Transcript Slide 1
РОССИЙСКИЙ УНИВЕРСИТЕТ ДРУЖБЫ НАРОДОВ (РУДН) Факультет физико-математических и естественных наук Кафедра нелинейного анализа и оптимизации группа НМ-601 Пример модели ВВП с учетом старения производственных фондов, её исследование и применение к экономике США Выполнил: Сергоян Айк Самвелович Научный руководитель: доцент, к.ф-м.н. Оленев Николай Николаевич Purposes of the research • A GDP model description • Research of the model • Calculation of parameters • Description of US economy by the model Introduction Cobb-Douglas’s model – Q=𝐴 × 𝐿𝛼 × 𝐾𝛽 Q - output, L - labor costs, K - capital expenditures, and 𝐴 – technological coefficient There are no any theoretical base to take as a constant 𝛼 in different sectors of economy. In order to understand that fact let us consider that we have an economy, where 𝛼 + 𝛽 = 1. Now let us consider two sectors of that economy with the same technological coefficient 𝐴 : 𝑄1 = 𝐴 × 𝐿𝛼1 × 𝐾11−𝛼 𝑄2 = 𝐴 × 𝐿𝛼2 × 𝐾21−𝛼 𝑄1 +𝑄2 ≠ 𝐴 × (𝐿1 + 𝐿2 )𝛼 × (𝐾1 + 𝐾2 )1−𝛼 . 𝐿 Equality is possible only if - 1 = 𝐾1 /𝐾2 . 𝐿2 Description of a GDP model Hypothesis: The number of homogeneous jobs in a given production unit are fixed over time, and product output decreases with a constant rate µ, µ>0 . Considered formula of GDP: 𝐺(𝑡) = 𝑀(𝑡) × 𝑓(𝑡). 𝐺(𝑡) - the value of GDP in year t 𝑓(𝑡) called the loading capacity 𝑀 𝑡 - maximal total output 𝐺 𝑡 <𝑀 𝑡 ⟹𝑓 𝑡 <1 Description of a GDP model (2) 𝑛 𝑀 𝑡 = 𝑖=0 𝛷(𝑡 − 𝑖)𝑒 −𝑖𝜇 𝑝(𝑡 − 𝑖) × 𝑏 𝒏 – number of years, after which production unit becomes unusable due to aging b – coefficient of the capital intensity 𝛷 𝑡 – investment in the economy in year 𝒕 𝑝(𝑡) – price index in year t 𝑝 𝑡 = 𝐺(𝑡)𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑟𝑖𝑐𝑒 𝐺(𝑡)𝑝𝑟𝑖𝑐𝑒 𝑖𝑛 2005 Conclusions a. A model of GDP was described b. Model was researched c. The coefficient of capital intensity – 𝑏, 𝛼, product output decreasing rate – 𝜇, were obtained d. Model was used and parameters were obtained for US economy. Description of a GDP model (3) Share of new capacities are constant in the total output. = 𝛼 = 𝑐𝑜𝑛𝑠𝑡 > 0, for each year t ⟹ 𝛷 𝑡 𝑝 𝑡 ×𝑏×𝑀 𝑡 𝜇 1−𝛼 𝐿 𝑡 𝑓 𝑡 =1− 1− × 𝜈 𝑡 𝑀 𝑡 𝝂 𝒕 – labor input in year 𝒕. 𝑳 𝒕 – Number of jobs in year t. 1 𝜇 1− 𝛼 Research of the model As the function 𝑓 𝑡 < 1, the value of 1 − 𝑓(𝑡) > 1. Note also that 1 − that, 0 < 𝜇 𝛼 < 1 ⟹ 𝜇 < 𝛼. 𝜇 𝛼 𝜇 𝛼 > 0, because otherwise – 𝜇 𝛼 < 1. So – 0 < 1 − < 1. It is easy to see, 49 𝐿 𝑡 = 𝑖=0 𝛷(𝑡 − 𝑖) 𝜐𝑖 (𝑡) × 𝑝(𝑡 − 𝑖) × 𝑏 Calculation of parameters The labor input appropriating to each production unit is constant by t. ⟹ 𝜈= 𝐿(𝑡) 𝛷(𝑡 − 𝑖) 49 𝑖=0 𝑝(𝑡 − 𝑖) × 𝑏 𝛷 𝑡 = 𝛷 𝑡 − 𝑖 −𝑖μ 𝑝 𝑡 × 49 𝑝 𝑡+𝑗 × 𝑖=0( 𝑝 𝑡 − 𝑖 ) × 𝑒 𝛷 𝑡+𝑗 =𝛼 𝛷 𝑡 + 𝑗 − 𝑖 49 −𝑖μ 𝑖=0( 𝑝 𝑡 + 𝑗 − 𝑖 ) × 𝑒 Solving the last equation we will obtain 𝛍. Than we can obtain 𝛼 Calculation of parameters(2) 𝑏= μ 1− 1− 1−α × 49 𝛷 𝑖=0 𝑝 𝑡−𝑖 𝑡−𝑖 ×𝑝 𝑡 ×𝛼 𝛷 𝑡 𝐺 𝑡 ×𝑝 𝑡 ×𝛼 Finally we obtain 𝒃 μ 1/1−α ×𝛷 𝑡 Description of US economy by the model Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 GDP (US) (∗ 1011 ) 97,648 100,76 104,176 109,08 116,308 123,641 133,362 139,95 142,969 140,439 145,82 4 9,666 10,034 10,165 11,023 12,078 13,013 14,104 15,334 16,731 18,29 19,822 Investment (∗ 1011 ) US economy μ = 0.00577 𝛼= 0.02383 b= 4.12341 Thank you for attention! ;)