Economics 320a Fall 2002 Instructor: Igor Livshits

Answer Key for Problem Set 5.

Notation: Yt Aggregate output ct Per capita consumption Lt Labour wt Wage Kt Capital rt Rental rate on capital

kt Per capita capital stock δ Depreciation rate Xt Investment it Interest rate

Nt Population size n Population growth rate

Economy: Commodities: Consumption good, Labour, Capital.

Households: Preferences are represented by , ( )∑∞

=0,

tttt

t lcuNβ

where ( ) ( )lclcu −+= 1loglog, θ is strictly increasing.

Production: Aggregate production function is ( ) αα −= 1, LKALKF tt . We model production through one representative competitive firm. Population Dynamics: Nt+1 = (1 + n) Nt Capital Stock Dynamics: Kt+1 = (1 - δ ) Kt + Xt The Rest of the Economy: is standard Neoclassical (see “Neoclassical Model” notes).

Parameters of the Economy: 06.0=δ 96.0=β 4.1=θ 3.0=α n = 0.015 A0 = 1

Questions: Use Minnesota MacroLab to answer the following questions. The software, instructions and a set of corresponding notes (note that the notation is different from that in class) can be downloaded from http://macrolab.econ.umn.edu/equilibriumpaths.html. Remember, that the initial condition of the economy is the steady state of the specified environment with settings of period 0, unless you change the setting of K0 to be different from 0.

1. What happens to labour, investment (as a share of income), if in period 2 productivity doubles and then stays at the new high level forever? Assume that taxes and government share are 0.

As can be seen from the following results, both labour and investment (as a fraction of output) are high (higher than in the steady state) as the economy converges (up) to the new steady state.

Answer Key for Economics 320a Page 2 of 5 Problem Set 5

Period N A c x y k l r w i x / y

0 1 1 0.582 -0.0475 0.534 1.835 0.3147 0.087 1.188 0.032 -0.0890 1 1.015 1 0.576 -0.0692 0.507 1.652 0.3055 0.092 1.162 0.1533 -0.1365 2 1.030 2 0.638 0.4014 1.039 1.462 0.4489 0.213 1.621 0.1253 0.3863 3 1.046 2 0.689 0.3914 1.081 1.749 0.4395 0.185 1.721 0.1063 0.3622 4 1.061 2 0.732 0.3799 1.112 2.006 0.4317 0.166 1.803 0.0927 0.3417 5 1.077 2 0.768 0.3682 1.136 2.232 0.4252 0.153 1.870 0.0826 0.3241 6 1.093 2 0.798 0.3571 1.155 2.430 0.4199 0.143 1.926 0.0749 0.3092 7 1.110 2 0.823 0.3468 1.170 2.602 0.4154 0.135 1.972 0.0689 0.2964 8 1.127 2 0.845 0.3376 1.182 2.751 0.4117 0.129 2.011 0.0642 0.2855 9 1.143 2 0.863 0.3293 1.192 2.881 0.4085 0.124 2.043 0.0604 0.2762

10 1.161 2 0.879 0.322 1.201 2.992 0.4059 0.120 2.071 0.0573 0.2682 11 1.178 2 0.892 0.3156 1.207 3.088 0.4037 0.117 2.094 0.0548 0.2614 12 1.196 2 0.903 0.31 1.213 3.171 0.4018 0.115 2.113 0.0527 0.2555 13 1.214 2 0.913 0.3052 1.218 3.242 0.4002 0.113 2.130 0.051 0.2506 14 1.232 2 0.921 0.301 1.222 3.303 0.3988 0.111 2.144 0.0495 0.2464 15 1.250 2 0.928 0.2973 1.225 3.356 0.3977 0.110 2.156 0.0483 0.2427 16 1.269 2 0.934 0.2942 1.228 3.401 0.3967 0.108 2.167 0.0473 0.2396 17 1.288 2 0.939 0.2915 1.230 3.439 0.3959 0.107 2.175 0.0465 0.2370 18 1.307 2 0.943 0.2892 1.232 3.472 0.3952 0.107 2.183 0.0457 0.2347 19 1.327 2 0.947 0.2872 1.234 3.501 0.3946 0.106 2.189 0.0451 0.2328

Last_pd_ss 0.968 0.2752 1.244 3.669 0.391 0.102 2.226 0.0417 0.2213

2. Keeping productivity constant over time, start the economy at the level of capital that is significantly higher than the steady state level (e.g., by 50 or 100%). How many periods does it take for that difference to be cut in half? Compare your results to those in Problem Set 2, Question II.3.

