The Solow Model (1956) Luca Spinesi November 6, 2014 Luca Spinesi () The Solow Model (1956) November 6, 2014 1 / 36
Motivation We ask if an economy can grow at a positive rate by simply saving and investing in its capital stock. Cross-country data from 1960 to 2000 show that the average GDP growth rate for 112 countries was 1.8%, and the average ratio of gross investment to GDP was 16%. Yet, there is some di erences across countries In general, data show a relationship between growth rate and the willingness of an economy to save and invest. Luca Spinesi () The Solow Model (1956) November 6, 2014 2 / 36
Production of GDP Real output (GDP) is produced according to some technology like this Y t = F (K t, L t ) where Y is the GDP, K is the stock of physical capital (stock variable), and L is the number of (hour) workers ( ow). Y and K are an homogeneous good. Very often we use a speci c function of the form named Cobb-Douglas Y t = K α t L 1 t α The Cobb-Douglas has constant returns to scale; if you double the amount of each input, you double output. (zk t ) α (zl t ) 1 α = z α+(1 α) Kt α L 1 t α = zkt α L 1 t α = zy t Luca Spinesi () The Solow Model (1956) November 6, 2014 3 / 36
Economy set-up We imagine a closed economy with no government purchase of goods and services: Y t = C t + I t In this way, saving S t equals gross investment I t Y t C t S t = I t Luca Spinesi () The Solow Model (1956) November 6, 2014 4 / 36
Capital Accumulation Output depends on capital and labor, so we need to know how those two things accumulate over time. Start with capital K = sy δk K is the change in the capital stock. It is the continuous time version of K t+1 K t, i.e., K 2013 K 2012, recall K is a stock variable. We know that total income is equal to total output (Y ). We assume that individuals - who work and own the capital - save a constant fraction, 0 < s < 1, of their income (Y ). sy = S = I, aggregate saving equals gross investment. K depreciates over time. A xed fraction, 0 < δ < 1, of the existing capital stock, K, breaks down at any given moment. Luca Spinesi () The Solow Model (1956) November 6, 2014 5 / 36
Capital Accumulation It will be useful to write this as the growth rate of capital, or divide through by K, K K = s Y K δ K K = s Y K δ Luca Spinesi () The Solow Model (1956) November 6, 2014 6 / 36
Population Growth We assume that the number of workers grows at the same rate as the population. Let L t = L 0 e nt or the number of workers exhibits exponential growth. The growth rate of the number of workers is thus L t L t = n which you can nd by taking logs and derivatives of the equation for L t. Luca Spinesi () The Solow Model (1956) November 6, 2014 7 / 36
We assume that rms are perfectly competitive There are many rms, all producing the same homogeous output Firms enter and exit freely They all produce using a similar Cobb-Douglas function They are all price-takers for the use of labor and capital, and for output For the representative rm, pro ts are Π = TR TC = Y rk wl = K α L 1 α rk wl where r is the market rate (per unit of time) for renting capital and w is the wage (per unit of time) of a worker. Luca Spinesi () The Solow Model (1956) November 6, 2014 8 / 36
The rm s pro t-maximizing decision ( rst-order conditions) are MPL = w = (1 α) Y L MPK = r = α Y K which just say that the rm sets the marginal product of a factor equal to its marginal cost. Luca Spinesi () The Solow Model (1956) November 6, 2014 9 / 36
From the rm rst-order conditions MPL = w = (1 α) Y L MPK = r = α Y K note that total payments to factors are equal to total output wl + rk = (1 α) Y L + α Y K = (1 {z L} {z} K w r α) Y + αy = Y This means that rms all have zero pro ts Π = TR TC = Y rk wl = Y Y = 0 This is consistent with our assumption that rms are perfectly competitive with each other, and rms enter and exit freely. If there were pro ts, more rms would enter and compete them away. This is a long-run equilibrium condition in a perfectly competitive market. Luca Spinesi () The Solow Model (1956) November 6, 2014 10 / 36
