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MIT Center for Real Estate Week 12: Real Estate and Regional Economic Growth • Exports, transfers, investments and the determinants of regional growth: demand. • Population growth and migration: supply • 3-Q model of regional response. Factor supply elasticity and the role of real estate. • Wages, productivity and real estate costs – across MSAs. MIT Center for Real Estate Income and Product Accounts in States Summary of Output and Income Accounts for Florida and Pennsylvania, 1991 Florida ($ billions) Pennsylvania ($ billions) Income Accounts* Income (Y) 262 242 Wages (w) 126 127 Other Income (y + G) 136 115 Consumption (C) 260 193 Private 214 161 Government 46 32 Federal Taxes (T) 38 41 Savings (S) -36 8 Output Accounts** Output (Q) 219 211 Wages (w) 126 127 Profits and Rents (π) 93 84 Consumption (C) 260 193 Investment (I) 44 27 Imports (M) 175 153 Exports (X) 92 144 INCOME (Y) - OUTPUT (Q) 43 31 MIT Center for Real Estate Regional Accounts: Flow of Funds • Regions do not have to have individually balanced accounts. Surpluses in goods can be balanced by deficits in capital or government flows: the following cross border flows however must sum to zero. Trade surplus: X-M [exports - imports] Gov. surplus (Federal): G-T [spending – taxes] Capital surplus: I – S [investment - savings] Profits surplus: y - π [received - earned] • Notice the in Florida, huge trade deficit is made up with huge negative savings. MIT Center for Real Estate Sources of Regional Demand . • Some variables are determined directly by the size of a state’s economy (Income or Output) : imports (M), Federal Taxes (T), consumption or savings (S) and profits earned in the state (π). • Other variables are determined by forces largely outside of the region and serve to bring money into the state, generating growth and ultimately determining state size (level of income or output): - Exports (X) - Investment (I) - Federal spending (G) - Unearned income: SS, retirement…(y) MIT Center for Real Estate Characterizing Export growth and Investment? ∑ e i n i = ∑ e i N+ ∑ e i (N i -N) + ∑ e i (n i -N i ) i i i i Share | Mix Competitive | Shift (i): industry n,e: regional growth in activity, level of activity N: national growth of activity • Share: a matter of timing • Mix: Historic industrial structure • Competitive: “our” companies versus “theirs” [innovation –vs- production costs: “product cycle] MIT Center for Real Estate Study of impact of each Demand factor on the Boston Area Economy over time (Coulson) Mix effect Share effect Competitive effect Impact on Region 0 1 2 3 4 years since start 8 9 10 11 MIT Center for Real Estate Regional Supply shifts are as important • Migration into a region that results from factors in the origin and not destination. [US history 1820-1920]. • Birth rates in the state – 20 years earlier! (Mass –vs- California Net Reproduction Rates). • Recent immigration from Mexico and Asia. MIT Center for Real Estate P Output Market Simultaneous Equilibrium in a region’s product, labor and structures markets. 1. Product Demand=production costs. 2. Costs = average of wages and rents. 3. Wages equilibrate labor supply with labor demand (proportional to output). 4. Rents do the same in structures market. Q D Q C=α K R + α L W W/P Labor Market L D =α L Q L S R Real Estate Market L K K D =α K Q W K S R MIT Center for Real Estate Changes in Regional output, prices, wages and rents in reaction to shift in product demand Qd to Qd’. 1). Prices (and costs) must rise. Ditto output. 2). Wages and employment rise. 3). Likewise for rents and stock of structures. 4). Reverse for downward demand shifts 5). Supply Elasticity and the Magnitude of price versus quantity changes? P Output Market Q Q D C=α Ecology of Fungi Ecology of Fungi Bởi: OpenStaxCollege Fungi play a crucial role in the Rotation of Axes Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone The way in which we slice the cone will determine the type of conic section formed at the intersection A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone See [link] 1/33 Rotation of Axes The nondegenerate conic sections Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in [link] A degenerate conic results when a plane intersects the double cone and passes through the apex Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines 2/33 Rotation of Axes Degenerate conic sections Identifying Nondegenerate Conics in General Form In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections In this section, we will shift our focus to the general form 3/33 Rotation of Axes equation, which can be used for any conic The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below Ax2 + Bxy + Cy2 + Dx + Ey + F = where