Exchange Equilibrium
We’ve already looked at linear supply, demand, and equilibrium with an agent (buyer) and a firm (seller). Now we turn to exchange equilibrium, when both parties are agents with some endowment of goods 1 and 2 and some preferences over those goods. The agents can exchange goods between them until an exchange equilibrium results.
Net Demand
For each agent in an exchange equilibrium-style problem, they have some net demand for each good based on the relative prices of both goods–a positive demand if they’d like more of the good, and a negative demand if they’d like to sell that good. On the y-axis we graph the price of good 1, our independent vairable, and on the x-axis we graph the amount of good 1 that the consumer would buy, the dependent variable. We hold the price of good 2 constant to see how demand for good 1 varies as a function of its price alone.
Let’s look at a very simple example of how EconNetDemandCurve works, and then we’ll add draggability and other features to bring it to life.
schema: EconSchema
layout:
OneGraph:
graph:
xAxis:
title: Demand for Good 1
min: -10
max: 30
yAxis:
title: Price of Good 1
min: 0
max: 30
objects:
- EconNetDemandCurve:
utilityFunction:
CobbDouglas: {alpha: 0.5}
budgetLine:
p2: 8
x: 40
y: 50
In the graph above, EconNetDemandCurve relies on the following objects:
- Utility Function: the utility function tells the net demand curve what the agent would buy at any price point
p2 given his or her preferences. The agent will maximize utility for any value p2, traced out along the net demand curve.
- Budget Line: the budget line includes a price for good 2,
p2, which will stay constant as p1 changes, so it is supplied to the budget line. It also includes an x and y value, the initial endowments of Good 1 and Good 2, respectively. This effectively determines how much the agent can buy of either good by selling the other.
Of course, the EconNetDemandCurve is not very useful as a lone, static object. Let’s add a draggable point describing the amount of Good 1 the agent demands at different prices:
schema: EconSchema
params:
- name: p1
value: 7
min: 5
max: 15
round: .01
calcs:
xCoord: ((200/params.p1) - 20)
layout:
OneGraph:
graph:
xAxis:
title: Demand for Good 1
min: -10
max: 30
yAxis:
title: Price of Good 1
min: 0
max: 30
objects:
- EconNetDemandCurve:
utilityFunction:
CobbDouglas: {alpha: 0.5}
budgetLine:
p2: 8
x: 40
y: 50
- Point:
coordinates: [calcs.xCoord, params.p1]
drag:
- vertical: p1
label:
text: "`Demand = ${calcs.xCoord.toFixed(2)}`"
Now we are able to see how the demand for the good changes as we drag the point across different prices. We can also allow the endowment to be shiftable, which will change the shape of the net demand curve:
schema: EconSchema
params:
- name: p1
value: 7
min: 5
max: 15
round: .01
- name: w1
value: 40
min: 20
max: 60
round: 0.1
- name: w2
value: 50
min: 30
max: 70
round: 0.1
calcs:
xCoord: ((params.w1/2) + ((params.w2*8)/(params.p1*2)) - params.w1)
layout:
OneGraphPlusSidebar:
graph:
xAxis:
title: Demand for Good 1
min: -10
max: 30
yAxis:
title: Price of Good 1
min: 0
max: 30
objects:
- EconNetDemandCurve:
utilityFunction:
CobbDouglas: {alpha: 0.5}
budgetLine:
p2: 8
x: params.w1
y: params.w2
- Point:
coordinates: [calcs.xCoord, params.p1]
drag:
- vertical: p1
label:
text: "`Demand = ${calcs.xCoord.toFixed(2)}`"
sidebar:
controls:
- title: Utility Function
sliders:
- param: w1
label: "\\omega_1"
- param: w2
label: "\\omega_2"
The true power of the EconNetDemandCurve is best captured when viewed simultaneously with a traditional EconOptimalBundle type graph representing an endowment, budget line, and indifference curve. Below, we show a more complicated two-graph panel that connects an EconOptimalBundle to the EconNetDemandCurve:
schema: EconSchema
params:
- name: a
value: 0.5
min: 0.01
max: 0.99
round: 0.01
- name: p1
value: 4
min: 0.1
max: 10
round: 0.01
- name: p2
