Unformatted text preview: Class notes concerning the Solow growth model. This topic will be treated in a slightly different way than the previous ones we have studied so far. First, there is no reference to this topic in the book, so it is very important you come to class and go to your TAs OH and to my OH if you do not understand. Second, instead of slides, we will go through these notes of mine step by step in class. In these notes I also embedded numerical exercises on the Solow model which are the highlighted part in yellow in the text. You will have extra exercises on the Solow model for you to practice posted in Bb. The Solow model was introduced first by Robert Solow an MIT economist who won the 1987 Nobel prize precisely for this contribution to Economics. It is a workhorse for modern macroeconomics as more complicated variants of the model are still widely used in current research papers. We will study the simplest stripped down version of the model. The one is tractable with your math knowledge. We make the simplifying assumption that there is no government or exchange of goods and services with the rest of the world. Hence GDP (the demand side of the economy) is given simply by Y = C + I as G = NX = 0. You might be asked to think about what happens if there is government and exchange with the rest of the world at some point though. So you have to fully understand the model to be able to tweak it, in case and answer that question. We'll look at an economy with given "structural characteristics":
Page 1 of 22 A given production function ==> the Cobb-Douglas. This represents the supply side of the economy. A given exogenous savings rate: s A given population growth rate: n A given depreciation rate of capital: d With this info we analyze the economy long run behavior...that's what growth is all about. In particular, we'll try to analyze changes in the economy through time: How do these structural factors interact to determine the investment level, and the evolution of the capital stock? How does the evolution of the capital stock interact with population in determining the change in production? We'll discuss how these factors determine the behavior of the economy period after period, and the implication of this for its long run evolution. - What are the level of physical capital, output, investment and consumption in the long run for a certain economy? We have 5 basic ingredients (equations) in the Solow model (you need to memorize those and be able to work the math out). Thankfully, we have seen 4 of these 5 equations previously at some point during this course so it is just a matter of putting them together, and understanding how they interact: 1. The production function: for all practical purposes we will use the Cobb-Douglas production function in this course. We have seen this equation concerning the production function
Page 2 of 22 already both in class and in the slides for chapter 12 and in the handout about production function. 1 A is the technology. is physical capital at period t is labor at period t is called the capital share you should know this from handout on production function. 1- is called the labor share you should know this from handout on production function. Only 2 factors of productions (K, L) are considered jointly with technology (A). This is for simplicity. At this stage, it's also easier to think of the economy as producing a single homogenous good, and using just one type of labor. But it is possible to modify this. 2. At any point in time total national savings are just a constant fraction of total national income. 2 is national savings s is the saving rate and it is expressed as a percentage. We have seen this equation and the saving rate already in the class notes for chapter 13. 3. Investment is equal to savings In a closed economy (without trade and relationships with the rest of the world), everything that is not consumed must be invested, so investment has to be equal to savings. We
Page 3 of 22 have seen this equation in chapter 13 already and in the class notes concerning math for saving. 3 4. The expression for capital accumulation 4 1 is the future amount of capital is the depreciation rate and it is expressed as a percentage. is the current amount of capital Hence, this expression simply says that future capital stock ) is equal to investment in new capital plus current ( capital ( ) minus the capital that breaks down and needs to ) (machines/structures that break down or be replaced ( become obsolete). We have seen the equation above in the handout about investment. 5. Population/Labor force. To keep things simple, we assume that the whole population of the country works labor force is equal to the total population. There is no unemployment or people outside the labor force. 5 1 This is the only equation of the Solow model that you have not seen already. But it is very simple to memorize.
