Model for rate of return to capital mathematical spiciness: ********** 10 stars (this appendix uses some advanced calculus) 1 Introduction

Transcription

1 Model for rate of return to capital mathematical spiciness: ********** 10 stars (this appendix uses some advanced calculus) 1 Introduction The purpose of this model is to investigate how different values of the growth rate (g) and the interest rate (r) affect the income to capital and labor (the issue discussed in Piketty s Capital). In this model, there are two inputs to production: labor and capital, each growing at a fixed rate. The quantity (Q) of the output product is produced according to a constant-elasticity-of-substitution (CES) production function: Q = Q 0 (al ρ + bk ρ ) 1/ρ The wage for labor and the rental price of capital are both equal to the value of their marginal product. The price of the output product is equal to its marginal cost. In addition to the one output product, some of the resources can be used to produce capital. Capital production is based on the same CES production function, except capital is measured by a different scale, as determined by the capital output conversion factor. For example, when this factor is equal to 1/2, it means that 1 unit of output can be converted to one half of a unit of capital. The Tobin q ratio is always equal to 1, which means that the purchase price of a unit of capital is equal to its marginal cost of production. The rate of return to capital is equal to the ratio of the rental price of capital divided by the purchase price of capital. If you buy one unit of capital, then the amount you have to pay is equal to the purchase price, and the amount you receive every year is the rental price. In this model, the exogenous variables are the production function parameters a, b, and ρ; the growth rate of labor g L ; the growth rate of capital g K ; and the capital conversion factor θ. This six-parameter abstract model is far removed from reality, but it is more versatile than the model in Piketty s Capital which only has two parameters (r and g). 1

2 2 Details on the CES production function For a two-input CES function: Q = Q 0 (al ρ + bk ρ ) 1/ρ the marginal products (partial derivatives) are: Q L = After cancelling the ρ and 1/ρ: ( ) 1 Q 0 (al ρ + bk ρ ) 1/ρ 1 (ρal ρ 1 ) ρ Q L = Q 0(aL ρ + bk ρ ) 1/ρ 1 al ρ 1 Do the same for K: ( ) Q 1 K = Q 0 (al ρ + bk ρ ) 1/ρ 1 (ρbk ρ 1 ) = Q 0 (al ρ + bk ρ ) 1/ρ 1 bk ρ 1 ρ The parameter ρ determines the elasticity of substitution (e) between the two inputs: e = 1 ρ 1 Although the CES production function is a special case of a completely general production function because it makes the assumption of the constant elasticity of substitution, it is general enough that includes three common types of production functions as special cases: 1. perfect substitutability, linear isoquants, e = ; ρ = 1, Q = Q 0 (al + bk) 2. Cobb-Douglas technology, curved isoquants, some substitutability, e = 1; ρ = 0, Q = Q 0 L a K b (assume a + b = 1) 3. no substitutability, recipe or Leontieff technology, L-shaped isoquants; e = 0; ρ = 2

3 Set the scale so the initial amount of labor equals 1, the initial stock of capital equals 1, and the initial amount of output equals 1. Then the scaling parameter Q 0 = 1. The profit equals: π = P Q wl r K K (where w is the wage and r K is the rental price of capital). The profit maximization conditions require finding the derivatives and setting them equal to zero: π L = P Q L w = 0 π K = P Q K r K = 0... Rearrange these equations to show that the price of each input will equal its marginal revenue product (also called the value of the marginal product), which is equal to the output price times the derivative: P Q L = w (1) P Q K = r K... Inserting the formula for the derivative Q L into equation 1: P (al ρ + bk ρ ) 1/ρ 1 (al ρ 1 ) = w (2) Do the same calculation for K: P (al ρ + bk ρ ) 1/ρ 1 (bk ρ 1 ) = r K (3)... Divide equation (2) by equation (3), and note how P (al ρ + bk ρ ) 1/ρ 1 cancels. The result is the profit-maximizing condition: 3

4 al ρ 1 bk ρ 1 = w r K (4) Let u represent the labor to capital ratio: u = L/K Let v represent the ratio of wage to capital rental cost: v = w/r k Use these definitions to rewrite equation 4: u ρ 1 = bv a Apply the exponent 1/(ρ 1) to both sides: u = ( ) 1/(ρ 1) bv (5) a Define e: e = 1 ρ 1 Rewrite equation for u using this definition: Find the ratio: Find the derivative: u v = u = ( ) e bv a ( ) e b v e 1 (6) a ( ) e 1 du bv dv = e b a a (7) 4

5 Find the ratio v/u from equation 6: ( ) v a e u = v 1 e b (8) The elasticity of substitution is defined to be: [elast of subst] = du v dv u Inserting the formulae from equations 7 and 8: [elast of subst] = e ( ) e 1 ( ) bv b a e v 1 e a a b Cancel the factors involving exponents for v, since v e 1 v 1 e = 1: [elast of subst] = e Combine the factors with b/a: ( b a [elast of subst] = e ) e 1 ( b a ) 1 ( ) e b a ( ) e 1+1 e b a The b/a factor cancels out since its exponent is 0: [elast of subst] = e ( ) 0 b = e a The result is that the expression we called e is in fact the elasticity of substitution, which is related to ρ by this formula: [elast of subst] = e = 1 ρ 1 Note that elasticity of substitution is constant, and depends on ρ. 5

