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6

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): $$ \...


5

In this example, "mass" means the same as "length". Let me try to explain it with a simple example. If you work in a continuous setting, you have an infinite number of agents, which lie on a line between $0$ and $M$. Let's say individual consumption is given by the function $C(i)$, for all agents $i \in [0,M]$. For total consumption, you ...


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

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

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 ...


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 ...


4

Since, I was urged to present an answer for didactic reasons to which I totally agree, I will provide the full set of corrections to avoid any ambiguity. % Model parameters alpha = 1/3; % s = 0.2; % Investment rate delta = 0.2; % Depreciation rate n = 0.02; % Growth of labor force g = 0.01; % technological progress eps = 1; k(1) = ...


4

These two assumptions are not necessarily contradictory. Just check whether the assumptions are satisfied by any candidate function. For example, take $F(K,N) = K^{\alpha}N^{1-\alpha}$, with $\alpha \in (0,1)$. Constant returns to scale: For any scaling factor $c \in (0, \infty)$: $$F(cK,cN) = (cK)^{\alpha}(cN)^{1-\alpha} = c^{\alpha}c^{1-\alpha} K^{\alpha}N^...


4

Yes, there are DSGE models that can be used for forecasting. These models typically have a particular kind of steady-state, which is, more precisely, called balanced growth path (BGP). On the BGP (in the absence of shocks), key indicators growth at the same constant rate. For example, GDP, household consumption, investment all grow at 2% a year. This is ...


4

We have that: $$ \dot K = s K^\alpha L_0^b e^{nbt} $$ Rewriting the differential equation gives: $$ K^{-\alpha} \frac{dK}{dt} = s L_0^b e^{nbt} $$ Integrate both sides with respect to $t$ from $0$ to $T$ gives: $$ \frac{1}{b} [K^b]^T_0 = s L_0^b \frac{1}{nb}[e^{nbt}]^T_0 $$ So: $$ K^b_T = K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT} $$ Equivalently:...


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

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

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

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

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}}{...


3

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 $\...


3

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,...


3

The story in the other answer is not fundamentally wrong but incomplete and bit inaccurate. Saving does actually affect capital stock through investment in Solow model (assuming based on the Solow tag that is the model you want intuition for), but there is more complexity to it. What is saving Investment is actually equal not just to private saving but both ...


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

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

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.


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.


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