Page 4 of 22 is the population growth rate and it is expressed as a percentage. You can think of this as the net difference between the birth rate and the death rate of the economy. So, the rate of growth of the labor force will be equal to the rate of growth of the population. We can reduce the 5 equations of the Solow model just down to 2 by doing some substitutions, here are the precise steps: First plug 2) into 3): 6 Now plug 6) into 4): 7 Finally, use 1) into 7): 8 1 1 This equation 8) together with equation 5) can be used to fully characterize the Solow model. Before we proceed further, let me try to show you numerically why the above steps are important in practice. A simple numerical example using the Solow model: Suppose that periods are years, and that an economy is characterized in the initial year by the following "parameters":
Page 5 of 22 Y = K0.5L0.5; the initial level of capital and labor are K1 = 10 and L1 = 5 s = 0.2 (20%), d = 0.03 (3%), n = 0.02 (2%) Year 1: plug K1 and L1 into the production function to obtain Y1 = 100.550.5 = 7.1 Year 2: Use the accumulation of capital equation and the population equation to obtain the new capital stock and new labor force K2 = sY1 + (1-d)*K1 = 0.2x7.1 +(1- 0.03)x10 = 1.41 + 9.7 = 11.11 L2 = (1+n)*L1 = 1.02x5 = 5. 1 Use these data to reiterate the process. Y2 = 18.104.22.168.5 = 7.53 Growth rate of GDP between years 2 and 1 = (7.537.1)/7.1 = 0.43/7.1 = 0.065 (6.5%) What do we do now? Well, to begin with we would like to rewrite the model in per capita or per workers terms. Why? Because it allows us to understand what are the key determinants of productivity, , in the Solow model. To this end, first find the expression for the general CobbDouglas production function in per capita terms: Adopting the convention of small caps variables to represent per-capita variables, we write:
Page 6 of 22 And the production function in per capita terms is simply: This expression tells us that productivity in the Solow model depends on technology A, the capital share , and physical capital per person This per capita (or "per worker") production function has the property of diminishing returns to capital (see handout on production function for more details on the topic). The following graph illustrates the property. Page 7 of 22 Since A and are given numbers to us in this model, from the expression of the production function in per capita terms we can understand that it is crucial to study physical capital per person to understand productivity in the Solow model. How can we study physical capital per person in more details? We start by dividing the left hand side and the right hand side of equation 8) by . This gives us the expression for capital accumulation in per capita (or per worker) terms. 1 Which becomes: 1 Now look at the left hand side of the expression above for a second, we have sort of a "miss-match in timing" on the left hand side because the numerator of that expression is future physical capital while in the denominator there is current population so we need to use a little trick and rewrite it as follow: 1 Page 8 of 22 What was the trick? I just divided and multiplied the left . That is the trick. hand side of the expression by At this point we need to recall equation 5) that tells us: 1 Using this fact we can finally write the expression for capital accumulation in per capita (or per worker) terms as: 1 1 So using our small caps variable convention, the expression becomes: 1 1 Or: 1 1 This expression is called the fundamental equation of the Solow model. It describes the evolution of capital per worker over time. In particular, this expression says that: Capital per capita (or per worker) increases with increases in savings per worker. Capital per capita is negatively affected by n, since for k to properly grow we have to compensate for the growth in population (labor force).
Page 9 of 22 Capital per capita is negatively related to the depreciation rate (d). Whether physical capital per person will grow or shrink will depend on the above mentioned forces. If savings per worker are more than enough to compensate for population growth and depreciation (this is called sometime the break-even capital), we have that capital stock per worker grows through time If savings per worker are not enough to compensate for population growth and depreciation, we have that capital stock per worker shrinks over time Now distribute the terms of the fundamental equation of the Solow model and do some algebra: With this latest expression we can study the concept of "steady state", and graph the Solow diagram. Finally we can make predictions for the future of the economy. The concept of steady state Basically, we say that any variable is in steady state when this variable does not grow: the variable stays constant over time this means that the value of a variable in period t is equal to the value of the variable in period t+1 for any t of your choice. In our fundamental equation of the Solow model, when is the capital per capita reaching the steady state?
