macroeconomics chapter 7 Capital Accumulation and Population Growth
approaching the Steady State
Assume production function = Y = k^(1/2)L^(1/2) to derive the per-worker production function f(k) divide both sides of the production function by the labor force L Y/L = K^(1/2)L^(1/2)/L ===> Y/L = (K/L)^1/2 Because y = Y/L and k = K/L, this equation becomes, y = k^1/2 This form of the production function states that output per worker equals the square root of the amount of capital per worker .According to the production function y = k^1/2, 4 units of capital per worker equals 2 units of output per worker Because 30 percent of output is saved and invested and 70 percent is consumed (with an s=.3), i = .6 (s x y), and c = 1.4 (1-s)y Because 10 percent of capital depreciates d = .4 (.1 x 4) Thus , the economy begins its second year with 4.2 units of capital per worker. change in k = sf(k)- d(k) Steady state is when k does not change change = 0 0 = sf(k*) - d(k*), or k*/f(k*) = s/d substitute numbers from production function k*/k*^1/2 = s/d s/d = .3/.1 = 3, k*/k*^1/2 = k*^1/2
The golden rule of capital is described by the equation
MPK = d, or when MPK-d = 0 note: the economydoes not automatically gravitate toward the golden rule steady state, there needsto be a specific saving rate to get there
Solow Growth model
Shows us how saving, population growth, and technological progress affect the level of an economy's output and its growth overtime. Designed to show how growth in the capital stock growth in the labor force, and advances in technology interact in an economy as well as how they affect a nation's total output of good's and services. We first assume that the labor force and technology are fixed. We then relax these assumptions by introducing changes (later)
The consumption Function
The Solow model assumes each year people save a fraction s of their income and consume a fraction (1 - s). We can express this idea with the consumption function c = (1 - s)y to see what this function implies for investment, substitute for c in the national income accounts identity equation y = (1 - s)y + i rearranged: i = sy
The Demand for Goods and the Consumption function
The demands for goods in the Solow growth model comes from consumption and investment. In other words, output per worker y is divided between consumption per worker c and investment per worker, i.
The golden rule level of capital
The steady state of k that maximizes consumption is called the golden rule level of capital and is denoted k*gold first determine steady state consumption per worker begin with national income identity equation y = c + i, rearrange to c = y + i consumption is output minus investment. Steady-state output per worker is f(k*) and investment is dk* because depreciation and investment are equal in the steady-state and not changing, therefore c* = f(k*) - dk* Steady state consumption is the gap between f(k*) and d(k*) when this gap is largest it is the k*gold, aka where consumption is largest
The supply and demand for goods
The supply and demand for goods played a central role in our static model of the closed economy, the same is true for the Solow growth model. by considering the supply and demand for goods, we can see what determines how much output is produced at any given time and how this output is allocated among alternative uses. The supply of goods in the Solow model is based on the production function, which states that output depends on captial stock and the labor force
How savings effect growth
When savings rate increases, the investment curve sf(k) curve shifts upward, this causes the new steady state, where the investment curve and depreciation curve meet, aka when the change in k is 0. therefore when the saving rate is low the economy will have a small capital stock and output in the steady state (in the long run).
Starting with too little capital
When the economy begins above the Golden rule, reaching the golden rule produces higher consumption at all points in time. When the economy begins below the golden rule, reaching the golden rule requires reducing consumption initialy to increase it in the future.
The Production function
Y = F(K, L) has constant returns to scale if zY = F(zK, ZL), for any positive z, that is if both capital and labor are multiplied by z, the amount the amount of output is also multiplied by z production functions with constant returns to scale allow us to analyze all quantities in the economy relative to the size of the labor force, to see that this is true, set z=1/l in the preceding equation to obtain: Y/L = f(K/L, 1) THIS EQUATION: shows that the amount of output per worker Y/L is a function of the amount of capital per worker. (the number 1 is constant and thus can be ignored) The assumption of constant returns to scale implies that the size of the economy- as measured by the number of workers- does not affect the relationship between output per worker and capital per worker Because of the size of the economy it it convenient to denote all quantities in per-worker terms (/L) we do this with lowercase letters, so y = Y/L is output per worker for example, and k = K/L is capital per worker, we can then write the production function as y = f(k) (so output per worker is a function of capital per worker) THE SLOPE of this production function shows how much extra output a worker produces when given an extra unit of capital, this amount is the marginal product of labor MPK = f(k+1) - f(k) As the amount of capital increases, the production function becomes flatter, indicating diminishing marginal product of capital. (When k is low, the average worker has only a little capital to work with, so an extra unit of capital is very useful and produces a lot of additional output. When k is high, the average worker has a lot of capital already, so an extra unit increases production only slightly.)
