## 3.1 Standard growth decomposition ## Source details **Canonical URL:** [3.1 Standard growth decomposition](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023170-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023170-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023170-print-pdf.pdf.json) --- ### Overview and central puzzle - In several rapidly growing economies (for example, Singapore, Malaysia, Türkiye, Panama) TFP levels over the past half century have declined or been estimated as lower despite large GDP per capita increases. - Puzzle: technological progress appears spectacular, yet standard growth accounting reports lower or negative TFP growth. - Core claim of the paper: standard growth decompositions overestimate the contribution of capital by not accounting for land rents, thereby underestimating TFP growth. ### Key conceptual points - Standard production function used in growth accounting: - Y = A F(K,L) (equation 3.1.1) - Total differential with perfect competition and constant returns yields: - dY/Y = dA/A + α dK/K + (1−α) dL/L (equation 3.1.5) - Estimated TFP growth: dˆA/ˆA = dY/Y − α dK/K − (1−α) dL/L (equation 3.1.6) - When land is included: - Real production function: Y = A F(K,L,T) (equation 3.2.1) - Define γ as the share of GDP going to land rents. - Correct decomposition with fixed land: - dA/A = dY/Y − (α − γ) dK/K − (1−α) dL/L (equation 3.2.3) - Difference between standard estimate and true TFP growth: - dˆA/ˆA = dA/A − γ dK/K (equation 3.2.4) - Implication: standard decompositions use α as the capital coefficient, whereas with land present the coefficient on capital should be α−γ. The omission of land causes TFP to be underestimated by γ dK/K. ### Empirical signal - Capital deepening (increase in the capital-output ratio) is strongly negatively associated with changes in Penn World Tables TFP estimates (Figure 3.2). - Capital deepening is positively associated with GDP growth (Figure 3.3), which argues against reverse causality (i.e., low TFP causing capital deepening via lower GDP growth). ### Numeric examples and magnitudes - Singapore: - Capital stock in 2019 was 39 times as high as in 1970. - Capital stock growth between 1970 and 2019: 7.5 percent annually. - Average capital income share during this period: 56 percent of GDP. - Assuming land rents were 25 percent of GDP, TFP was underestimated by 0.25*7.5=1.75 percent annually. - This would imply TFP did not decline by 0.2 percent annually between 1970 and 2019, but increased by 1.55 percent. - More precise annual estimate using annual data is 1.70 percent. - United States example: - Average annual growth rate of the capital stock between 1970 and 2019: 3.2 percent. - Average capital income share between 1970 and 2019: 39 percent. - Assuming land rents were 15 percent of GDP, underestimation of TFP was 0.15*3.2=0.5 percent per year. ### Interpretation difference: declining TFP vs. declining returns to scale - Declining TFP: implies more inputs needed to produce same output—signals declining efficiency or regressing technology. - Declining returns to scale (with land fixed): an increase in inputs leads to less-than-proportionate output growth but does not imply falling efficiency per se. - Example: With unchanged TFP and constant returns (capital income share 60 percent), Singapore’s observed inputs increases would predict output 27 times higher; observed output was 24 times higher → interpreted as negative TFP growth. But with capital coefficient 35 percent (accounting for land), same inputs predict output 11 times higher, implying substantial TFP contribution. ### Role of land in growth accounting and existing literature - Modern growth theory often omits land, focusing on capital and labor with constant returns (Solow technology). But urban land values have risen sharply in the second half of the twentieth century. - Rising land prices explain about 80 per cent of the global house price boom (Knoll et al. (2017), as cited). - Urban density is strongly linked to higher land values (Barr et al. (2018); Duranton and Puga (2020); Glaeser and Gyourko (2018), as cited). - Prior literature has not, to the authors’ knowledge, incorporated the sharp increase in land prices into growth accounting in the way proposed here. ### Re-estimation implication - The relationships and adjustments above underpin recalculations of TFP for economies where land rents may be substantial relative to GDP. --- ### 4.1–4.2 Determining land rents from the value of land; OECD calculations ### Theory and required variables - Share of land rents in GDP, γ: - γ = (r_T P_T T) / (P_Y Y) (equation 4.1.1) - Assume net return on land equals net return on capital: r_T = r_K − δ = r (equation 4.1.2) - Relationship between returns and asset values: - r P_T T + (r + δ) P_K K = α P_Y Y (equation 4.1.3) - Hence r = [α P_Y Y − δ P_K K] / [P_T T + P_K K] (equation 4.1.4) - Then γ = (r P_T T) / (P_Y Y) (equation 4.1.5) - Five variables needed to calculate land rents: - The value of land (P_T T) - The value of the capital stock (P_K K) - Nominal GDP (P_Y Y) - The