Appendix: Mathiness in the Theory of Economic Growth by Paul Romer

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1 Appendix: Mathiness in the Theory of Economic Growth by Paul Romer This appendix exists as both a Mathematica notebook called Mathiness Appendix.nb and as two different pdf print-outs. The notebook is a combination of statements intended to communicate with people and statements that communicate with the computational enine in Mathematica. One of the pdf printouts is called Mathiness Appendix.pdf. It prints only the plots and the typeset text that are intended as communication with people. The second pdf, called Mathiness Appendix Expanded.pdf prints those materials plus the supportin input to and detailed output from Mathematica that enerates the plots and checks some of the alebra. If you are readin one of these and would like any other, all three are both available for download from my website, paulromer.net. Appendix A: Scale Effects As a function of individual consumption q, the individual inverse demand function is p = q - a, so as a function of total quantity Q, the inverse demand function in a market with N identical individuals ives the demand price p D as this function of Q and N : p D = The market inverse supply function is Q N - a. 1

2 p S = Q b. Implicitly, these expressions assume units for measurin both output Q and the numeraire ood that set any multiplicative constant in these demand functions equal to 1. The market clearin condition with a mark-up m 1 that could arise from a tax or a monopoly markup is which implies Q N p D = m p S, This yields a solution for Q of the form - a = m Q b. Q = m - 1 a+b N a a+b. This fiure illustrates an equilibrium with a mark-up m = 2, a = 0.5, b = 1.5, yields a value of Q rouhly equal to 4: Demand Demand price Supply Supply price The shaded area represents the surplus enerated in this case. To calculate the value that this area represents, we start with the area under 2

3 calculate the value that this area represents, we start with the area under the individual demand curve from 0 to q, which takes the form u(q) = a q1- a. As a function of the total quantity consumed, the sum across individuals of the area under the demand curves is U(Q) = N 1 - a Total cost, the area under the supply curve, is Q N 1- a. C(Q) = b Q1+b. The surplus, equal to the shaded area in the fiure, then can be calculated as S = U(Q) - C(Q) = N 1 - a Q N 1- a b Q1+b. The Mathematica calculations demonstrate that the implied expression for the surplus S as a function of the underlyin parameters a, b, m, and N is S = (a + m b m) m- (1 - a) (1 + b) 1+b a+b N a (1+b) a+b. As noted in the main text, at a = 1 2 and b = 0 this reduces to S = 2 m - 1 m 2 N. Calculations Generatin the Fiure that Illustrates the Surplus 3

4 Appendix B: Growth in the Lucas-Moll Economies To justify such statements as the frequency at which innovations arrive, β, does not affect the rowth rate, (Lucas and Moll 2014, p. 29), the authors rely on a limit arument about a limit does not exist. If we let β (t) denote the rowth rate for a iven value of β and date t, the rate referenced in this quote is a limit, = lim t β (t) = 2 %, which indeed, does not depend on β. However, this number is not a ood uide to the behavior specified in the model at any date T. The proof, available in the Expanded version of this appendix, shows that lim β 0 β (T) = 0. In fact, it establishes a stroner result, that the rowth rate converes uniformly to zero on any finite interval: Proposition: In the family of B economies, for any T > 0 and any ϵ > 0, there exists a β > 0 such that for all β 0, β and all t [0, T], 0 β (t) < ϵ. So at every date T, the rowth rate does depend on β and it converes to zero as β oes to zero. This is exactly what one should expect from the model itself. When β is set equal to 0, the model implies that the rowth rate is also equal to zero. The difference between these two types of limit implies that the order one uses to calculate the double limit matters: lim T lim β 0 β (T) lim β 0 lim T β (T). This dependence on the order tells us that if we reconize (β, T) as a function from R 2 R, the limit of as (β, T) oes to (0, ) does not exist. Lucas and Moll use limit aruments about this limit that does not exist to claim that two types of economy, labeled here the P and B economies, are so similar as to be observationally equivalent. Simple numerical 4

5 are so similar as to be observationally equivalent. Simple numerical calculations show the rowth rate in these two types of economy differ in ways that are plainly observable. For example, this fiure shows a plot over time of the rowth rate in a version of the P economy. Growth Rate P Economy Year Contrast this with the next fiure, which shows the plot over time of the rowth rate in a collection of B economies with decreasin arrival rates β, where curves displaced to the riht have a lower value of β. It shows how the rowth rate at any fixed date such as T = 300 falls from a value close to 2% to a value close to zero as β decreases. 5

