Solow Growth ModelStylized Facts:\(\frac{Y}{L}\), or income per capita, shows continuing growth with no tendency for the rate of growth to slow. In the long run, economies grow over time. There are wide disparities in the growth rate of \(\frac{Y}{L}\). There are vast differences in the standard of living across countries. Factors of productions' share of total income are roughly constant, \(K \approx \frac{1}{3}\)\(\frac{K}{Y}\), or capital to output ratio, is roughly constant, and marginal returns to capital are nearly constant. \(\frac{K}{L}\), or capital per capita, shows continuing growth. Productivity tends to increase over the very long run. Share of consumption in GDP has remained constant, \(\approx 70\) percent in the United State. Financial development precedes economic growth and development.There are structural transformations in the development process that a generalized production function cannot capture, thus growth "miracles" and growth "disasters." Summary of the Solow ModelThe Solow Model is a closed model of economic growth, assuming a standard neoclassical production function with decreasing returns to capital. Taking the rates of savings and population growth as exogenous, Solow showed how these two variables determine the steady state level of income per capita. What the model explains: Savings has a positive relationship with income per capita. Population growth has a negative relationship with income per capitaWhat the model does not explain: The accumulation of physical capital, which is the only endogenous variable, cannot account for the differences in income per capita. Savings has only a level effect, not a growth effect, because of diminishing returns to capital. Sources of long run growth are exogenous (A or g) or not explained by the model. Production Function (Firm Behavior)\(Y_t = F(K_t, A_t L_t)\), where output equals the function of physical capital multiplied by the effectiveness of labor.There are constant returns to scale (doubling K and AL doubles output produced: \(cY_t = F(cK_t, cA_t cL_t) \Longrightarrow cF(K_t, A_t L_t)\)This has three implications: inputs other than capital, labor, and technology are relatively unimportant; the economy is large enough that gains from specialization have already been exhausted; and there are no scale effects for the production function, meaning income per capita is determined by physical capital per capita. Diminishing returns to capital means:\(\frac{\partial F}{\partial K_t} > 0, MPK > 0\) \(\frac{\partial ^2 F}{\partial ^2 K_t} > 0, \frac{\partial MPK}{\partial K_t} > 0\)\(\)Inada Conditions guarantee a finite, positive stock of capital and labor to achieve equilibrium:\(\lim _{K \rightarrow 0} \frac{\partial F}{\partial K} = + \infty\)\(\lim _{K \rightarrow \infty} \frac{\partial F}{\partial K} = 0\)Factor AccumulationLabor grows at a constant, exogenous rate: \(\frac{\dot L_t}{L_t} = n > 0\) where \(\frac{dL_t}{dt} \equiv \dot L_t = n L_t\)Technology grows at a constant, exogenous rate:\(\frac{\dot A_t}{A_t} = g > 0\) where \(\frac{dA_t}{dt} \equiv \dot A_t = g A_t\)Capital Accumulation, Investment and SavingsIn the Solow model, the stock of capital, K, is endogenously determined in the model whereas savings rate is constant and exogenous. The capital accumulation equation is:\(\dot K_t = I_t - \delta K_t\), meaning the change in capital equals investment minus the deprecation rate times the amount of capital (depreciated capital)\(K_{t + 1} = K_t (1 - \delta) + I_t\), meaning the amount of capital in the next period equals the previous period's capital minus depreciation plus investment In a closed economy, investment is equal to the output minus consumption, which equals