capital income share (α) - The depreciation rate on the capital stock (δ) ### Empirical approach for OECD countries - Data sources combined: - OECD data: value of capital stock, aggregate value of land, and GDP. - Penn World Tables 10.0: capital income share and depreciation rate. - Procedure: - Calculate r using equation (4.1.4). - Calculate γ from equation (4.1.5). ### OECD results - Land rents as a share of GDP (Figure 4.1 results cited): - Korea: about 24 percent of GDP. - Great Britain: 21 percent of GDP. - Germany: about 10 percent of GDP. - Note: Estimates for Australia are similar to Putland (2018), who estimates land rents in Australia in 2018 at slightly over 20 percent of GDP. --- ### 5 Determining land rents from population density ### Rationale and empirical links - Land values are much higher in densely populated areas; GDP is higher too but less sharply, implying the ratio of land values to GDP rises with population density. - Empirical evidence cited: - Five US urban agglomerations (New York, Los Angeles, San Francisco, Washington, DC, Chicago) account for 48% of all urban land value in the United States (Albouy et al. (2018), as cited). - In 2018, those five agglomerations accounted for 22.3 percent of US GDP—less than half their share of US land values. - Manhattan developable land in 2014: approximately $1.74 trillion (Barr et al. (2018), as cited). - Implication: If return on land is assumed constant, the share of land rents in GDP depends positively on population density. ### Strategy for economies without land-value data - For Singapore and Hong Kong SAR (no aggregate land-value data available), estimate a lower bound for the land-rents-to-GDP ratio using relationships observed in: - US states: link between population per developed area and the value of developed land relative to GDP. - US states: link between population-weighted density and land rents. - OECD countries with aggregate land-value data: link between weighted population density and share of land rents in GDP. - Conservative assumption for Singapore and Hong Kong SAR: - Korea has the highest weighted population density among OECD economies with land-value data, and land rents slightly less than 25 percent of GDP there. - Given Hong Kong SAR and Singapore have much higher weighted population density, authors conservatively assume land rents in Hong Kong SAR and Singapore are 25 percent of GDP. --- ### 5.1 Population per developed area and land rents: Evidence from US states ### Land values by category (Larson (2015)) - Developed land: $106,000 per acre (average) - Non-developed land: $6,500 per acre (average) - Agricultural land: $2,000 per acre (average) - Non-agricultural land: $21,000 per acre (average) ### Relationship between population per developed acre and land prices (US states) - For every 10 percent increase in population density, land prices increase by 51 percent. - California: 5.5 persons per developed acre → land price per developed acre is 450 thousand dollars. - South Dakota: 0.6 persons per developed acre → land price per developed acre is 8 thousand dollars. ### Population density and GDP per acre - Among US states, a 10 percent higher population per developed acre is associated with an 11 percent higher GDP per acre. ### Developed land value relative to GDP - Among US states, a 10 percent higher population per developed acre is associated with a 41 percent higher developed land value to GDP ratio. - California: developed land is worth 160 percent of GDP. - South Dakota: developed land is worth 30 percent of GDP. - Implication: Assuming the return on developed land is the same across states, the share of land rents on developed land in GDP is higher in more densely populated states. ### Population-weighted density as a density measure - Population-weighted density: mean of densities of subareas weighted by populations of those subareas (Ottensmann (2018)). - Example calculation in text: two areas, area 1 area = 1, population = 99; area 2 area = 99, population = 1. Country population density = 1. Weighted population density = 98.01. ### Weighted population density and land prices (US states) - Strong link between weighted population density and developed land price. - California: weighted population density is 8,213 per square mile → land price per developed acre is 450 thousand dollars. - South Dakota: weighted population density is 1,045 per square mile → land price per developed acre is 8 thousand dollars. - Result: ratio of value of developed land increases as weighted population density increases; hence share of land rents on developed land in GDP higher in more densely populated states (assuming identical returns on developed land). ### Footnote on data coverage - OECD data used in subsequent sections is for the value of all land—not broken down by category. --- *Content derived from "3.1 Standard growth decomposition" and adjacent sections of the provided IMF Working Paper PDF (wpiea2023170-print-pdf).