value: 8
min: 0.1
max: 10
round: 0.01
- name: w1
value: 40
min: 0
max: 100
round: 1
- name: w2
value: 30
min: 0
max: 50
round: 1
- name: showIndW
value: false
- name: showPOC
value: false
- name: showExplanation
value: false
calcs:
bottomGraphLeft: (-1*params.w1)
bottomGraphRight: (100-params.w1)
cutoffPrice: (calcs.endowmentBundle.mrs*params.p2)
wealth: (params.w1*params.p1 + params.w2*params.p2)
layout:
TwoVerticalGraphsPlusSidebar:
topGraph:
xAxis:
title: Units of Good 1
orient: bottom
min: 0
max: 100
ticks: 10
yAxis:
title: Units of Good 2
orient: left
min: 0
max: 50
objects:
- EconOptimalBundle:
name: optimum2
budgetLine:
p1: 2
p2: params.p2
x: params.w1
y: params.w2
inMap: true
utilityFunction:
CobbDouglas: {alpha: params.a}
indifferenceCurve:
inMap: true
label:
droplines: {}
r: 4
color: colors.offer
show: params.showPOC
- EconOptimalBundle:
name: optimum4
utilityFunction:
CobbDouglas: {alpha: params.a}
budgetLine:
p1: 4
p2: params.p2
x: params.w1
y: params.w2
inMap: true
indifferenceCurve:
inMap: true
label:
droplines: {}
r: 4
color: colors.offer
show: params.showPOC
- EconOptimalBundle:
name: optimum6
utilityFunction:
CobbDouglas: {alpha: params.a}
budgetLine:
p1: 6
p2: params.p2
x: params.w1
y: params.w2
inMap: true
indifferenceCurve:
inMap: true
label:
droplines: {}
r: 4
color: colors.offer
show: params.showPOC
- EconOptimalBundle:
name: optimum8
utilityFunction:
CobbDouglas: {alpha: params.a}
budgetLine:
p1: 8
p2: params.p2
x: params.w1
y: params.w2
inMap: true
indifferenceCurve:
inMap: true
label:
droplines: {}
r: 4
color: colors.offer
show: params.showPOC
- EconPriceOfferCurve:
lineStyle: dashed
utilityFunction:
CobbDouglas: {alpha: params.a}
budgetLine:
p1: params.p1
p2: params.p2
x: params.w1
y: params.w2
good: 1
show: params.showPOC
- EconBundle:
name: endowmentBundle
coordinates:
- params.w1
- params.w2
droplines:
vertical: "\\omega_1"
color: gray
label:
text: W
position: tr
utilityFunction:
CobbDouglas: {alpha: params.a}
drag:
- horizontal: w1
- vertical: w2
label:
text:
color: colors.endowment
show: params.showIndW
- EconOptimalBundle:
name: optimum
label:
text: X
droplines:
vertical: x_1^*
utilityFunction:
CobbDouglas: {alpha: params.a}
color: colors.utility
budgetLine:
p1: params.p1
p2: params.p2
x: params.w1
y: params.w2
draggable: false
handles: false
set: false
color: colors.budget
label:
text: "`BL_{m=${calcs.wealth.toFixed(0)}}`"
indifferenceCurve: {}
- Segment:
a: [0,8]
b: [params.w1,8]
endArrow: true
color: colors.endowment
label:
text: "`\\\\text{You have ${params.w1.toFixed(0)} units of good 1...}`"
location: 0.5
yPixelOffset: 15
show: params.showExplanation
- Segment:
a: [0,5]
b: [calcs.optimum.x,5]
endArrow: true
color: colors.utility
label:
text: "`\\\\text{...and want to consume ${calcs.optimum.x.toFixed(0)}...}`"
location: 0.5
yPixelOffset: -15
show: params.showExplanation
bottomGraph:
xAxis:
title: Net Demand for Good 1
orient: bottom
min: (calcs.bottomGraphLeft)
max: (calcs.bottomGraphRight)
ticks: 10
yAxis:
title: Price of Good 1
orient: left
min: 0
max: 10
objects:
- EconNetDemandCurve:
name: demandCurve
utilityFunction:
CobbDouglas: {alpha: params.a}
budgetLine:
p2: params.p2
x: params.w1
y: params.w2
max: calcs.cutoffPrice
- EconNetDemandCurve:
name: demandCurve
utilityFunction:
CobbDouglas: {alpha: params.a}
budgetLine:
p2: params.p2
x: params.w1
y: params.w2
color: colors.supply
min: calcs.cutoffPrice
- Point:
coordinates:
- calcs.optimum2.x - params.w1
- 2
r: 4
color: colors.demand
show: params.showPOC
- Point:
coordinates:
- calcs.optimum4.x - params.w1
- 4
r: 4
color: colors.demand
show: params.showPOC
- Point:
coordinates:
- calcs.optimum6.x - params.w1
- 6
r: 4
color: colors.demand
show: params.showPOC
- Point:
coordinates:
- calcs.optimum8.x - params.w1
- 8
r: 4
color: colors.demand
show: params.showPOC
- Segment:
a:
- calcs.bottomGraphLeft
- params.p1
b:
- calcs.bottomGraphRight
- params.p1
color: colors.budget