As can be seen from the following results, it takes a little more than 4 periods to cut the distance to the steady state in half (it took 6 periods in Solow model). The reason for increased speed of convergence is the adjustment of investment rates and labour supply of households along the transition path. Period N A c x y k l r w i k / kSS % diff

0 1 1 0.619 0.0486 0.667 3 0.3504 0.067 1.333 0.0103 1.63523 63.52 1 1.015 1 0.600 0.0623 0.662 2.8262 0.3556 0.070 1.304 0.0137 1.54050 54.05 2 1.0302 1 0.584 0.0739 0.658 2.6788 0.3603 0.074 1.278 0.0168 1.46015 46.02 3 1.0457 1 0.570 0.0836 0.654 2.5536 0.3644 0.077 1.255 0.0196 1.39191 39.19 4 1.0614 1 0.558 0.0918 0.650 2.4473 0.368 0.080 1.236 0.0223 1.33397 33.40 5 1.0773 1 0.548 0.0988 0.646 2.3569 0.3711 0.082 1.219 0.0246 1.28469 28.47 6 1.0934 1 0.539 0.1046 0.643 2.2801 0.3739 0.085 1.204 0.0267 1.24283 24.28 7 1.1098 1 0.531 0.1096 0.640 2.2147 0.3762 0.087 1.191 0.0286 1.20718 20.72 8 1.1265 1 0.524 0.1138 0.638 2.159 0.3783 0.089 1.180 0.0303 1.17682 17.68 9 1.1434 1 0.519 0.1173 0.636 2.1115 0.3801 0.090 1.171 0.0318 1.15093 15.09

10 1.1605 1 0.514 0.1203 0.634 2.0711 0.3816 0.092 1.163 0.0331 1.12891 12.89 11 1.1779 1 0.509 0.1229 0.632 2.0366 0.383 0.093 1.156 0.0343 1.11011 11.01 12 1.1956 1 0.506 0.125 0.631 2.0071 0.3841 0.094 1.150 0.0353 1.09403 9.40

Last_pd_ss 0.484 0.1376 0.622 1.8346 0.391 0.102 1.113 0.0417

Answer Key for Economics 320a Page 3 of 5 Problem Set 5

3. Repeat the previous question in the environment with technology growth, taking the rate of technology growth to be 1.4%: Start the economy at the level of capital that is significantly higher than the balanced growth path level (e.g., by 50 or 100%). How many periods does it take for that difference to be cut in half? Compare your results to those in Problem Set 2, Question II.6. Be cautious of the results you get (there may be a problem with the MacroLab).

As can be seen from the following results, it takes less than 4 periods to cut the distance to the balanced growth path in half (it took 5 periods in Solow model). The reason for increased speed of convergence is the adjustment of investment rates and labour supply of households along the transition path.

Period N A c x y k l r w i k BGP k / kBGP % diff

0 1 1 0.638 0.013 0.650 3 0.338 0.065 1.348 0.010 1.5256 1.9665 96.65 1 1.015 1.014 0.618 0.035 0.653 2.791 0.346 0.070 1.323 0.015 1.5469 1.8043 80.43 2 1.03 1.028 0.603 0.054 0.656 2.619 0.353 0.075 1.303 0.020 1.5686 1.6698 66.98 3 1.046 1.043 0.590 0.069 0.659 2.478 0.359 0.080 1.287 0.024 1.5905 1.5581 55.81 4 1.061 1.057 0.580 0.083 0.663 2.364 0.364 0.084 1.276 0.028 1.6128 1.4655 46.55 5 1.077 1.072 0.572 0.095 0.667 2.271 0.368 0.088 1.268 0.032 1.6354 1.3885 38.85 6 1.093 1.087 0.567 0.105 0.672 2.196 0.372 0.092 1.264 0.035 1.6583 1.3244 32.44 7 1.11 1.102 0.563 0.114 0.677 2.137 0.375 0.095 1.263 0.038 1.6815 1.2711 27.11 8 1.127 1.118 0.561 0.122 0.683 2.092 0.378 0.098 1.264 0.041 1.7050 1.2267 22.67 9 1.143 1.133 0.561 0.128 0.689 2.057 0.381 0.101 1.268 0.043 1.7289 1.1896 18.96