A, B, and C are not all zero We can use the values of the coefficients to identify which type conic is represented by a given equation You may notice that the general form equation has an xy term that we have not seen in any of the standard form equations As we will discuss later, the xy term rotates the conic whenever B is not equal to zero Conic Sections Example ellipse 4x2 + 9y2 = circle 4x2 + 4y2 = hyperbola 4x2 − 9y2 = parabola 4x2 = 9y or 4y2 = 9x one line 4x + 9y = intersecting lines (x − 4)(y + 4) = parallel lines (x − 4)(x − 9) = a point 4x2 + 4y2 = no graph 4x2 + 4y2 = − A General Note General Form of Conic Sections A nondegenerate conic section has the general form Ax2 + Bxy + Cy2 + Dx + Ey + F = where A, B, and C are not all zero [link] summarizes the different conic sections where B = 0, and A and C are nonzero real numbers This indicates that the conic has not been rotated ellipse Ax2 + Cy2 + Dx + Ey + F = 0, A ≠ C and AC > 4/33 Rotation of Axes circle Ax2 + Cy2 + Dx + Ey + F = 0, A = C hyperbola Ax2 − Cy2 + Dx + Ey + F = or − Ax2 + Cy2 + Dx + Ey + F = 0, where A and C are positive parabola Ax2 + Dx + Ey + F = or Cy2 + Dx + Ey + F = How To Given the equation of a conic, identify the type of conic Rewrite the equation in the general form, Ax2 + Bxy + Cy2 + Dx + Ey + F = Identify the values of A and C from the general form If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse If A and C are equal and nonzero and have the same sign, then the graph is a circle If A and C are nonzero and have opposite signs, then the graph is a hyperbola If either A or C is zero, then the graph is a parabola Identifying a Conic from Its General Form Identify the graph of each of the following nondegenerate conic sections 4x2 − 9y2 + 36x + 36y − 125 = 9y2 + 16x + 36y − 10 = 3x2 + 3y2 − 2x − 6y − = − 25x2 − 4y2 + 100x + 16y + 20 = Rewriting the general form, we have A = and C = −9, so we observe that A and C have opposite signs The graph of this equation is a hyperbola Rewriting the general form, we have 5/33 Rotation of Axes A = and C = We can determine that the equation is a parabola, since A is zero Rewriting the general form, we have A = and C = Because A = C, the graph of this equation is a circle Rewriting the general form, we have A = −25 and C = −4 Because AC > and A ≠ C, the graph of this equation is an ellipse Try It Identify the graph of each of the following nondegenerate conic sections 2 16y2 − x2 + x − 4y − = 16x2 + 4y2 + 16x + 49y − 81 = hyperbola ellipse Finding a New Representation of the Given Equation after Rotating through a Given Angle Until now, we have looked at equations of conic sections without an xy term, which aligns the graphs with the x- and y-axes When we add an xy term, we are rotating the conic about the origin If the x- and y-axes are rotated through an angle, say θ, then every point on the plane may be thought of as having two representations: (x, y) on the Cartesian plane with the original x-axis and y-axis, and (x ′ , y ′ ) on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis See [link] 6/33 Rotation of Axes The graph of the rotated ellipse x2 + y2 – xy – 15 = We will find the relationships between x and y on the Cartesian plane with x ′ ... MIT Center for Real Estate Week 12: Real Estate and Regional Economic Growth • Exports, transfers, investments and the determinants of regional growth: demand. • Population growth and migration: supply • 3-Q model of regional response. Factor supply elasticity and the role of real estate. • Wages, productivity and real estate costs – across MSAs. MIT Center for Real Estate Income and Product Accounts in States Summary of Output and Income Accounts for Florida and Pennsylvania, 1991 Florida ($ billions) Pennsylvania ($ billions) Income Accounts* Income (Y) 262 242 Wages (w) 126 127 Other Income (y + G) 136 115 Consumption (C) 260 193 Private 214 161 Government 46 32 Federal Taxes (T) 38 41 Savings (S) -36 8 Output Accounts** Output (Q) 219 211 Wages (w) 126 127 Profits and Rents (π) 93 84 Consumption (C) 260 193 Investment (I) 44 27 Imports (M) 175 153 Exports (X) 92 144 INCOME (Y) - OUTPUT (Q) 43 31 MIT Center for Real Estate Regional Accounts: Flow of Funds • Regions do not have to have individually balanced accounts. Surpluses in goods can be balanced by deficits in capital or government flows: the following cross border flows however must sum to zero. Trade surplus: X-M [exports - imports] Gov. surplus (Federal): G-T [spending – taxes] Capital surplus: I – S [investment - savings] Profits surplus: y - π [received - earned] • Notice the in Florida, huge trade deficit is made up with huge negative savings. MIT Center for Real Estate Sources of Regional Demand . • Some variables are determined directly by