Page 10 of 22 Formally, the steady state for capital per capita happens when: If we plug this into the fundamental equation of the Solow model we get (see the equation at page 10): 0 Solving for : This is the expression for the capital per capita level of steady state. NUMERICAL EXAMPLE: Consider the Solow model we are studying. The production function is given by . . The depreciation rate of capital is 3%, the population growth rate is 2% and the saving rate is 20%. Compute the steady state value of capital per capita. Try to repeat yourself the steps we have just done to obtain the fundamental equation of the Solow model for this specific example. If you do them correctly, you should get (please double check this for practice): 0.2
. 0.02 0.03 Page 11 of 22 We know that in steady state: Hence it must be that: 0 0.2 Solving for : 0.2
. . . 0.05 0.05 0.05 0.2 0.05 0.2 . 0.05 . 7.246 0.2 This is the level of physical capital per capita in the steady state of the economy. At this level the physical capital per capita will stop growing. With the level of physical capital per capita in steady state, we can compute also output per capita in steady state and consumption per capita in steady state. How? We just plug back in our production function in per capita terms. In steady state, we can plug the expression for the capital per capita level of steady state In the per capita production function and find:
Page 12 of 22 Going back to our numerical example: 0.05 . 1.811 0.2 At this level the output per capita will stop growing. What about consumption per capita in steady state? In our Solow model consumption is given by: 1 Consumption in per capita terms: 1 Consumption in per capita terms in steady state: 1 Going back to our numerical example: 1 0.2 1.811 1.4488 What about saving per capita in steady state? They are simply given by: 0.2 1.811 0.362 Building the Solow Diagram We now want to have a graphical representation of the economy. To simplify our life and make things easy and we assume that the difference between current and future . I am aware physical capital per person is "small", we are actually making an approximation error here, but most likely not a big one. Actually, this way we can have a very rough idea about the magnitude of the error we are
Page 13 of 22 . making. Go back to the expression for capital accumulation in per capita (or per worker) terms. It was: on the right If we now use our approximation hand side only we have an idea of the approximation error: The left hand side is a proxy for the growth of physical capital in per capita terms (and for the approximation error). The first term on the right hand side is savings in per capita terms (=investment in per capita terms). The second term on the right hand side is the amount of physical capital per person needed to replace the one that breaks down and the one needed for new workers. This second term is called the "break even investment". All this is very useful to draw the economic situation. If we were to represent all this math into a graph we would be drawing the Solow Diagram. Page 14 of 22 yt Solow Diagram (n d)kt yt Ak t y syt sAk t k kt Note that the graph is just expressing in pictures what we have done with the math previously. On the horizontal axis there is physical capital in per capita terms. There are 3 curves in the graph that you need to familiarize yourself with:
Page 15 of 22 I) The straight line from the origin is simply This is called "break-even investment" the reason why it is called so is because it is the amount of new machines per person that need to be bought/produced to make up for all the machines that break down and for the extra machines needed to keep up with the increase in population (i.e. new workers coming into the economy). II) The curve line starting from the origin that is "higher" out This curve represents total output per person It is a concave function that has that shape in the diagram because of the decreasing marginal return to capital as we have seen previously. III) The curve line starting from the origin that is "lower" in This curve represents total savings per person It has precisely the same shape as the curve representing total output, but it is just sort of shifted closer to the horizontal axis because it is total output is multiplied by s which is a percentage. Graphical visualization of the steady state: When the curve representing total savings crosses precisely with the line representing "break-even investment" (i.e. the two expressions are precisely the same) then we find the steady state of capital per person of the economy we have seen earlier that point is Page 16 of 22 And, again just to reiterate this once more, with the precise numerical value for you can then plug it in the expression for output per capita and obtain the output per capita of steady state and subtract savings per capita of steady state from output per capita of steady state to obtain consumption per capita of steady state 1 1 Note 2 more things: there are actually two points at which the curve crosses precisely representing total savings with the line representing "break-even investment" in the graph. The first point is, however, the origin of the axis. Since it does not have a very meaningful economic interpretation to have zero capital per person and zero saving per person and zero output per person, we disregard that steady state. The second point of intersection is . Page 17 of 22 is a stable steady state. This means that if by any chance for whatever reason the economy gets away from it, it tends to go back to it. Why? To see this, suppose the economy is starting at a point to the right of , on the horizontal axis. Just looking at the graph, this implies that This in turn implies that, from the fundamental equation of the Solow model: 0 0 physical capital per person is So decreasing over time. Until when? Until it goes precisely 0. back to where The opposite happens when the economy is starting at a point to the left of : the physical capital per person will grow toward . Bottom line: the economy reverts to always. And that's why we say that steady state. is a stable Important lesson(s) from this model: The sort of unpleasant conclusion of this model is that there is a limit to growth and that limit is precisely defined by (which in turn implies a limit to output per person and consumption per person). When the economy reaches the steady state, we are sort of stuck there forever.