10 1.161 1.149 0.561 0.135 0.696 2.031 0.383 0.103 1.273 0.045 1.7531 1.1587 15.87 11 1.178 1.165 0.563 0.140 0.703 2.014 0.384 0.105 1.280 0.047 1.7777 1.1328 13.28 12 1.196 1.182 0.566 0.145 0.711 2.003 0.386 0.107 1.289 0.048 1.8026 1.1113 11.13 13 1.214 1.198 0.569 0.150 0.719 1.998 0.387 0.108 1.300 0.049 1.8278 1.0932 9.32 14 1.232 1.215 0.573 0.154 0.727 1.998 0.388 0.109 1.311 0.050 1.8534 1.0781 7.81 15 1.25 1.232 0.578 0.158 0.736 2.002 0.389 0.110 1.324 0.051 1.8793 1.0654 6.54 16 1.269 1.249 0.583 0.162 0.745 2.010 0.390 0.111 1.338 0.052 1.9056 1.0549 5.49 17 1.288 1.267 0.589 0.166 0.754 2.021 0.390 0.112 1.353 0.053 1.9323 1.0460 4.60 18 1.307 1.284 0.595 0.169 0.764 2.035 0.391 0.113 1.368 0.053 1.9594 1.0385 3.85 19 1.327 1.302 0.602 0.172 0.774 2.051 0.391 0.113 1.384 0.054 1.9868 1.0323 3.23

Last_pd_ss 0.593 0.177 0.770 1.987 0.394 0.116 1.369 0.056

Answer Key for Economics 320a Page 4 of 5 Problem Set 5

4. Compare welfare effects of and the present value of the revenue generated by

a. 50% labour income tax imposed for one period only in period 1, b. 3% labour income tax imposed for 9 periods starting in period 1.

All other taxes and labour income tax in all other periods are 0. Use environment with no technological growth (for simplicity). What can you conclude about “smooth” taxation?

As can be seen from the following results, “smoothing” taxes over time reduces distortions these taxes impose. One time tax Utility = -55.64

t N A tax_l c x y k l r w i T 0 1 1 0 0.471 0.164 0.6347 1.835 0.4028 0.104 1.103 -0.01 0 1 1.015 1 0.5 0.448 -0.125 0.3235 1.861 0.1529 0.052 1.482 0.054 0.113 2 1.03 1 0 0.454 0.154 0.6075 1.600 0.4011 0.114 1.06 0.052 0 3 1.046 1 0 0.458 0.152 0.6097 1.634 0.3996 0.112 1.068 0.05 0 4 1.061 1 0 0.462 0.150 0.6115 1.663 0.3984 0.11 1.075 0.049 0 5 1.077 1 0 0.465 0.148 0.6131 1.687 0.3973 0.109 1.08 0.048 0 6 1.093 1 0 0.468 0.147 0.6144 1.708 0.3964 0.108 1.085 0.047 0 7 1.11 1 0 0.470 0.145 0.6155 1.727 0.3956 0.107 1.089 0.046 0 8 1.127 1 0 0.472 0.144 0.6164 1.742 0.3949 0.106 1.093 0.046 0 9 1.143 1 0 0.474 0.143 0.6172 1.755 0.3943 0.106 1.096 0.045 0

10 1.161 1 0 0.475 0.142 0.6179 1.767 0.3939 0.105 1.098 0.044 0 11 1.178 1 0 0.477 0.142 0.6184 1.777 0.3934 0.104 1.1 0.044 0 12 1.196 1 0 0.478 0.141 0.6189 1.785 0.3931 0.104 1.102 0.044 0 Last_pd_ss 0.484 0.138 0.6217 1.835 0.391 0.102 1.113 0.042 0 Smooth taxes Utility = -55.46

t N A tax_l c x y k l r w i T PV of T 0 1 1 0 0.482 0.142 0.6241 1.835 0.3931 0.102 1.111 0.04 0 1 1.015 1 0.03 0.481 0.131 0.6117 1.839 0.3816 0.1 1.122 0.04 0.013 0.0987 2 1.03 1 0.03 0.480 0.131 0.6111 1.832 0.3817 0.1 1.121 0.04 0.013 0.0894 3 1.046 1 0.03 0.479 0.131 0.6107 1.826 0.3819 0.1 1.119 0.041 0.013 0.0796 4 1.061 1 0.03 0.479 0.131 0.6102 1.821 0.3819 0.101 1.118 0.041 0.013 0.0695 5 1.077 1 0.03 0.479 0.131 0.6096 1.815 0.3819 0.101 1.117 0.041 0.013 0.0591 6 1.093 1 0.03 0.478 0.131 0.6091 1.81 0.3819 0.101 1.117 0.041 0.013 0.0482 7 1.11 1 0.03 0.478 0.131 0.6085 1.806 0.3818 0.101 1.116 0.041 0.013 0.0368 8 1.127 1 0.03 0.478 0.130 0.6078 1.801 0.3816 0.101 1.115 0.041 0.013 0.025 9 1.143 1 0.03 0.478 0.130 0.6071 1.796 0.3814 0.101 1.114 0.044 0.013 0.0127

10 1.161 1 0 0.479 0.141 0.6193 1.791 0.3929 0.104 1.103 0.043 0 0 11 1.178 1 0 0.479 0.140 0.6196 1.797 0.3926 0.103 1.105 0.043 0 12 1.196 1 0 0.480 0.140 0.6199 1.803 0.3924 0.103 1.106 0.043 0 Last_pd_ss 0.484 0.138 0.6217 1.835 0.391 0.102 1.113 0.042 0

Answer Key for Economics 320a Page 5 of 5 Problem Set 5

5. Compare welfare effects of and the present value of the revenue generated by

a. 90% capital income tax imposed for one period only in period 1, b. 12% capital income tax imposed for 9 periods starting in period 1.