the size of a state’s economy (Income or Output) : imports (M), Federal Taxes (T), consumption or savings (S) and profits earned in the state (π). • Other variables are determined by forces largely outside of the region and serve to bring money into the state, generating growth and ultimately determining state size (level of income or output): - Exports (X) - Investment (I) - Federal spending (G) - Unearned income: SS, retirement…(y) MIT Center for Real Estate Characterizing Export growth and Investment? ∑ e i n i = ∑ e i N+ ∑ e i (N i -N) + ∑ e i (n i -N i ) i i i i Share | Mix Competitive | Shift (i): industry n,e: regional growth in activity, level of activity N: national growth of activity • Share: a matter of timing • Mix: Historic industrial structure • Competitive: “our” companies versus “theirs” [innovation –vs- production costs: “product cycle] MIT Center for Real Estate Study of impact of each Demand factor on the Boston Area Economy over time (Coulson) Mix effect Share effect Competitive effect Impact on Region 0 1 2 3 4 years since start 8 9 10 11 MIT Center for Real Estate Regional Supply shifts are as important • Migration into a region that results from factors in the origin and not destination. [US history 1820-1920]. • Birth rates in the state – 20 years earlier! (Mass –vs- California Net Reproduction Rates). • Recent immigration from Mexico and Asia. MIT Center for Real Estate P Output Market Simultaneous Equilibrium in a region’s product, labor and structures markets. 1. Product Demand=production costs. 2. Costs = average of wages and rents. 3. Wages equilibrate labor supply with labor demand (proportional to output). 4. Rents do the same in structures market. Q D Q C=α K R + α L W W/P Labor Market L D =α L Q L S R Real Estate Market L K K D =α K Q W K S R MIT Center for Real Estate Changes in Regional output, prices, wages and rents in reaction to shift in product demand Qd to Qd’. 1). Prices (and costs) must rise. Ditto output. 2). Wages and employment rise. 3). Likewise for rents and stock of structures. 4). Reverse for downward demand shifts 5). Supply Elasticity and the Magnitude of price versus quantity changes? P Output Market Q Q D C=α Types of Societies Types of Societies Bởi: OpenStaxCollege Maasai men are hunting with shepherd’s MIT Center for Real Estate Week 12: Real Estate and Regional Economic Growth • Exports, transfers, investments and the determinants of regional growth: demand. • Population growth and migration: supply • 3-Q model of regional response. Factor supply elasticity and the role of real estate. • Wages, productivity and real estate costs – across MSAs. MIT Center for Real Estate Income and Product Accounts in States Summary of Output and Income Accounts for Florida and Pennsylvania, 1991 Florida ($ billions) Pennsylvania ($ billions) Income Accounts* Income (Y) 262 242 Wages (w) 126 127 Other Income (y + G) 136 115 Consumption (C) 260 193 Private 214 161 Government 46 32 Federal Taxes (T) 38 41 Savings (S) -36 8 Output Accounts** Output (Q) 219 211 Wages (w) 126 127 Profits and Rents (π) 93 84 Consumption (C) 260 193 Investment (I) 44 27 Imports (M) 175 153 Exports (X) 92 144 INCOME (Y) - OUTPUT (Q) 43 31 MIT Center for Real Estate Regional Accounts: Flow of Funds • Regions do not have to have individually balanced accounts. Surpluses in goods can be balanced by deficits in capital or government flows: the following cross border flows however must sum to zero. Trade surplus: X-M [exports - imports] Gov. surplus (Federal): G-T [spending – taxes] Capital surplus: I – S [investment - savings] Profits surplus: y - π [received - earned] • Notice the in Florida, huge trade deficit is made up with huge negative savings. MIT Center for Real Estate Sources of Regional Demand . • Some variables are determined directly by the size of a state’s economy (Income or Output) : imports (M), Federal Taxes (T), consumption or savings (S) and profits earned in the state (π). • Other variables are determined by forces largely outside of the region and serve to bring money into the state, generating growth and ultimately determining state size (level of income or output): - Exports (X) - Investment (I) - Federal spending (G) - Unearned income: SS, retirement…(y) MIT Center for Real Estate Characterizing Export growth and Investment? ∑ e i n i = ∑ e i N+ ∑ e i (N i -N) + ∑ e i (n i -N i ) i i i i Share | Mix Competitive | Shift (i): industry n,e: regional growth in activity, level of activity N: national growth of activity • Share: a matter of timing • Mix: Historic industrial structure • Competitive: “our” companies versus “theirs” [innovation –vs- production costs: “product cycle] MIT Center for Real Estate Study of impact of each Demand factor on the Boston Area Economy over time (Coulson) Mix effect Share effect Competitive effect Impact on Region 0 1 2 3 4 years since start 8 9 10 11 MIT Center for Real Estate Regional Supply shifts are as important • Migration into a region that results from factors in the origin and not destination. [US history 1820-1920]. • Birth rates in the state – 20 years earlier! (Mass –vs- California Net Reproduction Rates). • Recent immigration from Mexico and Asia. MIT Center for Real Estate P Output Market Simultaneous Equilibrium in a region’s product, labor and structures markets. 