* ### 3.1 Standard growth decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 16 ### 3.1 Standard growth decomposition ### Overview and central puzzle - In several rapidly growing economies (for example, Singapore, Malaysia, Türkiye, Panama) TFP levels over the past half century have declined or been estimated as lower despite large GDP per capita increases. - Puzzle: technological progress appears spectacular, yet standard growth accounting reports lower or negative TFP growth. - Core claim of the paper: standard growth decompositions overestimate the contribution of capital by not accounting for land rents, thereby underestimating TFP growth. ### Key conceptual points - Standard production function used in growth accounting: - Y = A F(K,L) (equation 3.1.1) - Total differential with perfect competition and constant returns yields: - dY/Y = dA/A + α dK/K + (1−α) dL/L (equation 3.1.5) - Estimated TFP growth: dˆA/ˆA = dY/Y − α dK/K − (1−α) dL/L (equation 3.1.6) - When land is included: - Real production function: Y = A F(K,L,T) (equation 3.2.1) - Define γ as the share of GDP going to land rents. - Correct decomposition with fixed land: - dA/A = dY/Y − (α − γ) dK/K − (1−α) dL/L (equation 3.2.3) - Difference between standard estimate and true TFP growth: - dˆA/ˆA = dA/A − γ dK/K (equation 3.2.4) - Implication: standard decompositions use α as the capital coefficient, whereas with land present the coefficient on capital should be α−γ. The omission of land causes TFP to be underestimated by γ dK/K. ### Empirical signal - Capital deepening (increase in the capital-output ratio) is strongly negatively associated with changes in Penn World Tables TFP estimates (Figure 3.2). - Capital deepening is positively associated with GDP growth (Figure 3.3), which argues against reverse causality (i.e., low TFP causing capital deepening via lower GDP growth). ### Numeric examples and magnitudes (preserve source values exactly) - Singapore: - Capital stock in 2019 was 39 times as high as in 1970. - Capital stock growth between 1970 and 2019: 7.5 percent annually. - Average capital income share during this period: 56 percent of GDP. - Assuming land rents were 25 percent of GDP, TFP was underestimated by 0.25*7.5=1.75 percent annually. - This would imply TFP did not decline by 0.2 percent annually between 1970 and 2019, but increased by 1.55 percent. - More precise annual estimate using annual data is 1.70 percent. - United States example: - Average annual growth rate of the capital stock between 1970 and 2019: 3.2 percent. - Average capital income share between 1970 and 2019: 39 percent. - Assuming land rents were 15 percent of GDP, underestimation of TFP was 0.15*3.2=0.5 percent per year. ### Interpretation difference: declining TFP vs. declining returns to scale - Declining TFP: implies more inputs needed to produce same output—signals declining efficiency or regressing technology. - Declining returns to scale (with land fixed): an increase in inputs leads to less-than-proportionate output growth but does not imply falling efficiency per se. - Example: With unchanged TFP and constant returns (capital income share 60 percent), Singapore’s observed inputs increases would predict output 27 times higher; observed output was 24 times higher → interpreted as negative TFP growth. But with capital coefficient 35 percent (accounting for land), same inputs predict output 11 times higher, implying substantial TFP contribution. --- ### Role of land in growth accounting and existing literature (selected points) - Modern growth theory often omits land, focusing on capital and labor with constant returns (Solow technology). But urban land values have risen sharply in the second half of the twentieth century. - Rising land prices explain about 80 per cent of the global house price boom (Knoll et al. (2017), as cited). - Urban density is strongly linked to higher land values (Barr et al. (2018); Duranton and Puga (2020); Glaeser and Gyourko (2018), as cited). - Prior literature has not, to the authors’ knowledge, incorporated the sharp increase in land prices into growth accounting in the way proposed here. --- ### 4.1–4.2 Determining land rents from the value of land; OECD calculations ### Theory and required variables - Share of land rents in GDP, γ: - γ = (r_T P_T T) / (P_Y Y) (equation 4.1.1) - Assume net return on land equals net return on capital: r_T = r_K − δ = r (equation 4.1.2) - Relationship between returns and asset values: - r P_T T + (r + δ) P_K K = α P_Y Y (equation 4.1.3) - Hence r = [α P_Y Y − δ P_K K] / [P_T T + P_K K] (equation 4.1.4) - Then γ = (r P_T T) / (P_Y Y) (equation 4.1.5) - Five variables needed to calculate land rents: - The value of land (P_T T) - The value of the capital stock (P_K K) - Nominal GDP (P_Y Y) - The capital income share (α) - The depreciation rate on the capital stock (δ) ### Empirical approach for OECD countries - Data sources combined: - OECD data: value of capital stock, aggregate value of land, and GDP. - Penn World Tables 10.0: capital income share and depreciation rate. - Procedure: - Calculate r using equation (4.1.4). - Calculate γ from equation (4.1.5). ### OECD results (preserve reported values) - Land