Stone-Geary utility and calculation of elasticities

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Paul Verheyen

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Created this as a copy of SG_Elasticiteiten_2020.

Name	Status	Author	Last Modified
Demand_functions_from_ a_ utility_function	Ready to use	Paul Verheyen	15/04/2020 13:08
Stone-Geary utility and calculation of elasticities	Ready to use	Paul Verheyen	15/04/2020 11:36
Stone-Geary utility maximization for a given budget	Ready to use	Paul Verheyen	10/10/2024 08:33
Budget minimization for a given Stone-Geary utility function	Ready to use	Paul Verheyen	10/10/2024 08:35

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Question statement

An individual has as utility function

$U(q_1,q_2)={\var{c1}} \cdot (q_1-{\var{a1}})^{{1}/{\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}}/{\var{n1}}}$

where $q_1$ and $q_2$ denote the quantities consumed of respectively the products 1 and 2.

Let $P^*$ be the point $(p_1 , p_2 , y) = ( \var{gegp1} , \var{gegp2} , \var{gegy} ) .$

Give any introductory information the student needs.

			Name	Type	Generated Value

n1	integer	2
theta	integer	6
c1	integer	3
s	integer	6
gegp2	integer	10
t1	integer	1
v	integer	37
eta	integer	42
n1min1	integer	1
a1	integer	12
a2	integer	504
t2	integer	1
n2	integer	2
n1min1maala1	integer	12
a1maalt2	integer	12
gegp1	integer	60
gegy	integer	10200
q1	integer	49
q2	integer	726
epsq1p1	rational	-43/49
epsq1p2	rational	-6/7
epsq2p1	rational	-6/121
epsq2p2	rational	-79/121
epsq1y	rational	85/49
epsq2y	rational	85/121
benepsq1p1	decimal	dec("-0.878")
benepsq2p2	decimal	dec("-0.653")
benepsq1y	decimal	dec("1.735")
benepsq2y	decimal	dec("0.702")
benepsq1p2	decimal	dec("-0.857")
benepsq2p1	decimal	dec("-0.050")

Automatically regenerate variables?

Name

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Value

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

→ Used by:

a1
a2
epsq1p1
epsq1p2
epsq1y
epsq2p1
epsq2p2
epsq2y
eta
gegy
n1min1
n2
q2
t2

Statement
Advice

Advice

Step 1: Formulate the maximum problem in order to find the demand functions.
Let $p_1$ and $p_2$ be the prices for respectively the products 1 and 2, and let $y$ be the available budget (suppose you spend this amount $y$ completely on these products). Then the problem can be stated as

$\begin{eqnarray} \max_{q_1,q_2} U(q_1,q_2) &=& {\var{c1}} \cdot (q_1-{\var{a1}})^{1 / {\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}} / {\var{n1}}} \mbox{ if } && p_1 \cdot q_1 + p_2 \cdot q_2 = y \end{eqnarray}$
or equivalently

$\begin{eqnarray} \max_{q_1,q_2} ln U(q_1,q_2) &=& ln({\var{c1}}) + {\frac{1}{\var{n1}}} \cdot ln(q_1-{\var{a1}}) + {\frac{\var{n1min1}}{\var{n1}}} \cdot ln(q_2-{\var{a2}}) \\ \mbox{ if } && p_1 \cdot q_1 + p_2\cdot q_2 = y \end{eqnarray}$

Step 2: Write down the corresponding Lagrange function for this constrained maximum problem.

$\begin{eqnarray} L(q_1, q_2, \lambda) &=& ln U(q_1,q_2) + \lambda \cdot (y - p_1 \cdot q_1 - p_2 \cdot q_2) &=& ln({\var{c1}}) + {\frac{1}{\var{n1}}} \cdot ln(q_1-{\var{a1}}) + {\frac{\var{n1min1}}{\var{n1}}} \cdot ln(q_2-{\var{a2}}) + \lambda \cdot ( y - p_1 \cdot q_1 - p_2\cdot q_2 ) \end{eqnarray}$

Step 3: Determine the first order conditions in order to calculate critical points of the Lagrangian.

$\begin{eqnarray} \mbox{(1) } \quad \frac{\partial {L}}{\partial q_1} & = & {\frac{1}{\var{n1}}} \cdot {\frac {1} {q_1-{\var{a1}}} } - \lambda \cdot p_1 = 0 \\ \mbox{(2) } \quad \frac{\partial {L}}{\partial q_2} & = & {\frac{\var{n1min1}}{\var{n1}}} \cdot {\frac {1} {q_2-{\var{a2}}}} - \lambda \cdot p_2 = 0 \\ \mbox{(3) } \quad \frac{\partial {L}}{\partial \lambda} & = & y - p_1 \cdot q_1 - p_2 \cdot q_2 = 0 \end{eqnarray}$

Step 4: Find the solution of this system of equations in $q_1$ , $q_2$ and $\lambda$ .
Calculating $\lambda$ in both equations (1) and (2) leads to