Factor Shares Calculate the fraction of total output that is paid to each factor. and wl Y = (1 α) Y L r K Y = αy K L Y = (1 α) K Y = α Factor shares of output are thus constant when we use the Cobb-Douglas, regardless of the amount of K or L. Consistent with the stylized facts on factor shares. Those facts suggest that α = 1/3 and (1 α) = 2/3, roughly. Luca Spinesi () The Solow Model (1956) November 6, 2014 11 / 36
Production per worker We typically care about output per worker, not just total output. Why? We do not think that India is richer than the Netherlands, even if India produces a lot more GDP. Once we divide by population or labor force, income per person in India is much lower than in the Netherlands. Divide total output by L to get y = Y L = 1 L K α L 1 α = K α L 1 α 1 = K α L α = K α = k α L where we are using y = Y L and k = K L to represent per-worker values. So y = k α is GDP per worker Luca Spinesi () The Solow Model (1956) November 6, 2014 12 / 36
This technology implies that capital per worker has a diminishing marginal output (recall α < 1). If k rises, output per worker rises, but the size of the increase falls as k increases. Luca Spinesi () The Solow Model (1956) November 6, 2014 13 / 36
Capital per Worker Accumulation How do capital per worker accumlates? Take logs and derivatives. ln k t t k = K L ln k = ln K ln L = 1 k t k t t = 1 k t k t = ln k t t = k t k t = 1 K t K t t k t k t = K t K t From the prior slide, we know that 1 L t L t t L t L t = K t K t L t L t so that K K = s Y K δ k t k t = K t K t L t L = s Y t K t δ L t L t = s Y t K t δ n Luca Spinesi () The Solow Model (1956) November 6, 2014 14 / 36
Solving the Solow Model First, what do we mean by solve? We mean that we want to be able to express the endogenous variables in terms of only exogenous ones. Endogenous variables - things we are trying to explain 1 Output Y, and/or output per worker y 2 Capital K, and/or capital per worker k Exogenous variables - things we take as given in the model α, capital s share in output s, the savings rate n, the population growth rate δ, the depreciation rate k 0, the initial level of capital per worker Luca Spinesi () The Solow Model (1956) November 6, 2014 15 / 36
Solve the Solow Model Getting a precise equation for k or y is not terribly easy. However, we can see the solution in some simple diagrams k t = K t K t L t L t k t = sy t (δ + n) k t k t = 0, sy t = (δ + n) k t Luca Spinesi () The Solow Model (1956) November 6, 2014 16 / 36
The diagram shows us that If k < k, then sy > (δ + n)k, and k > 0 If k < k, then sy < (δ + n)k, and k < 0 The Solow model predicts that capital per worker will stabilize at some value k where investment sk just o sets depreciation and population growth (δ + n)k. We refer to k as the steady state of the Solow Model. We have solved the Solow model in a general sense. If you tell me k 0, the initial starting point for capital per worker, and the parameters α, δ, s, n, I can tell you whether capital per worker will grow or shrink. α, δ, s, n describe the two lines, and k 0 tells me where along the x-axis we start. Luca Spinesi () The Solow Model (1956) November 6, 2014 17 / 36
The Steady State No matter what, k eventually ends up at k. We can give a precise answer to what determines k. It is the value of k such that This solves to k = 0 0 = sk α (δ + n)k. sk α = (δ + n)k k α 1 = δ + n s s k = δ + n 1 1 α which is the capital per worker at steady state. Luca Spinesi () The Solow Model (1956) November 6, 2014 18 / 36
Implications The steady state is k = s δ + n 1 1 α which depends only on the parameters of the model. Note that it does not depend on k 0. This implies that 1 k (and so y = (k ) α ) is rising with the savings rate, s 2 k (and so y = (k ) α ) is declining with the population growth rate, n, and with the depreciation of capital, δ Luca Spinesi () The Solow Model (1956) November 6, 2014 19 / 36
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Statics What if the savings rate, s, changes? Could be a policy change, or some di erence in individuals willingness to save for the future. Let s go to s 0 > s. Luca Spinesi () The Solow Model (1956) November 6, 2014 23 / 36
Saving Increase What is the e ect of s rising to s 0? The steady state rises to k. The economy will be richer eventually. Immediately after the change, k < k, so k > 0, capital starts growing Output per worker grows until the economy reaches the new steady state y = (k ) α, y > y because k > k What about consumption? Luca Spinesi () The Solow Model (1956) November 6, 2014 24 / 36
