Neoclassical Growth Models. Solow Model

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1 Neoclassical Growth Models Model - mathematical representation of some aspect of the econom; best models are often ver simple but conve enormous insight into how the world wors "ll theor depends on assumptions which are not quite true. That is what maes it theor. The art of successful theorizing is to mae the inevitable simplifing assumptions in such a wa that the final results are not ver sensitive." - Robert Solow, 1956 Solow Model ssumptions 1. Single, homogeneous good (output) - this also implies that there is no international trade (have to have at least two different goods for an trade to tae place) 2. Technolog is exogenous - technolog available to firms is unaffected b the actions of the firms, including R&D (we'll relax this later); implication is that production function is not shifting 3. enesian ssumption - individuals save a constant fraction of their income (i.e., people consume in proportion to income)... this assumption is consistent with the data... let the savings rate be s total savings is s 4. abor Force - population is same as labor force (good enough to have constant labor force participation rate); population grows at constant rate n (t) 0 e nt 5. Perfect Competition - zero economic profit; price taers in labor maret (w) and capital maret (r) 6. Constant Returns to Scale - can use an constant returns production function (i.e., double input produces double output); we'll focus on Cobb-Douglas Two Inputs - capital and labor; capital is accumulated endogenousl through savings Constant Elasticit of Substitution (CS) - production function with constant returns to scale; generalization of Cobb-Douglas; (t) F(,) [b ρ + c ρ ] 1/ρ Cobb-Douglas Function - (t) F(,) 1-, (0,1) Test Constant Returns - F(2,2) (2) (2) F(,)... ep 7. Constant Depreciation Rate - capital depreciates at a constant rate δ Basic Model - built around two equations: production function & capital accumulation equation I. Production Function - describes how capital inputs, (t), combine with labor, (t), to produce output, (t) Capital - loo at corn... some ou eat (consumption), some ou plant to eat more in the future (capital); capital postpones consumption b certain amount in hopes of higher (but uncertain) future consumption... definition of investment Max Profits - hire capital and labor at maret prices to maximize profits (Note: we can consider our good to be a numeraire (P 1) which means r and w are in terms of ) Max F(,) - r - w First Order Conditions - dπ F r 0... marginal product of capital r d 1 of 7

2 dπ F w 0... marginal product of labor w d Using Cobb-Douglas - 1 F 1 1 r r 1 (1 F ) (1 ) (1 ) (1 ) 1 w w... we can use this to get the demand for labor Finding Stead State - in order to find some form of the production function to give us a stead state, we need to identif variables that won't change over time; in this case, we can focus on r & w because (1) empirical data sas the're stable over time [fact 5 from Introduction to Growth], (2) based on the wa the production function is set up, if either of these variables grows, the other has to decline and if we're taling growth rates, that means the other variable will eventuall reach zero... not realistic ( 1 ) From first order conditions we have w is also constant; that's called the output per worer: We also have r is also constant; to put things in same terms (per / worer or per capita), we'll use a tric: ; since is constant that / means capital per worer: is also constant Results for Production Function - 1. Zero Profit - results from constant returns assumption; (1 ) Profit r w (1 ) 0 paments to inputs completel exhaust the value of output produced 2. Constant abor & Capital - as fractions of GDP; results from Cobb-Douglas r w production function: and 1 ; this agrees with empirical data (see fact 5 from Introduction to Growth) 3. Growth Rates Equal - tae ln of both sides of capital per worer: / ln ln ln 1 d 1 d 1 d Totall differentiate wrt t: or Those are growth rates; is a stable variable in stead state so d/ 0; that means... the growth rate of capital equals the growth rate of labor in the stead state 2 of 7

3 Per Capita Production - now that we have stead state variables we can wor with ( and ), we need to convert the production function to incorporate these variables: F(, )... because of constant returns to scale, we can write F, F(,1)... introduce per capita production function f : f () 1 f Using Cobb-Douglas - ( ) ' 1 > 0 output per worer alwas increases with increases '' ( 1) 2 < 0 (because < 1) we have diminishing returns to capital (the added output per worer b increasing capital per worer declines as we add more capital) II. Capital ccumulation - the change in the capital stoc over time is equal to the infusion of new capital from savings (s) minus the depreciation of capital d s δ Example - econom starts with output of 100 and capital base of 200; the capital depreciate rate is 5 percent and the savings rate is 30 percent Savings (gross investment) s 0.3(100) 30 Depreciation δ 0.05(200) 10 Capital accumulation (net investment) s - δ Per Capita Capital ccumulation - just lie we did with the production function, we want to get this equation to incorporate variables that are constant in the stead state ( and ) d / Divide all terms b : s δ s δ d / Where did we see that term before? We got it from differentiation ln ln ln wrt t when we showed that the growth rates of capital and labor are equal in the stead state (on previous page); the general (non stead state) d / d / d / result was d / d / From assumption 4 substitute the labor growth rate n: n d / d / d / Solve that for : + n d / Substitute that into the first equation we had: + n s δ To get into the equation, use the / tric again: Solve for capital accumulation per worer: d d / + n s s ( n + δ ) Stead State - in stead state, must be constant so d/ 0 / / δ s ( n + δ ) s δ f() 3 of 7