savings: \(I_t = Y_t - C_t = S_t\)\(S_t = sY_t\), where \(s\) is the fraction of output not consumed, \(C_t = (1 - s) Y_t\), thus,\(I_t = S_t = sY_t\), and now we can substitute into capital accumulation:\(\dot K_t = sF(K, AL) - \delta K_t\)Rewriting Equations in Intensive (per capita) FormWe begin with\(Y = F(K, AL) = K^{\alpha} (AL)^{1 - \alpha}\)\(cY = F(cK, cAL) = c^{\alpha} c^{1 - \alpha} K^{\alpha} (AL)^{1 - \alpha}\), using constant returns to scaleMake \(c = \frac{1}{AL}\) and \(y = cY = \frac{Y}{AL}\) and \(k = \frac{K}{AL}\)\(y = F(\frac{K}{AL}^\alpha) \equiv f(k) ^\alpha\)Deriving the Capital Accumulation EquationThis is the Fundamental Differential Equation of Solow Model, since it identifies how the rate of capital changes due to the savings minus the growth in AL (N + g) and minus capital depreciation:\(\frac{d_t}{dt} = \frac{d \frac{K_t}{A_t L_t}}{dt} \) using the quotient rule we have,\(\frac{(A_t L_t) (\frac{d K_t}{dt}) - K_t (\frac{dA_t L_t}{dt})}{(A_t L_t)^2}\) Taking the definitions of \(\frac{d K_t}{dt} = \dot K_t; \frac{d L_t}{dt} = \dot L_t\) and using the product rule:\(\frac{(A_t L_t) \dot K_t - K_t (A_t \dot L_t + \dot A_t L_t)}{(A_t L_t)^2}\)Then separating terms and canceling:\(\frac{(A_t L_t) \dot K_t}{A_t L_t ^2} - \frac{K_t (A_t \dot L_t + \dot A_t L_t)}{(A_t L_t)^2} = \frac{\dot K_t}{A_t L_t} - \frac{K_t}{A_t L_t} \cdot (\frac{A_t \dot L_t}{A_t L_t} + \frac{\dot A_t L_t}{A_t L_t})\)Substituting in the definition of \(\dot K_t = sY_t - \delta K_t\) and canceling: \(= \frac{sY_t - \delta K_t}{A_t L_t} - (\frac{K_t}{A_t L_t})(\frac{\dot L_t}{L_t} + \frac{\dot A_t}{A_t})\)Thus we have the definitions of the growth rates: \(= sy - \delta k_t - k_t (n + g)\)And finally, the capital accumulation (change in capital) per unit of effective labor:\(\dot k_t = sf(k_t) - (n +g + \delta)k_t\)Steady State: Comparative Statics and DynamicsAt steady state, \(\frac{dk_t}{dt} = \dot k_t \rightarrow k*\)Thus, break-even investment (where you are investing/saving just enough to cover costs of old and new capital) will be: \(\dot k_t = 0 \rightarrow I_t = (n + g + \delta)k_t\). The amount of actual capital investment converges with the break-event equation.To solve for break-even, we set the \(\dot k_t\) equation to zero:\(0 = sf(k_t) - (n + g+ \delta) k_t\)\(sf(k_t) = (n + g+ \delta) k_t\)\(k* = \frac{sf(k_t)}{n +g + \delta}\)Then, we solve for when actual investment equals zero:\(0 = sk^\alpha - (n + g + \delta)k_t\)\( sk^\alpha = (n + g + \delta)k_t\)\( s = (n + g + \delta)k^{1 - \alpha}\)\(k^{1 - \alpha} = \frac{s}{n + g+ \delta}\)\(k*= \frac{s}{n + g+ \delta} ^{\frac{1}{1 - \alpha}} \)And now we have the steady state equilibrium capital stock:Balanced Growth PathWhen \(k\) converges to \(k*\) in the steady state is also called the balanced growth path. We can also find the growth rate of effective labor:\(\frac{d(A_t L_t)}{dt} = A_t \dot L_t + \dot A_t L_t\)\(\frac{A_t \dot L_t + \dot A_t L_t}{A_t L_t} = \frac{\dot L_t}{L_t} + \frac{\dot A_t}{A_t} = n + g\)Since \(k \) is constant at \(k*\) , total capital (K) grows at rate \(n + g\):\(\frac{\dot K}{K} = n + g\)On the balanced growth path, the growth rate of output per worker, \(\frac{Y}{L}\) is determined solely by the rate of technological progress. In the Solow Model, only changes in technological progress have growth effects; all other changes have level effects. Change in Savings RateThe increase in the savings rate (s) shifts the investment line upward and so the steady state converges to a new equilibrium in which output increases and \(k*\) increases as well:Response Functions: Change in Savings RateGolden Rule Savings RateThe Golden Rule is the level of capital that maximized