Growth Rate The Solow model predicts that growth is faster, the farther away from steady state is an economy. Look at the growth rate of k k k = sy k (n + δ) = sk α k (n + δ) = s k 1 α (n + δ) As k rises, the growth rate of k falls. Luca Spinesi () The Solow Model (1956) November 6, 2014 25 / 36
Growth Rate k k = 0, s = (n + δ) k1 α Luca Spinesi () The Solow Model (1956) November 6, 2014 26 / 36
Trend Growth Growth rates go to zero in the Solow model. y does not grow in steady state. To see this, take logs and derivatives of y = k to nd lny = lnk ẏ y = α k k Because k = 0 in steady state, it must be that ẏ y = 0 in steady state. No trend growth in the long run, only level e ect recall e ect of a higher saving (investment): y = (k ) α, y > y because k > k Yet, data show that GDP per worker has grown at a positive constant rate. Luca Spinesi () The Solow Model (1956) November 6, 2014 27 / 36
Trend Growth To allow for trend growth in the Solow model, we ll incorporate productivity improvements. Let production be Y = K α (AL) 1 α where A is labor-augmenting technological progress. In per worker terms this is y = k α A 1 α Growth in A will allow there to be trend growth in output per worker even in steady state. Luca Spinesi () The Solow Model (1956) November 6, 2014 28 / 36
Technology Growth We assume that A t = A 0 e gt which implies that technology growth rate is Ȧ t A t = g Luca Spinesi () The Solow Model (1956) November 6, 2014 29 / 36
Trend Growth We can also rconsider growth in output per worker. Take logs and derivatives of y = k α A 1 α to get ẏ y = α k k + (1 α) Ȧ A We re looking for the situation when ẏ/y is constant. This requires k/k to be constant. From k k = s y k (n + δ) capital per worker growth will be constant only if y/k is constant. y/k will be constant only if ẏ y = k k. Luca Spinesi () The Solow Model (1956) November 6, 2014 30 / 36
Trend Growth Putting this all together ẏ y (1 α) ẏ y ẏ y = αẏ y + (1 = (1 α) Ȧ A = Ȧ A = g α) Ȧ A y must grow at rate g if it is going to grow at a constant rate. Luca Spinesi () The Solow Model (1956) November 6, 2014 31 / 36
Balanced Growth Path The steady state of the Solow model will be exactly a situation where growth in y is constant at g. Further, y and k and A will all grow at the same rate. This is called balanced growth path, a time path where variables grow at a constant rate. What is the steady state when there is technology? We saw above that ẏ k y = g is the constant growth rate. This also implies k = g. Luca Spinesi () The Solow Model (1956) November 6, 2014 32 / 36
Balanced Growth Path So in steady state g = s y k (n + δ) Plug in for y to get α g = sa1 k 1 α (n + δ) Solve for k t = A t s n + δ + g 1 1 α Luca Spinesi () The Solow Model (1956) November 6, 2014 33 / 36
Steady State with Technology We said that k t = A t s n + δ + g 1 1 α in steady state. The ratio k/a is constant because k and A grow at the rate g. Output per worker in this steady state is k α y = k α A 1 α = A A y = A s n + δ + g α 1 α where the y and A are there to be explicit that both are growing over time. Luca Spinesi () The Solow Model (1956) November 6, 2014 34 / 36
Level E ect versus Growth E ect s y t = A t n + δ + g α 1 α The growth rate of y is equal to g along the balanced growth path - which is the steady state of the Solow model. But the level of y depends on the term in the parentheses. The parameters for savings, s, and population growth, n, have e ects on the level of output per worker, but don t alter the growth rate of output per worker. Changes in those parameters will induce a temporary surge (or decline) in growth relative to g as the economy shifts to the new trend line. Luca Spinesi () The Solow Model (1956) November 6, 2014 35 / 36
Level E ect versus Growth E ect Consider an increase in savings from s to s. You can use the old Solow model without technology to consider what happens in the transition output per worker grows rapidly for a while until we reach the new steady state. Then just add trend growth. Luca Spinesi () The Solow Model (1956) November 6, 2014 36 / 36