Question

come up with a solow growth model steady state equation for an economy where there is population growth and also a lump-sum tax which is put onto all individuals

Answer 1

Solow Growth Model: Steady-State Growth Path

Concepts of dynamic equilibrium

* What is an appropriate concept of equilibrium in a model where variables like Y and K grow over time?

+ Must consider a growth path rather than a single, constant equilibrium value

+ Stable equilibrium growth path is one where

* If the economy is on the equilibrium path it will stay there

* If the economy is off the equilibrium path it will return to it

* Equilibrium growth path could be constant K, constant rate of growth of K, or something completely different (oscillations, explosive/accelerating growth, decay to zero, etc.)

+ We build on the work of Solow and others who determined the nature of the equilibrium growth path for our models. + As long as we can demonstrate existence and stability, we know we have solved the problem.

* In Solow model (and others), the equilibrium growth path is a steady state in which “level variables” such as K and Y grow at constant rates and the ratios among key variables are stable.

+ I usually call this a “steady-state growth path.”

+ Romer tends to use “balanced growth path” for the same concept.

Finding the Solow steady state:

* In the Solow model, we know that L grows at rate n and A grows at rate g. The growth of K is determined by saving. Since Y depends on K, AL, it seems highly unlikely that output is going to be unchanging in steady state (a “stationary state”).

* Easiest way to characterize Solow steady state is as a situation where y and k are constant over time.

Breakeven investment line:

* How big a flow of new capital per unit of effective labor is necessary to keep existing K/AL constant?

* Must offset shrinkage in numerator through depreciation and increase in denominator through labor growth and technological progress:

* Need for each unit of k to replace depreciating capital

* Need n for each unit of k to equip new workers

* Need g for each unit of k to “equip” new technology

* The more capital each effective labor unit has the bigger the new flow of capital that is required to sustain it:

breakeven investment is linear in capital per effective worker.

+ At k1 the amount of new investment per effective worker (on curve) exceeds the amount required for breakeven (on the line) by the gap between the curve and the line, so k is increasing ( k > 0).

+ At k2 the amount of new investment per effective worker falls short of the amount required for breakeven, so k is decreasing ( k < 0 ).

+ At k* the amount of new investment per effect worker exactly balances the need for breakeven investment, so k is stable: k = 0 .

* At this level of k the economy has settled into a steady state in which k will not change.

+ Show graph with k on vertical axis.

* In this graph, k1 and k2 have same interpretation as in earlier graph.

* Existence and stability

+ Will there always be a single, unique intersection of the line and curve?

Yes.

* Diminishing returns assumption assures that curve is concave downward.

* Inada conditions assure that curve is vertical at origin and horizontal in limit.

* For any finite slope of the breakeven line, there will be one intersection with curve

* if economy begins at any level of k other than k* it will converge over time toward k*.

* Steady-state growth path exists, is unique, and is stable.