1. Product Demand=production costs. 2. Costs = average of wages and rents. 3. Wages equilibrate labor supply with labor demand (proportional to output). 4. Rents do the same in structures market. Q D Q C=α K R + α L W W/P Labor Market L D =α L Q L S R Real Estate Market L K K D =α K Q W K S R MIT Center for Real Estate Changes in Regional output, prices, wages and rents in reaction to shift in product demand Qd to Qd’. 1). Prices (and costs) must rise. Ditto output. 2). Wages and employment rise. 3). Likewise for rents and stock of structures. 4). Reverse for downward demand shifts 5). Supply Elasticity and the Magnitude of price versus quantity changes? P Output Market Q Q D C=α Laws of Inheritance Laws of Inheritance Bởi: OpenStaxCollege Mendel generalized the results MIT Center for Real Estate Week 12: Real Estate and Regional Economic Growth • Exports, transfers, investments and the determinants of regional growth: demand. • Population growth and migration: supply • 3-Q model of regional response. Factor supply elasticity and the role of real estate. • Wages, productivity and real estate costs – across MSAs. MIT Center for Real Estate Income and Product Accounts in States Summary of Output and Income Accounts for Florida and Pennsylvania, 1991 Florida ($ billions) Pennsylvania ($ billions) Income Accounts* Income (Y) 262 242 Wages (w) 126 127 Other Income (y + G) 136 115 Consumption (C) 260 193 Private 214 161 Government 46 32 Federal Taxes (T) 38 41 Savings (S) -36 8 Output Accounts** Output (Q) 219 211 Wages (w) 126 127 Profits and Rents (π) 93 84 Consumption (C) 260 193 Investment (I) 44 27 Imports (M) 175 153 Exports (X) 92 144 INCOME (Y) - OUTPUT (Q) 43 31 MIT Center for Real Estate Regional Accounts: Flow of Funds • Regions do not have to have individually balanced accounts. Surpluses in goods can be balanced by deficits in capital or government flows: the following cross border flows however must sum to zero. Trade surplus: X-M [exports - imports] Gov. surplus (Federal): G-T [spending – taxes] Capital surplus: I – S [investment - savings] Profits surplus: y - π [received - earned] • Notice the in Florida, huge trade deficit is made up with huge negative savings. MIT Center for Real Estate Sources of Regional Demand . • Some variables are determined directly by the size of a state’s economy (Income or Output) : imports (M), Federal Taxes (T), consumption or savings (S) and profits earned in the state (π). • Other variables are determined by forces largely outside of the region and serve to bring money into the state, generating growth and ultimately determining state size (level of income or output): - Exports (X) - Investment (I) - Federal spending (G) - Unearned income: SS, retirement…(y) MIT Center for Real Estate Characterizing Export growth and Investment? ∑ e i n i = ∑ e i N+ ∑ e i (N i -N) + ∑ e i (n i -N i ) i i i i Share | Mix Competitive | Shift (i): industry n,e: regional growth in activity, level of activity N: national growth of activity • Share: a matter of timing • Mix: Historic industrial structure • Competitive: “our” companies versus “theirs” [innovation –vs- production costs: “product cycle] MIT Center for Real Estate Study of impact of each Demand factor on the Boston Area Economy over time (Coulson) Mix effect Share effect Competitive effect Impact on Region 0 1 2 3 4 years since start 8 9 10 11 MIT Center for Real Estate Regional Supply shifts are as important • Migration into a region that results from factors in the origin and not destination. [US history 1820-1920]. • Birth rates in the state – 20 years earlier! (Mass –vs- California Net Reproduction Rates). • Recent immigration from Mexico and Asia. MIT Center for Real Estate P Output Market Simultaneous Equilibrium in a region’s product, labor and structures markets. 1. Product Demand=production costs. 2. Costs = average of wages and rents. 3. Wages equilibrate labor supply with labor demand (proportional to output). 4. Rents do the same in structures market. Q D Q C=α K R + α L W W/P Labor Market L D =α L Q L S R Real Estate Market L K K D =α K Q W K S R MIT Center for Real Estate Changes in Regional output, prices, wages and rents in reaction to shift in product demand Qd to Qd’. 