To answer the next question, copy the growth model below into a spreadsheet. The spreadsheet contains the values from a simple growth model. Please reconstruct the underlying formulas from this model and then use them to answer the questions below.

Set up the spreadsheet so that you can change the values for s and for the growth rate of the labor force in cells C1 and C2. Remember to use exponential growth rates and natural logs (ln). The three key relationships you need to fill in the formulas are the production function and the rules for calculating capital and labor next year from values of capital, labor, and output (GDP) this year. The production function is the one from the note: "Diminishing Returns": F(K,L)=K^(1/3)*L^(2/3). (Note: K^x means K raised to the power x) . The rule for labor, L, in given year is that it is equal to exp(g) times labor in the previous year, where g is the growth rate of the labor force, and exp is the exponential function described in "On Overview of Macroeconomics." The capital stock this year is equal to the capital stock yesterday plus s times output yesterday. You also need to be able to calculate the marginal product of an input and the growth rate of any variable.

6. Between years 99 and 100, the increase in the savings rate from 20% to 40% causes the growth rate of output to be higher by how much per year?

Investment share s =		20%
Growth of labor force =		1.50%
						Output			Growth in
Year	Capital	Labor	Output	MPK	MPL	per Worker		GDP growth	output per worker
1	1000	1000	1000.000	0.333	0.667	1.000
2	1200	1015.11306461572	1073.338	0.298	0.705	1.057		7.08%	5.58%
3	1414.66769307126	1030.45453395352	1145.260	0.270	0.741	1.111		6.49%	4.99%
4	1643.71966262528	1046.02785990872	1216.106	0.247	0.775	1.163		6.00%	4.50%
5	1886.9408551814	1061.83654654536	1286.149	0.227	0.807	1.211		5.60%	4.10%
6	2144.17061023934	1077.88415088463	1355.609	0.211	0.838	1.258		5.26%	3.76%
7	2415.29244784827	1094.17428370521	1424.670	0.197	0.868	1.302		4.97%	3.47%
8	2700.22642432472	1110.71061035571	1493.484	0.184	0.896	1.345		4.72%	3.22%
9	2998.9232934009	1127.49685157938	1562.183	0.174	0.924	1.386		4.50%	3.00%
10	3311.35996795985	1144.53678435131	1630.880	0.164	0.950	1.425		4.30%	2.80%
11	3637.5359388373	1161.83424272828	1699.672	0.156	0.975	1.463		4.13%	2.63%
12	3977.47041121286	1179.39311871139	1768.648	0.148	1.000	1.500		3.98%	2.48%
13	4331.19998803823	1197.21736312181	1837.884	0.141	1.023	1.535		3.84%	2.34%
14	4698.77677670775	1215.31098648973	1907.450	0.135	1.046	1.570		3.72%	2.22%
15	5080.26682757666	1233.67805995674	1977.410	0.130	1.068	1.603		3.60%	2.10%
16	5475.74883581362	1252.32271619186	2047.821	0.125	1.090	1.635		3.50%	2.00%
17	5885.31305451341	1271.2491503214	2118.737	0.120	1.111	1.667		3.40%	1.90%
18	6309.06037899076	1290.46162087289	2190.206	0.116	1.131	1.697		3.32%	1.82%
19	6747.10157105389	1309.96445073325	2262.275	0.112	1.151	1.727		3.24%	1.74%
20	7199.5565987121	1329.76202812147	2334.987	0.108	1.170	1.756		3.16%	1.66%
21	7666.55407182099	1349.858807576	2408.383	0.105	1.189	1.784		3.09%	1.59%