drag:
- directions: y
param: p1
expression: params.p1 + drag.dy
label:
text: p_1
location: 0.05
- Segment:
a:
- calcs.bottomGraphLeft
- 2
b:
- calcs.bottomGraphRight
- 2
color: colors.budget
lineStyle: dotted
show: params.showPOC
- Segment:
a:
- calcs.bottomGraphLeft
- 4
b:
- calcs.bottomGraphRight
- 4
color: colors.budget
lineStyle: dotted
show: params.showPOC
- Segment:
a:
- calcs.bottomGraphLeft
- 6
b:
- calcs.bottomGraphRight
- 6
color: colors.budget
lineStyle: dotted
show: params.showPOC
- Segment:
a:
- calcs.bottomGraphLeft
- 8
b:
- calcs.bottomGraphRight
- 8
color: colors.budget
lineStyle: dotted
show: params.showPOC
- Point:
coordinates:
- calcs.optimum.x - params.w1
- params.p1
label:
text: d_1(p_1|p_2)
color: colors.demand
droplines:
vertical: (calcs.optimum.x - params.w1).toFixed(0)
drag:
- vertical: p1
show: (params.p1 < calcs.cutoffPrice)
- Point:
coordinates:
- calcs.optimum.x - params.w1
- params.p1
label:
text: s_1(p_1|p_2)
position: tr
color: colors.supply
droplines:
vertical: (calcs.optimum.x - params.w1).toFixed(0)
drag:
- vertical: p1
show: (params.p1 > calcs.cutoffPrice)
- Segment:
a:
- 0
- 1
b:
- calcs.optimum.x-params.w1
- 1
endArrow: true
color: colors.supply
label:
text: "`\\\\text{...so you want to sell ${(params.w1-calcs.optimum.x).toFixed(0)}}`"
location: 0.5
yPixelOffset: -15
show: (params.showExplanation && (calcs.optimum.x < params.w1))
- Segment:
a:
- 0
- 1
b:
- calcs.optimum.x-params.w1
- 1
endArrow: true
color: colors.demand
label:
text: "`\\\\text{...so you want to buy ${(calcs.optimum.x-params.w1).toFixed(0)}
more}`"
location: 0.5
yPixelOffset: -15
show: (params.showExplanation && (calcs.optimum.x > params.w1))
sidebar:
controls:
- title: Utility Function
sliders:
- param: a
label: "\\alpha"
divs:
- html: "`$$u(x_1,x_2) = x_1^\\\\alpha x_2^{1 - \\\\alpha} = x_1^{${params.a.toFixed(2)}}x_2^{${(1-params.a).toFixed(2)}}$$`"
- title: Budget Parameters
sliders:
- param: p1
label: p_1
- param: p2
label: p_2
- title: Options
checkboxes:
- param: showPOC
label: "\\text{Show offer curve and bundles for }p_1 = 2,4,6,8"
- param: showIndW
label: "\\text{Show indifference curve through $W$}"
- param: showExplanation
label: "\\text{Show explanation}"
divs:
- html: "`
You start with an endowment of $\\\\color{${colors.endowment}}{\\\\omega_1
= ${params.w1.toFixed(0)}}$ units of good 1 and $\\\\color{${colors.endowment}}{\\\\omega_2
= ${params.w2.toFixed(0)}}$ of good 2. With $p_1 = ${params.p1.toFixed(2)}$
and $p_2 = ${params.p2.toFixed(2)}$, this has a monetary value of $$\\\\color{${colors.budget}}{m
= p_1\\\\omega_1 + p_2\\\\omega_2 \\\\approx ${calcs.wealth.toFixed(0)}}$$`"
show: params.showExplanation
- html: "`With this Cobb-Douglas utility function, your gross demand
for good 1 is $$\\\\color{${colors.utility}}{x_1^* = \\\\frac{\\\\alpha
m}{p_1} \\\\approx \\\\frac{${params.a.toFixed(2)} \\\\times ${calcs.wealth.toFixed(0)}}{${params.p1.toFixed(2)}}
\ \\\\approx ${calcs.optimum.x.toFixed(0)}}$$`"
show: params.showExplanation
- html: "`Since ${calcs.optimum.x.toFixed(0)} is more than your endowment
of ${params.w1.toFixed(0)}, you want to buy the difference: that is, your
net demand is $$\\\\color{${colors.demand}}{d_1 = x_1^* - \\\\omega_1
\\\\approx ${calcs.optimum.x.toFixed(0)} - ${params.w1.toFixed(0)} \\\\approx
${(calcs.optimum.x - params.w1).toFixed(0)}}$$`"
show: (params.showExplanation && (params.p1 < calcs.cutoffPrice))
- html: "`Since ${calcs.optimum.x.toFixed(0)} is less than your endowment
of ${params.w1.toFixed(0)}, you want to sell the difference: that is,
your net supply is $$\\\\color{${colors.supply}}{s_1 = \\\\omega_1
- x_1^* \\\\approx ${params.w1.toFixed(0)} - ${calcs.optimum.x.toFixed(0)}
\\\\approx ${(params.w1 - calcs.optimum.x).toFixed(0)}}$$(That is, your
net demand is $ ${(calcs.optimum.x - params.w1).toFixed(0)}$.)`"
show: (params.showExplanation && (params.p1 > calcs.cutoffPrice))
Woohoo!