have a BGP; then use the production function to argue about the growth rate of 2.]
- What is the BGP growth rate of 2, 4 and %? What is the ratio %⁄4 along the BGP? Show in the phase diagram what the BGP equilibrium path looks like.
[If you know what the ratio %⁄4 is along the BGP, you can draw the BGP path in the (%, 4) graph]2
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 4$, 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 4 from the dynamic equation for 4; check what happens next if you start with % above or below the BGP; would you converge to the BGP or not?]