6 These plots show that there at every date T is no value for the rowth rate that the P economy shares with the collection of all the B economies. The only value that they have in common is the limit of the rowth rate for fixed β as T oes to infinity. As noted in the text of the accompanyin article, this kind of limit is not an observable. No feasible set of observations on the rowth rate, which must of course be taken at finite times, can ever falsify an assertion about a limit as time oes to infinity. Plottin Growth in the P Economy Plottin Growth in the B Economies 6

7 Appendix C: Calculatin the Ratio of Wealth to Income Let denote income measured in ross terms and let denote income measured in net terms. (The mnemonic is amma for ross and nu for net. ) Symmetrically, let and s ν denote the ross and net measures of savin. Let δ be the depreciation rate. Let be the rate of rowth of ross output. Let be the stock of capital. By definition, = - δ, < = - δ = s ν. For to be constant in a steady state, < implies must be equal to, which = < = - δ = s ν. This yields the two expressions for the steady-state ratio of capital to output, = From the definitions, the ratio of to is equal to = + δ + δ, (1) = s ν. (2) = 1 + δ = 1 + δ s ν By takin the ratio of equation (2) to equation (1), we have a second expression for the ratio of ross to net income Combinin this with equation (3) yields (3) = s ν + δ. (4) 7

8 Solvin for yields = s ν + δ δ s ν 1 + δ s ν = s ν + δ. = s ν + δ + δ s ν = s ν + δ + δ s ν. (5) Let be an initial rowth rate and * be a lower rowth rate. In the example considered by Piketty and Zucman (2014), * = 1. Followin 2 their approach, assume that s ν stays constant when chanes. Let other variables with an asterisk denote the values implied by the new savin rate. For example, let s * γ denote the new ross savin rate implied by *. Consider the ratio * * =. To decompose this into the three terms described in the text, rewrite the numerator and denominator in terms of ratios that involve * * = *. Then use the expression for from equation (1) to rewrite the first term, * = and multiply and divide by to et * * +δ +δ * * = * +δ +δ * Y γ * = T Y 1 T 2 T 3. γ 8

9 The first of these three terms, T 1 = * +δ, +δ is the chane in the ratio of capital to ross income that we would observe if the ross savin rate remained constant at its initial value when the rowth rate falls from to *. The second term, T 2 = * = * +δ * +δs ν +δ +δs ν captures the chane in the ross savin rate that is implied by the chane in when s ν remains constant. The final term, T 3 = * = 1 + δ s ν * 1 + δ s ν is the chane in the ratio of ross to net income implied by the chane in the rowth rate. The parameter values in the paper, δ = 3 %, s ν = 10 %, = 3 %, and * = 1.5 % imply values T 1 = 1.333, T 2 = 1.375, and T 3 = In this example, because falls to half its previous value, the ratio of to doubles. The calculations verify that with these values, the product T 1 T 2 T 3 is equal to 2. Calculations for steady state relationships Dynamics Because it is an easy exercise, it may also be of some interest to explore the time path for these variables. Consider a case of an economy that starts in a steady state with = 3 % and s ν = 10 % so the implied value, 9

10 starts in a steady state with so the implied value for is approximately 18 %. We can calculate the capital to ross output ratio as = + δ. Let k = denote this ratio and consider the expression for its percentae rate of chane which implies k < k = < - Y < γ = - δ -, k < = - (δ + ) k, (6) where is related to s ν by equation (9) from above, = s ν + δ + δs ν. (7) Suppose that at time zero, the economy is at its steady state at a rowth rate = 3 % per year. At time zero, the rowth rate falls to from 3 % to 1.5 %. In an approach that maintains that the savin rate is fixed when the rowth rate chanes, there are now two possibilities -- either the ross savin rate remains constant or the net savin rate remains constant. To capture the first case, we can use a version of equation (6) uses * = 1.5 % but leaves unchaned at rouhly 18 %. In the second case, in equation (6) we insert both the new value * = 25 % implied by equation (7) and the new rowth rate *. This plot compares the evolution of the capital to ross output ratio under these two scenarios. 10

11 Capital to Gross- Output Ratio Year s νν constant γ constant Calculations for the dynamic paths 11

Phase Diagrams: construction and comparative statics

Phase Diagrams: construction and comparative statics 1 / 11 Phase Diarams: construction and comparative statics November 13, 215 Alecos Papadopoulos PhD Candidate Department of Economics, Athens University of Economics and Business papadopalex@aueb.r, https://alecospapadopoulos.wordpress.com