how population affects golden rule
c = y - i because steady-state output is f(k*) and steady state investment is (d+n)k* we an express steady state consumption as c*= f(k*) - (d + n)k* we conclude that the level of k* that maximizes consumption (golden rule) is: MPK = d + n in the golden rule steady state, the marginal product of capital net of depreciation (MPK - rate of depreciation) equals the rate of population growth.
investment function
i = sy
figure 7-12 pg. 215
increasing d or n makes the break-even investment become steeper decreasing the steady state and investment. consumption per worker is. this predicts that countries with higher population growth will have lower levels of GDP per person. notice that a change in the population growth rate, like a change in the savings rate, has a level effect on income per person but not on steady state growth rate of income per person
population growth and technological progress
population and the labor force grow at a constant rate n labor force growth rate = population growth rate change in k = i - (d + n)k (investment increase capital per worker but depreciation and population growth decrease it) rewrite this equation as change k = sf(k) - (d + n)k to see what determines the steady state level of capital per worker, we use figure 7-11 so when sf(k) = (d + n)k note: if k is less that k*, i is greater than break-even investment, so k rises. If k is greater than k*, investment is less than break-even investment so k falls. When economy is in steady state, investment has two purposes, some of it (dk*) replaces depreciated capital, and the rest (nk*) provides new workers with the steady state amount of capital.
finding the golden rule steady state
production function y = f(k) when the economy is in steady state: k*/f(k*) = s/d, k*/k*^1/2 = s/.1 square both sides of equation because it rearranges to k*^1/2 =s/.1 (s/.1 = 10s) k* = 100s^2 THE ABOVE EQUATION WILL HELP YOU FIND THE STEADY STATE CAPITAL STOCK FOR ANY SAVING RATE Another way: find when MPK = d MPK of k^1/2 = 1/2k^-(1/2) or 1/2k^(1/2) then set it equal to the rate of depreciation and solve for k*
incorporating depreciation
t od o this we assume a certain fraction d of the capital stock k wears out each year. d is the depreciation rate. For example if capital lasts an average of 25 years, then the depreciation each year is 4 percent per year. Change in capital stock = investment - depreciation because i = sf(k) we can write it as change (k) = sf(k) - d(k) k* is the amount of investment that equals the amount of depreciation, this is the steady-state level of capital. THe change (k) = 0
Growth in the Capital Stock and the Steady State
the capital stock is a key determinant of the economy's output, but the capital stock can change over time, and those changes can lead to economic growth. Two forces that influence cpaital stock: investment and depreciation. Investment is a expenditure on plant and equipment, and it causes the capital stock to rise depreciation is the wearing out of old capital, and it causes the capital stock to fall. by substituting the production function for y, we can express investment per worker as a function of the capital stock per worker i = sf(k) PAGE 197 SEE FIGURE 7-2 the savings rate s determines the allocation of ouptut between consumption and investment. For any level of capital k, output is f(k), investment is sf(k) and consumption is f(k) - sf(k)
Steady-State level of capital
when change in k = 0 sf(k) - d(k) an economy at the steady state will stay there and an economy not at the steady state will go there regardless of the level of capital with which the economy begins, it ends up with the steady-state level of capital- it represents the LONG-RUN EQUILIBRIUM OF THE ECONOMY note: suppose that an economy starts with more than the steady state level of capital, in this case investment is less than depreciation: capital is wearing out faster than it is being replaced. The capital stock will fall, again approaching steady state level.
National income accounts identity equation
y = c + i