22	8148.2307580417	1370.259310957	2482.502	0.102	1.208	1.812		3.03%	1.53%
23	8644.73116649123	1390.96812846378	2557.380	0.099	1.226	1.839		2.97%	1.47%
24	9156.20718880745	1411.98991966766	2633.053	0.096	1.243	1.865		2.92%	1.42%
25	9682.81778920342	1433.32941456034	2709.555	0.093	1.260	1.890		2.86%	1.36%
26	10224.7287365579	1454.9914146182	2786.918	0.091	1.277	1.915		2.82%	1.32%
27	10782.1123727689	1476.98079388264	2865.175	0.089	1.293	1.940		2.77%	1.27%
28	11355.1474125479	1499.30250005677	2944.357	0.086	1.309	1.964		2.73%	1.23%
29	11944.0187706082	1521.96155561863	3024.493	0.084	1.325	1.987		2.69%	1.19%
30	12548.9174128295	1544.96305895134	3105.614	0.082	1.340	2.010		2.65%	1.15%
31	13170.040228504	1568.31218549017	3187.748	0.081	1.355	2.033		2.61%	1.11%
32	13807.5899211969	1592.0141888871	3270.925	0.079	1.370	2.055		2.58%	1.08%
33	14461.7749161128	1616.07440219289	3355.172	0.077	1.384	2.076		2.54%	1.04%
34	15132.8092821563	1640.49823905704	3440.517	0.076	1.398	2.097		2.51%	1.01%
35	15820.9126671271	1665.29119494588	3526.988	0.074	1.412	2.118		2.48%	0.98%
36	16526.3102447006	1690.45884837909	3614.612	0.073	1.425	2.138		2.45%	0.95%
37	17249.2326720223	1716.00686218486	3703.417	0.072	1.439	2.158		2.43%	0.93%
38	17989.9160568986	1741.94098477407	3793.429	0.070	1.452	2.178		2.40%	0.90%
39	18748.6019336944	1768.26705143373	3884.677	0.069	1.464	2.197		2.38%	0.88%
40	19525.5372471593	1794.9909856399	3977.185	0.068	1.477	2.216		2.35%	0.85%
41	20320.9743434998	1822.11880039051	4070.983	0.067	1.489	2.234		2.33%	0.83%
42	21135.1709680978	1849.65659955832	4166.097	0.066	1.501	2.252		2.31%	0.81%
43	21968.3902693452	1877.61057926434	4262.553	0.065	1.513	2.270		2.29%	0.79%
44	22820.9008081276	1905.98702927192	4360.379	0.064	1.525	2.288		2.27%	0.77%
45	23692.9765725443	1934.79233440203	4459.602	0.063	1.537	2.305		2.25%	0.75%
46	24584.896997496	1964.03297596984	4560.250	0.062	1.548	2.322		2.23%	0.73%
47	25496.946988817	1993.71553324308	4662.350	0.061	1.559	2.339		2.21%	0.71%
48	26429.4169516599	2023.84668492234	4765.929	0.060	1.570	2.355		2.20%	0.70%
49	27382.6028228773	2054.43321064388	4871.016	0.059	1.581	2.371		2.18%	0.68%
50	28356.8061071687	2085.48199250502	4977.639	0.059	1.591	2.387		2.17%	0.67%
51	29352.3339167895	2117.00001661267	5085.825	0.058	1.601	2.402		2.15%	0.65%
52	30369.4990146373	2148.99437465522	5195.604	0.057	1.612	2.418		2.14%	0.64%
53	31408.6198605534	2181.4722654982	5307.004	0.056	1.622	2.433		2.12%	0.62%
54	32470.0206606921	2214.44099680407	5420.054	0.056	1.632	2.448		2.11%	0.61%
55	33554.0314198288	2247.90798667647	5534.783	0.055	1.641	2.462		2.09%	0.59%
56	34660.9879964881	2281.8807653293	5651.221	0.054	1.651	2.477		2.08%	0.58%
57	35791.2321607895	2316.36697678109	5769.397	0.054	1.660	2.491		2.07%	0.57%
58	36945.1116549161	2351.3743805749	5889.343	0.053	1.670	2.505		2.06%	0.56%
59	38122.9802561245	2386.91085352427	6011.088	0.053	1.679	2.518		2.05%	0.55%
60	39325.1978422209	2422.98439148554	6134.663	0.052	1.688	2.532		2.03%	0.53%