rents as a share of GDP (Figure 4.1 results cited): - Korea: about 24 percent of GDP. - Great Britain: 21 percent of GDP. - Germany: about 10 percent of GDP. - Note: Estimates for Australia are similar to Putland (2018), who estimates land rents in Australia in 2018 at slightly over 20 percent of GDP. --- ### 5 Determining land rents from population density ### Rationale and empirical links - Land values are much higher in densely populated areas; GDP is higher too but less sharply, implying the ratio of land values to GDP rises with population density. - Empirical evidence cited: - Five US urban agglomerations (New York, Los Angeles, San Francisco, Washington, DC, Chicago) account for 48% of all urban land value in the United States (Albouy et al. (2018), as cited). - In 2018, those five agglomerations accounted for 22.3 percent of US GDP—less than half their share of US land values. - Manhattan developable land in 2014: approximately $1.74 trillion (Barr et al. (2018), as cited). - Implication: If return on land is assumed constant, the share of land rents in GDP depends positively on population density. ### Strategy for economies without land-value data - For Singapore and Hong Kong SAR (no aggregate land-value data available), estimate a lower bound for the land-rents-to-GDP ratio using relationships observed in: - US states: link between population per developed area and the value of developed land relative to GDP. - US states: link between population-weighted density and land rents. - OECD countries with aggregate land-value data: link between weighted population density and share of land rents in GDP. - Conservative assumption for Singapore and Hong Kong SAR: - Korea has the highest weighted population density among OECD economies with land-value data, and land rents slightly less than 25 percent of GDP there. - Given Hong Kong SAR and Singapore have much higher weighted population density, authors conservatively assume land rents in Hong Kong SAR and Singapore are 25 percent of GDP. --- *Italicized source attribution: Content derived from "3.1 Standard growth decomposition" and adjacent sections of the provided IMF Working Paper PDF (wpiea2023170-print-pdf).* ### 5.1 Population per developed area and land rents: Evidence from US states ### 5.1 Population per developed area and land rents: Evidence from US states ### Land values by category (Larson (2015)) - Developed land: $106,000 per acre (average) - Non-developed land: $6,500 per acre (average) - Agricultural land: $2,000 per acre (average) - Non-agricultural land: $21,000 per acre (average) ### Relationship between population per developed acre and land prices (US states) - For every 10 percent increase in population density, land prices increase by 51 percent. - California: 5.5 persons per developed acre → land price per developed acre is 450 thousand dollars. - South Dakota: 0.6 persons per developed acre → land price per developed acre is 8 thousand dollars. ### Population density and GDP per acre - Among US states, a 10 percent higher population per developed acre is associated with an 11 percent higher GDP per acre. ### Developed land value relative to GDP - Among US states, a 10 percent higher population per developed acre is associated with a 41 percent higher developed land value to GDP ratio. - California: developed land is worth 160 percent of GDP. - South Dakota: developed land is worth 30 percent of GDP. - Implication: Assuming the return on developed land is the same across states, the share of land rents on developed land in GDP is higher in more densely populated states. --- ### Population-weighted density as a density measure - Population-weighted density: mean of densities of subareas weighted by populations of those subareas (Ottensmann (2018)). - Example calculation in text: two areas, area 1 area = 1, population = 99; area 2 area = 99, population = 1. Country population density = 1. Weighted population density = 98.01. ### Weighted population density and land prices (US states) - Strong link between weighted population density and developed land price. - California: weighted population density is 8,213 per square mile → land price per developed acre is 450 thousand dollars. - South Dakota: weighted population density is 1,045 per square mile → land price per developed acre is 8 thousand dollars. - Result: ratio of value of developed land increases as weighted population density increases; hence share of land rents on developed land in GDP higher in more densely populated states (assuming identical returns on developed land). ### Footnote on data coverage - OECD data used in subsequent sections is for the value of all land—not broken down by category. --- ### Re-estimation implications highlighted in subsequent sections - The relationships above underpin recalculations of TFP for economies where land rents may be substantial relative to GDP. *Source: wpiea2023170-print-pdf - 5.1 Population per developed area and land rents: Evidence from US states* --- _Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023170-print-pdf.pdf_