All other taxes and capital income tax in all other periods are 0. Use environment with no technological growth (for simplicity). How do your conclusions compare to those of question 4? Why? Keep in mind that the distortions of the one period large tax are overestimated, since in the MacroLab the policy is foreseen (in period 0).

As can be seen from the following results, the implication of question 4 does not hold here. Even though the capital tax in period 1 is not a complete surprise (it is foreseen in period 0), the distortion from a (somewhat) unexpected one time large tax on capital income is no bigger than that from a stream of “smooth” capital income taxes, generating the same revenue. The key idea is that a surprise tax (that unexpected tax implemented after the relevant decisions have been made by the agents) does not introduce distortions. One time tax Utility = -55.46

t N A tax_k c x y k l r w i T 0 1 1 0 0.499 0.1096 0.608 1.8346 0.379 0.099 1.124 0.0043 0 1 1.015 1 0.9 0.481 0.1396 0.62 1.8071 0.392 0.103 1.107 0.4277 0.07 2 1.03 1 0 0.481 0.1393 0.62 1.8111 0.392 0.103 1.108 0.0426 0 3 1.046 1 0 0.482 0.139 0.621 1.8145 0.392 0.103 1.109 0.0425 0 4 1.061 1 0 0.482 0.1388 0.621 1.8174 0.392 0.103 1.109 0.0424 0 5 1.077 1 0 0.482 0.1387 0.621 1.8199 0.392 0.102 1.11 0.0423 0 6 1.093 1 0 0.483 0.1385 0.621 1.822 0.392 0.102 1.11 0.0422 0 7 1.11 1 0 0.483 0.1384 0.621 1.8238 0.392 0.102 1.111 0.0421 0 8 1.127 1 0 0.483 0.1383 0.621 1.8254 0.391 0.102 1.111 0.042 0 9 1.143 1 0 0.483 0.1382 0.621 1.8267 0.391 0.102 1.111 0.042 0

10 1.161 1 0 0.483 0.1381 0.621 1.8279 0.391 0.102 1.112 0.0419 0 11 1.178 1 0 0.483 0.138 0.621 1.8289 0.391 0.102 1.112 0.0419 0 12 1.196 1 0 0.484 0.138 0.622 1.8297 0.391 0.102 1.112 0.0419 0 Last_pd_ss 0.484 0.1376 0.622 1.8346 0.391 0.102 1.113 0.0417 0 Smooth taxes Utility = -55.4609

t N A tax_k c x y k l r w i T PV of T 0 1 1 0 0.493 0.1202 0.613 1.8346 0.383 0.1 1.12 0.0362 0 1 1.015 1 0.12 0.491 0.1222 0.613 1.8175 0.384 0.101 1.116 0.0419 0.009 0.0706 2 1.03 1 0.12 0.488 0.1241 0.612 1.8035 0.385 0.102 1.112 0.0425 0.009 0.0642 3 1.046 1 0.12 0.486 0.1261 0.612 1.7926 0.386 0.103 1.109 0.043 0.009 0.0574 4 1.061 1 0.12 0.484 0.1281 0.613 1.7843 0.387 0.103 1.107 0.0434 0.009 0.0504 5 1.077 1 0.12 0.483 0.1302 0.613 1.7786 0.388 0.103 1.105 0.0437 0.009 0.043 6 1.093 1 0.12 0.481 0.1325 0.614 1.7755 0.389 0.104 1.104 0.044 0.009 0.0352 7 1.11 1 0.12 0.48 0.1351 0.615 1.7749 0.391 0.104 1.102 0.0441 0.009 0.027 8 1.127 1 0.12 0.479 0.138 0.617 1.7769 0.392 0.104 1.102 0.0442 0.009 0.0184 9 1.143 1 0.12 0.477 0.1414 0.619 1.7816 0.393 0.104 1.101 0.0498 0.009 0.0094

10 1.161 1 0 0.478 0.1408 0.619 1.7892 0.393 0.104 1.103 0.0435 0 0 11 1.178 1 0 0.479 0.1404 0.62 1.7958 0.393 0.104 1.105 0.0432 0 12 1.196 1 0 0.48 0.14 0.62 1.8014 0.392 0.103 1.106 0.043 0 Last_pd_ss 0.484 0.1376 0.622 1.8346 0.391 0.102 1.113 0.0417 0