Page 18 of 22 Note, however, that the model predicts that there is a limit to growth of physical capital per person (and hence output per person and consumption per person) not to actual physical capital and output. Why? Well, think about it, if the economy gets stuck at a certain constant, and since population is the denominator of physical capital per person is growing at the rate of n, then it means that physical capital -which is the numerator of that variable- must be growing at the same rate as the denominator, n to maintain that constant level . This also implies that total output is growing at the rate n in steady state for the Solow model. And total consumption as well. What's next? More policy analysis: A policy that changes the saving rate s will shift the total saving curve up or down depending on whether the policy implies an increase or decrease in s. In the example below we increase the saving rate from a level s1 to s2 we reach a new steady statehigher physical capital per person and output per person than before. Page 19 of 22 Increase in Savings Rate: s2 > s1
t (n d )k t y y t Ak t s2 y t s 2 Ak t s 1 y t s 1 Ak t k k k t What happens to the growth rate of the economy with this policy change? 3 issues: A) In the long-run equilibrium (steady-steady), the growth rate of aggregate variables does not depend on s (only on n), so there's no change in the long run (steady-state) growth rate of the economy. Per capita variables do not grow in steady state. B) In the short run, savings per worker are larger than (n+d)*k, so the economy grows faster than usual. This increased growth rate will be reduced as the economy approaches the new steady-state, until, in the end, we're back to the same growth rate as before. C) Nevertheless, per capita capital and income levels are higher in the new steady state. This is the long run effect of increased savings in the Solow model. Page 20 of 22 A policy that changes n will change the slope of "break even investment" line. In the example below we increase the population growth rate from a level n1 to n2 we reach a new steady state lower physical capital per person and output per person than before.
Increase in Population Growth Rate: n2 > n1
t (n 2 d )k t (n1 d )k t y t Ak t y sy t sAk t k k k t What happens to the growth rate of the economy with this policy change? 4 Issues: i) In the long-run equilibrium (steady-steady), the growth rate of per capita variables is zero, so there's no change in the long run (steady-state) growth rate of per capita variables. ii) In the long-run equilibrium (steady-steady), the growth rate of aggregate variables is equal to the population growth rate, so there's an increase in the growth rate of aggregate variables.
Page 21 of 22 iii) In the short run, savings per worker are smaller than (n+d)k, so the economy grows slower than usual. This means that, in the short run, the growth rate of the per capita variables will be negative (y and k falling). Again, this growth rate approaches zero as the economy approaches the new steady-state, until, in the end, we're back to the same growth rate as before. iv) Nevertheless, per capita income level is lower in the new steady state. This is the long run effect of increased population growth on the per capita variables in the Solow model. - What about a policy that changes d? (think about this yourself! Draw a graph and think about it!!) - What about a policy that policy that changes A (think about this yourself! Draw a graph and think about it!!) Page 22 of 22 ...
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- Spring '08
- Steady State, per capita, Capital accumulation, Solow model