6 3 Decision formulas in the model Choose the price level scale so the initial wage is equal to 1: w = 1 The rental price of capital r k is found by rearranging equation 4: r k = wbkρ 1 al ρ 1 The quantity of output (Q) comes from the production function: Q = (al ρ + bk ρ ) 1/ρ Rearrange equation 4 to find a formula for capital K: K = L ( ) rk a 1/(ρ 1) wb Insert the formula for K into formula for output (Q): ( ) Q = (al ρ + bl ρ rk a ρ/(ρ 1)) 1/ρ wb Factor out L ρ : Q = [L ρ ( a + b ( ) )] rk a ρ/(ρ 1) 1/ρ wb Note that Q is proportional to L: Find the derivative: Q = L ( Now find the total cost (T C): a + b ( ) ) rk a ρ/(ρ 1) 1/ρ wb ( ( ) ) dq dl = rk a ρ/(ρ 1) 1/ρ a + b wb T C = wl + r k K 6

7 dt C dq From the definition of u, we have K = L/u: ( L T C = wl + r k u = L w + r ) k u From equation 5 for the profit-maximizing situation: Find the derivative: u = L K = ( bv a ) 1/(ρ 1) = dt C dl = w + r k ( bw ar k ( ) ark 1/(ρ 1) bw ) 1/(ρ 1) The marginal cost (which will equal the output price) is the derivative dt C dq, which can be found from the chain rule since we know and : dl dl P = MC = dt C dl dq dl = dt C dq P = ( w + r ( ) ar 1/(ρ 1) k bw a + b ( ar k bw ) ρ/(ρ 1) ) 1/ρ (9) Since this model only has one product produced by one type of worker, the formulae for national income and labor income are straightforward: national income = = P Q labor income = wl labor share of income: wl capital income: r k K capital share of income = r kk The purchase price of capital is equal to P/θ, where θ is defined as follows. If θ = 1, then one unit of output can be converted to one unit of capital, so the price of capital is the same as the price of output. If θ = 0.5, then one unit of output can be converted to one-half of a unit of capital, so one unit of capital will cost twice as much as a unit of output. If θ = 2, then one unit of output can be converted to two units of capital, so one unit of capital will cost half as much as a unit of output. 7

8 The total national wealth (ω) is equal to the purchase price of capital P k times the amount of capital K: ω = P k K The rate of return (interest rate) r r equals the ratio of rental price of capital to purchase price of capital: r r = r k P k The change in capital ( K) equals the growth rate of capital times the amount of capital: K = g k K The dollar amount of investment (I) equals the change in capital times the purchase price of capital: I = P k K Investment is equal to savings, so the saving rate (s) is equal to I. Note: by assuming that investment is equal to saving, this model does not represent short-term aggregate demand instability. This model is only intended to represent long-run growth trends. In this model, the growth rate of capital is set as a parameter to the model, and that determines the rate of saving. In reality, it works the other way around: the amount of savings determines the rate of growth of capital. The relation between them is the same either way. 8

9 4 Results of the model Here are some different examples of this model with different parameters. Also, see the online Excel model where you can experiment with the model by entering your own values for the parameters. In the model, L and K both start at 1 (because the units are defined this way), and they both grow over time at the specified rates. The quantity produced Q is determined by the production function Q = Q 0 (al ρ + bk ρ ) 1/ρ The price of the output product is determined by equation 9. 9

10 Model with equal growth rates for capital and labor; relatively high substitutability; result: constant savings rate, constant labor share, constant real wage ****************************** Model 1 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

11 Model 1 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 1 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

12 Model 1 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 1 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 12

13 Model where labor grows twice as fast as capital; relatively high substitutability; result: falling savings rate, rising labor share, falling real wage ****************************** Model 2 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

14 Model 2 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 2 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

15 Model 2 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 2 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 15

16 Model where capital grows twice as fast as labor; relatively high substitutability; result: rising savings rate, falling labor share, rising real wage ****************************** Model 3 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

17 Model 3 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 3 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

18 Model 3 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 3 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 18

19 Model with equal growth rates for capital and labor; moderate substitutability; result: constant savings rate, constant labor share, constant real wage ****************************** Model 4 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

20 Model 4 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 4 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

21 Model 4 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 4 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 21

22 Model where labor grows twice as fast as capital; moderate substitutability; result: falling savings rate, rising labor share, falling real wage ****************************** Model 5 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

23 Model 5 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 5 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

24 Model 5 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 5 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 24

25 Model where capital grows twice as fast as labor; moderate substitutability; result: rising savings rate, falling labor share, rising real wage ****************************** Model 6 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

26 Model 6 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 6 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

27 Model 6 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 6 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 27

28 Model with equal growth rates for capital and labor; low substitutability; result: constant savings rate, constant labor share, constant real wage ****************************** Model 7 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= -5 capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

29 Model 7 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 7 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

30 Model 7 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 7 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 30

31 Model where labor grows twice as fast as capital; low substitutability; result: falling savings rate, falling labor share, falling real wage ****************************** Model 8 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= -5 capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

32 Model 8 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 8 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

33 Model 8 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 8 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 33

34 Model where capital grows twice as fast as labor; low substitutability; result: rising savings rate, rising labor share, rising real wage ****************************** Model 9 labor growth rate: gl= capital growth rate: gk= production function parameters: a= , b= , ro= -5 capital output conversion factor: theta= Output Output time Labor Capital Quantity Price t L K Q P

35 Model 9 National rental purchase interest Income price of price of rate time P Q wage capital capital r k P K t w r k P K r r Model 9 total labor total capital labor income capital income time income share income share r k K w L t w L r k K

36 Model 9 increase investment saving in (in dollars) rate I time wealth capital P K K t P K K K I s Model 9 output Consumption Consumption per real Good per time capita wage Production capita Q L w Q t Q c P c L **************************** 36