61	40552.1304594381	2459.60311115694	6260.100	0.051	1.697	2.545		2.02%	0.52%
62	41804.150392655	2496.77525190488	6387.429	0.051	1.705	2.558		2.01%	0.51%
63	43081.6362379067	2534.50917761785	6516.684	0.050	1.714	2.571		2.00%	0.50%
64	44384.9729771405	2572.81337858832	6647.895	0.050	1.722	2.584		1.99%	0.49%
65	45714.5520551769	2611.69647342311	6781.097	0.049	1.731	2.596		1.98%	0.48%
66	47070.7714588413	2651.1672109826	6916.322	0.049	1.739	2.609		1.97%	0.47%
67	48454.0357982366	2691.23447234925	7053.603	0.049	1.747	2.621		1.97%	0.47%
68	49864.7563901301	2731.90727282592	7192.975	0.048	1.755	2.633		1.96%	0.46%
69	51303.3513434334	2773.19476396429	7334.472	0.048	1.763	2.645		1.95%	0.45%
70	52770.2456467569	2815.10623562406	7478.128	0.047	1.771	2.656		1.94%	0.44%
71	54265.8712580232	2857.65111806316	7623.980	0.047	1.779	2.668		1.93%	0.43%
72	55790.667196129	2900.83898405963	7772.062	0.046	1.786	2.679		1.92%	0.42%
73	57345.0796346448	2944.67955106551	7922.412	0.046	1.794	2.690		1.92%	0.42%
74	58929.561997547	2989.18268339335	8075.065	0.046	1.801	2.701		1.91%	0.41%
75	60544.5750569782	3034.35839443567	8230.060	0.045	1.808	2.712		1.90%	0.40%
76	62190.5870330344	3080.21684891802	8387.433	0.045	1.815	2.723		1.89%	0.39%
77	63868.0736955788	3126.76836518615	8547.224	0.045	1.822	2.734		1.89%	0.39%
78	65577.5184680853	3174.02341752759	8709.470	0.044	1.829	2.744		1.88%	0.38%
79	67319.4125335162	3221.99263852849	8874.212	0.044	1.836	2.754		1.87%	0.37%
80	69094.2549422397	3270.68682146594	9041.489	0.044	1.843	2.764		1.87%	0.37%
81	70902.5527219961	3320.11692273654	9211.341	0.043	1.850	2.774		1.86%	0.36%
82	72744.8209899201	3370.29406432159	9383.810	0.043	1.856	2.784		1.86%	0.36%
83	74621.5830666327	3421.22953628966	9558.938	0.043	1.863	2.794		1.85%	0.35%
84	76533.3705924128	3472.93479933681	9736.765	0.042	1.869	2.804		1.84%	0.34%
85	78480.7236454632	3525.42148736537	9917.336	0.042	1.875	2.813		1.84%	0.34%
86	80464.1908622863	3578.70141010157	10100.693	0.042	1.882	2.822		1.83%	0.33%
87	82484.3295601845	3632.78655575279	10286.882	0.042	1.888	2.832		1.83%	0.33%
88	84541.7058619033	3687.689093705	10475.945	0.041	1.894	2.841		1.82%	0.32%
89	86636.8948224365	3743.42137726085	10667.929	0.041	1.900	2.850		1.82%	0.32%
90	88770.480558011	3799.99594641925	10862.879	0.041	1.906	2.859		1.81%	0.31%
91	90943.0563772733	3857.42553069696	11060.843	0.041	1.912	2.867		1.81%	0.31%
92	93155.2249146999	3915.72305199271	11261.867	0.040	1.917	2.876		1.80%	0.30%
93	95407.598266252	3974.90162749473	11465.999	0.040	1.923	2.885		1.80%	0.30%
94	97700.7981273	4034.97457263219	11673.289	0.040	1.929	2.893		1.79%	0.29%
95	100035.455932842	4095.95540407116	11883.785	0.040	1.934	2.901		1.79%	0.29%
96	102412.213000041	4157.85784275599	12097.538	0.039	1.940	2.910		1.78%	0.28%
97	104831.720673107	4220.69581699653	12314.599	0.039	1.945	2.918		1.78%	0.28%
98	107294.640470554	4284.4834656021	12535.019	0.039	1.950	2.926		1.77%	0.27%
99	109801.644234852	4349.23514106272	12758.850	0.039	1.956	2.934		1.77%	0.27%
100	112353.414284518	4414.96541277856	12986.146	0.039	1.961	2.941		1.77%	0.27%