Thursday, July 28, 2022
HomeMacroeconomicsWith a Cobb-Douglas Manufacturing Perform, the Differential Equation for the Solow Progress...

With a Cobb-Douglas Manufacturing Perform, the Differential Equation for the Solow Progress Mannequin has a Closed-Type Answer—Tsering Sherpa and Miles Kimball — Confessions of a Provide-Aspect Liberal


The system for capital per employee okay is subsequently a constant-elasticity-of-substitution (CES) mixture between the preliminary degree of capital per employee and the steady-state degree of capital per employee okay* given above, with elasticity of substitution equal to 1/α > 1 and a weight on the preliminary degree of capital per employee that begins at 1 and exponentially decays on the charge (1-α)δ with the steady-state degree of capital per employee okay* having a complementary weight such that the 2 weights add to 1.

The system for capital per employee, which drives all the opposite evolving variables within the mannequin, implies that the convergence charge is the same as (1-α)δ. (That convergence charge generalize to instances with different manufacturing features, so long as α is interpreted as capital’s share on the steady-state degree of capital per employee.) It is a fairly gradual charge of convergence. For instance, even when δ is comparatively excessive, at a continuous-time charge of 10.5% per yr, convergence could be a continuous-time charge of seven% per yr if capital’s share is the same as 1/3. Which means by the rule of 70 that the half-life of exits from the steady-state could be ten years, because the financial system nears the regular state. (The rule of 70 is just a consequence of the the pure logarithm of two equaling roughly .7.)

On the regular state, capital per employee is unchanging over time. That additionally signifies that unchanging on the regular state. Intuitively, funding is sufficient to compensate for depreciation. If there’s inhabitants progress, or progress within the efficient variety of employees past inhabitants progress due to technological progress, the differential equation and its resolution above proceed to carry so long as okay is interpreted as capital per efficient employee and δ is interpreted as

δ = depreciation charge + inhabitants progress charge + charge of labor augmenting technological progress.

The ability

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