1). Prices (and costs) must rise. Ditto output. 2). Wages and employment rise. 3). Likewise for rents and stock of structures. 4). Reverse for downward demand shifts 5). Supply Elasticity and the Magnitude of price versus quantity changes? P Output Market Q Q D C=α Forms of Government Forms of Government Bởi: OpenStaxCollege Former Iraqi dictator MIT Center for Real Estate Week 12: Real Estate and Regional Economic Growth • Exports, transfers, investments and the determinants of regional growth: demand. • Population growth and migration: supply • 3-Q model of regional response. Factor supply elasticity and the role of real estate. • Wages, productivity and real estate costs – across MSAs. MIT Center for Real Estate Income and Product Accounts in States Summary of Output and Income Accounts for Florida and Pennsylvania, 1991 Florida ($ billions) Pennsylvania ($ billions) Income Accounts* Income (Y) 262 242 Wages (w) 126 127 Other Income (y + G) 136 115 Consumption (C) 260 193 Private 214 161 Government 46 32 Federal Taxes (T) 38 41 Savings (S) -36 8 Output Accounts** Output (Q) 219 211 Wages (w) 126 127 Profits and Rents (π) 93 84 Consumption (C) 260 193 Investment (I) 44 27 Imports (M) 175 153 Exports (X) 92 144 INCOME (Y) - OUTPUT (Q) 43 31 MIT Center for Real Estate Regional Accounts: Flow of Funds • Regions do not have to have individually balanced accounts. Surpluses in goods can be balanced by deficits in capital or government flows: the following cross border flows however must sum to zero. Trade surplus: X-M [exports - imports] Gov. surplus (Federal): G-T [spending – taxes] Capital surplus: I – S [investment - savings] Profits surplus: y - π [received - earned] • Notice the in Florida, huge trade deficit is made up with huge negative savings. MIT Center for Real Estate Sources of Regional Demand . • Some variables are determined directly by the size of a state’s economy (Income or Output) : imports (M), Federal Taxes (T), consumption or savings (S) and profits earned in the state (π). • Other variables are determined by forces largely outside of the region and serve to bring money into the state, generating growth and ultimately determining state size (level of income or output): - Exports (X) - Investment (I) - Federal spending (G) - Unearned income: SS, retirement…(y) MIT Center for Real Estate Characterizing Export growth and Investment? ∑ e i n i = ∑ e i N+ ∑ e i (N i -N) + ∑ e i (n i -N i ) i i i i Share | Mix Competitive | Shift (i): industry n,e: regional growth in activity, level of activity N: national growth of activity • Share: a matter of timing • Mix: Historic industrial structure • Competitive: “our” companies versus “theirs” [innovation –vs- production costs: “product cycle] MIT Center for Real Estate Study of impact of each Demand factor on the Boston Area Economy over time (Coulson) Mix effect Share effect Competitive effect Impact on Region 0 1 2 3 4 years since start 8 9 10 11 MIT Center for Real Estate Regional Supply shifts are as important • Migration into a region that results from factors in the origin and not destination. [US history 1820-1920]. • Birth rates in the state – 20 years earlier! (Mass –vs- California Net Reproduction Rates). • Recent immigration from Mexico and Asia. MIT Center for Real Estate P Output Market Simultaneous Equilibrium in a region’s product, labor and structures markets. 1. Product Demand=production costs. 2. Costs = average of wages and rents. 3. Wages equilibrate labor supply with labor demand (proportional to output). 4. Rents do the same in structures market. Q D Q C=α K R + α L W W/P Labor Market L D =α L Q L S R Real Estate Market L K K D =α K Q W K S R MIT Center for Real Estate Changes in Regional output, prices, wages and rents in reaction to shift in product demand Qd to Qd’. 1). Prices (and costs) must rise. Ditto output. 2). Wages and employment rise. 3). Likewise for rents and stock of structures. 4). Reverse for downward demand shifts 5). Supply Elasticity and the Magnitude of price versus quantity changes? P Output Market Q Q D C=α Types of Groups Types of Groups Bởi: OpenStaxCollege Most of us feel comfortable using the ... sections and rotation of axes • Introduction to Conic Sections 21/33 Rotation of Axes Key Equations General Form equation of a conic section Ax2 + Bxy + Cy2 + Dx + Ey + F = Rotation of a conic... coordinate plane where the axes of rotation are formed by rotating an angle θ from the positive x-axis, then 9/33 Rotation of Axes the coordinates of the point with respect to the new axes are (x ′ , y... shows the graph 11/33 Rotation of Axes Writing Equations of Rotated Conics in Standard Form Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn