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5

I don't quite understand what you mean by "share that goes into capital", but the common interpretation is that $\alpha$ is the share of income/output spent on capital. You can show that the following way: Since the factors will be compensated according to their marginal products, under the assumption of competitive markets, we have (for capital): $$\... 4 For completeness, let me illustrate this in the continuous time framework. The Solow equation, in the simplest of cases, is \dot{k} = s f(k) - \delta k = \phi(k) Then we have \frac{\partial \phi}{\partial k} = s f'(k) - \delta = \frac{sf'(k)k - \delta k }{k}. In steady state (i.e., \dot{k} = \phi(k^{\ast}) = 0), we have \delta k = s f(k), hence ... 4 Your reasoning is exactly correct and the answer should be C. We are told to begin "from a steady state which is below the golden rule of capital accumulation". So, s<s_g, where s is the initial savings rate and s_g is the golden-rule savings rate. Suppose that at period 250, our savings rate jumps from s to s_g. Then the evolution of per-... 3 Let$$Q = F(K,L)$$Assume a) F(K,L) exhibits consant returns to scale. We need this to aggregate from the individual firms to the total. b) Price taking behavior and c) Profit maximizing behavior from the part of the firms. Then, throughout the dynamic process,$$w = \frac {\partial F(K,L)}{\partial L} \equiv F_L$$i.e. the wage is equal to ... 3 Have you seen the GitHub Project Replicating Mankiw, Romer and Weil 1992? It seems to have both the data and a replication of the original results. For the curious, the paper is A Contribution to the Empirics of Economic Growth. Abstract: This paper examines whether the Solow growth model is consistent with the international variation in the standard ... 3 Yes the answer should be C. I have attached an image showing the variation with time of the variables y(per capita output), c(per capita consumption) and i(per capita investment). I am assuming at t=t_0 the savings rate is increased. The consumption per capita initially will fall because the savings rate has increased. Eventually it must go above ... 3 Take a look at the dynamics of the capital: k_{t+1}=sA_ty_t+(1-\delta-n)k_t. A sudden positive shock to TFP in period t increases the capital stock of the next period k_{t+1}. So, there is no contemporaneous effect on k, convergence to the new steady state will be gradual. The other variables have contemporaneous relationship with TFP. EDIT: A ... 3 Below please find a portion of a lecture slide a professor of mine used last year. Please note that \gamma_{\tilde{y}} denotes per-capita output growth, \gamma_{\tilde{k}} denotes per-capita capital growth and \alpha(t) denotes the income share of capital. R(t) - the Solow residual - can then be easily computed. The production function you have ... 3 Let y=Y/L and k=K/L be the per-worker levels of output and capital. Observe that y=Ak^\alpha. Steady state is given by:$$k^*=sy^*+(1-\delta)k^*,$$or$$k^*=sA(k^*)^\alpha+(1-\delta)k^*.$$Doing the algebra:$$k^*=\left(\frac{sA}{\delta}\right)^{\frac{1}{1-\alpha}}.$$And:$$y^*=A\left(\frac{sA}{\delta}\right)^{\frac{\alpha}{1-\alpha}}=A^{\frac{1}{1-...

3

For stability, we want $$\frac{\partial k_{t+1}}{\partial k_t}\Big|_{\bar k} <1 \implies \frac{(1-\delta) + \sigma A_0 f'(\bar k)}{1+n} <1$$ $$\implies f'(\bar k) < \frac {\delta+n}{\sigma A_0 } = \frac {f(\bar k)}{\bar k}$$ So we need the marginal product of capital to be smaller than the average product at the steady state. Equivalently,...

2

To simplistically answer your question, use the following: $Y = K^\alpha (AL)^{1 - \alpha}$ In order to prove all three can be equal: We will assume that technological progress is labor augmenting (Harrod Neutral) or "labor saving", while holding the following true: Returns to capital are roughly constant over time. Capital share of income is roughly ...

2

In the model with technological progress the capital per effective worker remains constant, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and Martin book, Chapter 1.

