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One interpretation: Think about the equation above in GDP accounting terms. We know that in an economy, Total Savings ($S(t)$) = Total Investment ($I(t)$) (standard GDP accounting exercise). How is this relevant here? The right hand side is the Total savings in the economy: $\underbrace{s}_{\text{savings rate}}\times\underbrace{Y(t)}_{\text{total output}}$....

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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): $$\... 4 For completeness, let me illustrate this in the continuous time framework. The Solow equation, in the simplest of cases, is \dot{k} = s f(k) - \delta k = \phi(k) Then we have \frac{\partial \phi}{\partial k} = s f'(k) - \delta = \frac{sf'(k)k - \delta k }{k}. In steady state (i.e., \dot{k} = \phi(k^{\ast}) = 0), we have \delta k = s f(k), hence ... 4 Your reasoning is exactly correct and the answer should be C. We are told to begin "from a steady state which is below the golden rule of capital accumulation". So, s<s_g, where s is the initial savings rate and s_g is the golden-rule savings rate. Suppose that at period 250, our savings rate jumps from s to s_g. Then the evolution of per-... 4 An answer along macro textbook lines is given by @1muflon1. A shorter answer is as follows. Consider an investor who borrows capital from household (who owns the capital, in growth models) to invest in the firm with production technology f(k). The rate of return r on capital for the household is the interest rate of borrowing for the investor. Each ... 4 I will assume that you are talking about the Solow, R. M. (1956). A contribution to the theory of economic growth. The quarterly journal of economics, 70(1), 65-94. as to my best knowledge Solow did not published paper entitled the "Theory of Economic Growth" in 1956 and the first equation you use in your question is from the paper. Moreover, I also think ... 3 Take a look at the dynamics of the capital: k_{t+1}=sA_ty_t+(1-\delta-n)k_t. A sudden positive shock to TFP in period t increases the capital stock of the next period k_{t+1}. So, there is no contemporaneous effect on k, convergence to the new steady state will be gradual. The other variables have contemporaneous relationship with TFP. EDIT: A ... 3 This is not proven in Romer but it is a well known result. To derive it mathematically you need to take the following steps: First, the capital as in Romer depreciates so the evolution of capital will be given:$$k_t = k_{t-1} + i_t- \delta k_{t-1} \tag{1}$$where k_t is the present stock of capital, k_{t-1} previous stock of capital, i_t is ... 3 Have you seen the GitHub Project Replicating Mankiw, Romer and Weil 1992? It seems to have both the data and a replication of the original results. For the curious, the paper is A Contribution to the Empirics of Economic Growth. Abstract: This paper examines whether the Solow growth model is consistent with the international variation in the standard ... 3 Yes the answer should be C. I have attached an image showing the variation with time of the variables y(per capita output), c(per capita consumption) and i(per capita investment). I am assuming at t=t_0 the savings rate is increased. The consumption per capita initially will fall because the savings rate has increased. Eventually it must go above ... 3 Let$$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$$i.e. the wage is equal to ... 3 Below please find a portion of a lecture slide a professor of mine used last year. Please note that \gamma_{\tilde{y}} denotes per-capita output growth, \gamma_{\tilde{k}} denotes per-capita capital growth and \alpha(t) denotes the income share of capital. R(t) - the Solow residual - can then be easily computed. The production function you have ... 3 Let y=Y/L and k=K/L be the per-worker levels of output and capital. Observe that y=Ak^\alpha. Steady state is given by:$$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-...

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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: $${k^*}^{\alpha-1} > \alpha{k^*}... 3 For stability, we want$$\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}$$So we need the marginal product of capital to be smaller than the average product at the steady state. Equivalently,... 2 To simplistically answer your question, use the following: Y = K^\alpha (AL)^{1 - \alpha} In order to prove all three can be equal: We will assume that technological progress is labor augmenting (Harrod Neutral) or "labor saving", while holding the following true: Returns to capital are roughly constant over time. Capital share of income is roughly ... 2 In the model with technological progress the capital per effective worker remains constant, implies that capital per worker grows at the rate of exogenous rate of technological progress. See Barro and Martin book, Chapter 1. 2 The steady-state value k^* must be a fixed point : (1+g)(1+n)k^* = s(k^*)^{\alpha} +(1 - \delta) k^* Taking the difference between this equation and the dynamic one : (1+g)(1+n)(k_{t+1} - k^*) = s((k_{t+1})^{\alpha} - (k^*)^{\alpha}) +(1 - \delta) (k_t - k^*) Now if you denote by d_t = k_t - k^* the distance to steady-state, this gives : (1+g)(... 2 if the population growth rate grows, why could the capital and income per capita decreases This is basically asking "why does higher population growth lower the steady state capital per worker"? Mathematics Let us assume a Cobb-Douglas production function with constant returns to scale. This is,$$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}}{...

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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 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. 2 Yes, for any A and B $$\text{A}\cdot(g+n) + \text{B}\cdot(g+n) = \text{(A+B)}\cdot(g+n)$$ does indeed hold. 1 Based on the steady state your production function is Cobb-Douglas.Taking logs and derivatives wrt time of$Y$,$\frac{Y}{L}$and$\frac{Y}{AL}$in the steady state yields the desired result:$K$grows with$n+g$on the BGP. It would be interesting to know what the supervisor's objection was. 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 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. 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+...

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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 ...

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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.

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