4 That is, the gross savings (investment in capital) per worer must equal the lost capital per worer (lost through depreciation and the increase in the number of worers) Solve for Growth Rates - use formulas with standard tric of taing ln of both sides, then differentiating ln ln... since 0, we must have 0 1 ln ln + (1 ) ln + ( 1 ) n + ( 1 ) n n... so output grows at same rate as labor force Graphicall - the Solow model is prett eas to solve graphicall... just loo at where s intersects (δ + n) Stable Equilibrium - if we have anwhere other than *, it'll eventuall adjust to be at *; for example, start with < *; this means savings outpaces effective depreciation so we're accumulating capital ( ); this continues until * lgebraicall - solving is a little more complicated; start with stead state equation: s ( n + δ ) Plug in production per worer function: s ( n + δ ) Solve for : s 1 s n + δ n + δ Plug that bac into the production per worer function to solve for : s * ( *) n + δ dvantage of lgebra - can chec model empiricall: ln * ln s ln( n + δ ) 1 1 regression should show parameters β 1 β 2 ; also can solve for and chec if < 1 Summar - started with two equations: d F(, ) and s ( n + δ ) Then introduced output per worer and capital per worer: / and / Modified original equations: Stead State Measure d Growth Rate f () and s ( n + δ ), 0 Stead state has d/ 0 so we now,, n s ( n + δ ) Other things we showed Growth rates of labor, capital, and output are equal: n * * 1 * * (n + δ ) s (n + δ ) s Capital accumulation 4 of 7

5 Comparative Statics Increase Savings Rate - s and Note: growth rate in long run doesn't change since growth rate of capital must equal the growth rate of labor (population); the result will be a higher (GDP per capita) that grows at the same rate as before the increased savings rate Decrease Population Growth Rate - n and 2 Decrease Depreciation Rate - δ and Increase Productivit of Capital - and (n 1 + δ) (n 2 + δ) s (n 1 + δ) s (n + δ) s 2 s 1 (n + δ 1) (n + δ 2) s s Transitional Dnamics - how model evolves (between equilibria) lgebraicall - we alread showed: (which we used to find 0 in stead state; we're not in stead state now so we have to wor with the general version; the other equation we need is the capital per worer accumulation equation which we'll divide b : s s ( n + δ ) ( n + δ ) 1 Now we have two differential equations to describe the transitional dnamics; since diffeq isn't fun, we'll just loo at the transitions geometricall b plotting that second equation to determine what happens to the growth rate of Geometricall - Increase Savings Rate - s shifts the s -1 curve; at the current capital per worer, we aren't in equilibrium so / increases (i.e., the rate of growth of capital per worer increases); note form the first equation we showed above that this means the output per worer is also increasing; eventuall, we'll settle bac down where s 2-1 n + δ so 1 / goes bac to zero (so does / ); the end result is a one time increase in capital per worer () and output per worer (), but the growth rate of these remains the same at zero Decrease Population - if (and n remains constant), we have a one time increase in capital per worer (); this shifts us to the right on the graph and there's a disparit between (n + δ) and s -1 which means / < 0 (which means / < 0) ; it remains so until capital per worer goes bac to the original level; the end result is a one 1 time decrease output per worer ()... this is wh we care about unemploment 2 2 s 2-1 s 1-1 (n + δ) s of 7

6 Problems - Model is Population Driven - growth rates of output () and capital () equal growth rate of labor () No Growth in Output per Worer - model sas doesn't grow in stead state, but data for U.S. sas otherwise... this is wh we introduce technological change Solow Model + Technolog Technolog - captured in the production function Hics-Neutral - F(, ) F(, ) Capital ugmenting - F(, ) abor ugmenting - F(, ) Results of technolog are most transparent with labor augmenting technolog so that s where we'll focus Effective abor - ; we use to represent the level of technolog; using the labor augmenting production function, if changes from 1 to 2, that means worers are twice as productive (we effectivel get the same wor as if we doubled the number of worers) Update Model - basicall have four equations that drive the model 1 1. F(, ) ( ) gt 2. ( t) 0e g (constant rate of technological advancement)... this means that technological progress is exogenous ("mana from heaven"; technolog descends upon the econom automaticall and regardless of whatever else is going on in the econom) nt 3. ( t) 0e n (constant labor growth rate)... same as before 4. s δ... same capital accumulation equation as before What's Constant? - divide equation 1 b capital to get output per capital: 1 1 ( ) Since both and grow at constant rates (g and n), must also grow at a constant rate or else / would either go to infinit or go to zero (both are unrealistic); if we invert / we get capital per effective worer: / / Similarl, we can define output per effective worer: equation 1: 6 of 7 / / ; now we can rewrite 1 ( ) Output per effective worer and capital per effective worer will be constant in stead state Growth Rates - Do the ln-differentiate tric on and ln ln ln 0 g capital per worer () grows at same rate as technological progress

7 ln ln ln ln 0 + g + n ln ln ln 0 g ln ln ln ln 0 + g + n Measure,,,, Stead State Growth Rate 0 n g g + n Solving the Model - tr to put capital accumulation equation in terms of and s Divide b : δ B using the ln-differentiate tric on, we found: s Solve that for and substitute it into the first equation: + + δ s s / Notice g and n ; multipl b : + g + n δ / Use definitions / s and / : ( n + g + δ ) Solve for : s ( n + g + δ ) Stead State - 0 so s ( n + g + δ ) * ( n + g + δ ) s Comparative Statics and Transitional Dnamics are similar to original model * 7 of 7