2

The steady-state value $k^*$ must be a fixed point : $(1+g)(1+n)k^* = s(k^*)^{\alpha} +(1 - \delta) k^*$ Taking the difference between this equation and the dynamic one : $(1+g)(1+n)(k_{t+1} - k^*) = s((k_{t+1})^{\alpha} - (k^*)^{\alpha}) +(1 - \delta) (k_t - k^*)$ Now if you denote by $d_t = k_t - k^*$ the distance to steady-state, this gives : $(1+g)(... 2 We want to prove that $$\frac{n+g+\delta}{s} > f'(k^*)$$ Replace the left hand side with the equivalent from the expression$sf(k^*)=(n+\delta+g)k^*$, and you get: $$\frac{f(k^*)}{k^*} > f'(k^*)$$ Cobb-Douglas case Without loss of generality, assume $$f(k^*) = {k^*}^{\alpha}$$ Then, the above inequality is: $${k^*}^{\alpha-1} > \alpha{k^*}... 2 if the population growth rate grows, why could the capital and income per capita decreases This is basically asking "why does higher population growth lower the steady state capital per worker"? Mathematics Let us assume a Cobb-Douglas production function with constant returns to scale. This is,$$Y=K^{\alpha}L^{1-\alpha}$$It can be shown that in the ... 2 It represents two things: (1) the elasticity of output with respect to capital, and (2) capital's share of output. To show (1), just take the natural log of the production equation, and then take the derivative of the logged equation with respect to time.$$\ln{Y} = \alpha \ln{K} + (1 - \alpha)(\ln{A} + \ln{L})\frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{... 2 Following Yorgos's alternative interpretation, (about$\alpha$which shows the percentage change of$Y$at$1\%$change in$K$), one intuition may also be to log-linearize your production function. As follows$\ln Y = \alpha \ln K + (1-\alpha) \ln L$. Then recalling that log may be used to compute continuous variations, you could substitute, e.g$K$for$\...

2

It seems you are confounding growth and growth rate. The difference between the blue and purple curves, i. e. growth, may be small when capital is small, but the ratio of the difference to capital, i. e. the growth rate will indeed be larger the closer capital is to zero.

2

The following graph shows a point at which the total amount of depreciation is greater than savings. This decreases the amount of capital per effective worker, over time, shown by the arrow pointing downwards and to the left.

1

What is the depreciation rate ($\delta$)? Assumptions: $Y_t = K_t^{\alpha} (A_tL_t)^{1-\alpha}$, $\alpha = 0.5$ ($\alpha$ = capital income share), $A_0 =1$ and $g = 0$ ($g$ = growth rate of technology), $s = 0.2$ ($s$ = savings rate), $n = 0.05$ ($n$ = growth rate of the labor force/population). Capital in the next period is equal to the capital from the ...

1

I do not directly see what intuition you are supposed to get from this growth rate equation, and the resulting math exercise but here is a sketch of the proof. First determine the steady state capital by setting $\Delta K_{t+1}=0$ and solving for $K^*$. Substitute this value for the $(K^*)^{(2/3)}$ in the fraction. Replace the $\frac{Y^*}{K^*}$ in the ...

1

Let production function $$F(K_t,L_t)=K_t^aL_t^{1-a}$$ $$max[c^*=f(k^*)-(δ+n)k^*]$$ with respect to $k^*$ Then $MPK=a(k^*)^{a-1}$ So $k_G= (\frac{a}{(δ+n)})^{1/1-a}$ which is golden rule level of capital stock per capita. Now calculate consumption at golden rule level $$c_G=f(k_G)-(δ+n)k_G$$ $$c_G=(k_G)^a-(δ+n)k_G$$ $$c_G=(\frac{a}{(δ+n)})^{a/1-a}-(δ+... 1 As I said in the comments, without additional details proving anything is either impossible or trivial. Impossible if I do not make any assumptions, trivial if I am allowed to make any assumption I like. Let me give you a hint on how to solve this in the framework how it is usually taught. The assumptions made are: There are decreasing returns to scale to ... 1 The approach developed by Barro and Lee has been used in many empirical analysis. They provide a detailed discussion of how to construct an aggregate human capital measure. 1 The Solow model was indeed the first model to account for technological progress. While models like the the Harrod-Domar model and the Neo-Marxian Feldman–Mahalanobis model conciser investment and saving as contributors to economic growth, they don't consider any measure of technology like total factor productivity to contribute to economic growth, like ... 1 I've tried to "fill the blanks", hope this helps.$$\tilde{k} = \frac{K}{AL}\frac{d[\tilde{k}]}{dt} = \frac{d}{dt}\left[\frac{K}{AL} \right] \quad (1) Now we use the quocient rule: to differentiate $f(x) = \frac{g(x)}{h{x}}$ (for $h(x) \neq 0$) we can apply the formula $f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}$. In your problem, \$f(x) = \dot{k}...

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