consumption. The Golden Rule savings rate is a social optimum savings rate that maximizes utility (derived from consumption) for everyone. This occurs where, \(f'(k) > 0 ; f''(k) < 0\)\(y_t = f(k_t)\)\(i_t = sf(k_t)\)\(c_t = (1 - s)f(k_t)\)\(\dot k_t = sf(k_t) - (n + g + \delta) k_t\)Taking the consumption function and expanding it for more understanding:\(c_t(s) = (1 - s)f(k_t(s))\) Multiplying this out gives,\(c_t(s) = f(k_t(s)) - sf(k_t(s))\)And since \(\dot k_t = sf(k_t) - (n + g + \delta) k_t\) and \(\dot k* = 0\), thus \(sf(k*) = (n + g + \delta)k\)\(c*_t(s) = f(k_t(s)) - (n + g + \delta)k*(s)\)In the steady state, consumption is maximized where savings rate is at a level in which \(f'(k)\), which is the slope of the production function or marginal product of capital, and \((n + g + \delta)\), or exogenous changes, are equal. So, we max \(c*(s)\), which is consumption as a function of savings, using the chain rule: \(\frac{\partial c*(s)}{\partial s} = \frac{\partial f}{\partial k} \frac{\partial k}{\partial s} - (n + g + \delta) \frac{\partial k}{\partial s} = 0\)\(f'(k) \frac{\partial k}{\partial s} - (n + g + \delta) \frac{\partial k}{\partial s} = 0\)\((f'(k) - (n + g + \delta)) \frac{\partial k}{\partial s} = 0\)\(f'(k) - (n + g + \delta)= 0\)\(\therefore c* = f'(k) = n + g + \delta\)Again, consumption is maximized where savings is at a level in which the first derivative of capital (slope of the line) and the marginal product of capital are equal. Now, the Solow Model is globally stable. Growth AccountingThis is a method of measuring how much each factor contributes to economic growth, which reveals to us the residual of the rate of technological progress. It decomposes the total output into that which is due to increases in labor and capital, and then a part that cannot be explained with those changes in factor utilization. Consider the production function: \(Y_t = F(K_t, A_t L_t)\), where there is the assumption of constant returns to scale and perfect competition. This implies that factors will earn their marginal products (real interest rate and wages respectively):\(\frac{d Y}{d K} = MPK = r\)\(\frac{d Y}{d L} = MPL = w\)Remember that the marginal product of capital signifies the additional unit(s) of output produced by one additional unit of capital, and the same for MPL. Now we are ready to solve:Step 1: Total Differentiation\(\dot Y_t = \frac{\partial Y_t}{\partial A_t} \dot A_t + \frac{\partial Y_t}{\partial K_t} \dot K_t + \frac{\partial Y_t}{\partial L_t} \dot L_t\)Step 2: Divide by Y\(\frac{\dot Y_t}{Y_t} = \frac{\partial Y_t}{\partial A_t} \frac{\dot A_t}{Y_t} + \frac{\partial Y_t}{\partial K_t} \frac{\dot K_t}{Y_t} + \frac{\partial Y_t}{\partial L_t} \frac{\dot L_t}{Y_t}\)Step 3: Rewrite\(\frac{\dot Y_t}{Y_t} = \frac{K_t}{Y_t} \frac{\partial Y_t}{\partial K_t} \frac{\dot K_t}{K_t} + \frac{L_t}{Y_t} \frac{\partial Y_t}{\partial L_t} \frac{\dot L_t}{L_t} + \frac{A_t}{Y_t} \frac{\partial Y_t}{\partial A_t} \frac{\dot A_t}{A_t}\)\(\therefore \frac{\dot Y_t}{Y_t} = \alpha_K \frac{\dot K_t}{K_t} + \alpha_L \frac{\dot L_t}{L_t} + R_t\)The alphas are the portions attributed to each factor, traditionally 1/3 to capital and 2/3 to labor. Growth accounting can be used as a diagnostic tool for figuring out how much we don't know about growth through the Solow residual. Restating Shortcomings with the Solow ModelImplied difference across countries in capital per worker is implausibly largeImplied qualitative discrepancy in capital stock across countries is implausibly largeImplied difference in the marginal return to capital across countries is also implausibly largeFinally, g is exogenous which means we don't understand growth as well as we'd like