$\begin{eqnarray} {\frac{1}{\var{n1}}}\cdot{\frac {1} {p_1 \cdot(q_1-{\var{a1}})} }& = & {\frac{\var{n1min1}}{\var{n1}}}\cdot{\frac {1} {p_2 \cdot(q_2-{\var{a2}})}}\\ \\ {p_2} \cdot {(q_2-{\var{a2}})} & = & {\var{n1min1}} \cdot {p_1} \cdot {(q_1-{\var{a1}})} \\ \\ \mbox{(4)} \quad \quad \quad {p_2} \cdot {q_2} & = & {\var{a2}} \cdot {p_2} + {\var{n1min1}} \cdot {p_1} \cdot {(q_1-{\var{a1}})} \end{eqnarray}$
i.e.

$\begin{eqnarray} {q_2} & = & {\var{a2}} + {\frac {1} {p_2}} \cdot {\var{n1min1}} \cdot {p_1} \cdot {(q_1-{\var{a1}})} \end{eqnarray}$
Substitute equation (4) in the budget restriction (3):

$\begin{eqnarray} {p_1} \cdot {q_1} + {\var{a2}} \cdot {p_2} + {\var{n1min1}} \cdot {p_1}\cdot {(q_1-{\var{a1}})} & = & y\\ {p_1} \cdot {q_1} + {\var{a2}} \cdot {p_2} + {\var{n1min1}} \cdot {p_1} \cdot{q_1} - {\var{n1min1}} \cdot {\var{a1}} \cdot {p_1} & = & y\\ {\var{n1}} \cdot {p_1}\cdot {q_1} & = & y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2} \\ {q_1} & = & {\frac {1}{\var{n1} \cdot {p_1}}} \cdot { ( y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2} ) } \\ {q_1} & = & {\frac {y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2}}{\var{n1} \cdot {p_1}}} \end{eqnarray}$
Substituting this last formula in equation (4), we can calculate

$q_2$ :

$\begin{eqnarray} {q_2} & = & \frac{\var{t2}\cdot y-\var{a1maalt2} \cdot {p_1} + \var{a2}\cdot{p_2}}{\var{n1} \cdot {p_2}} \end{eqnarray}$
The demand functions for the products are given by

$\begin{eqnarray} {q_1=q_1 \left( p_1,p_2,y \right) } & = & {\frac {y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2}}{\var{n1} \cdot {p_1}}}\\ {q_2=q_2 \left( p_1,p_2,y \right) } & = & \frac{\var{t2}\cdot y-\var{a1maalt2}\cdot{p_1} + \var{a2}\cdot {p_2}}{\var{n1} \cdot {p_2}} \end{eqnarray}$

Step 5: Calculate the elasticities of demand.

$\begin{eqnarray} {\epsilon}^{q_1}_{p_1} & = & {\frac {{\partial}{q_1}} {{\partial} {p_1}}} \cdot {\frac {p_1} {q_1}} = {\frac {1} {{\var{n1}} }} \cdot {\frac {{\partial}} {{\partial} {p_1}}} {\left( {\var{n1min1maala1}} + {\frac {y - {\var{a2}}\cdot {p_2}} {p_1}} \right) } \cdot {\frac {p_1} {q_1}} = - {\frac {y - {\var{a2}} \cdot {p_2}} {{\var{n1}} \cdot {p_1} \cdot {q_1} } } \quad, {\mbox{ evaluated in }}P^*: \var{epsq1p1} \\ {\epsilon}^{q_1}_{p_2} & = & {\frac {{\partial}{q_1}} {{\partial} {p_2}}} \cdot {\frac {p_2} {q_1}} = - {\frac {\var{a2}} {{\var{n1}} \cdot {p_1}} } \cdot {\frac {p_2} {q_1}} \quad, {\ } \mbox{ evaluated in } P^*: \var{epsq1p2} \\ {\epsilon}^{q_2}_{p_1} & = & {\frac {{\partial}{q_2}} {{\partial} {p_1}}} \cdot {\frac {p_1} {q_2}} = - {\frac {\var{n1min1maala1}} {{\var{n1}} \cdot {p_2}} } \cdot {\frac {p_1} {q_2}} \quad , \mbox{ evaluated in }P^*: \var{epsq2p1} \\ {\epsilon}^{q_2}_{p_2} & = & {\frac {{\partial}{q_2}} {{\partial} {p_2}}} \cdot {\frac {p_2} {q_2}} = {\frac {1} {{\var{n1}} }} \cdot {\frac {{\partial}} {{\partial} {p_2}}} { \left( {\var{a2}} + {\frac {{\var{n1min1}} \cdot y - {\var{n1min1maala1}} \cdot {p_1}} {p_2}} \right) } \cdot {\frac {p_2} {q_2}} = - {\frac {{\var{n1min1}} \cdot y - {\var{n1min1maala1}} \cdot {p_1}} {{\var{n1}} \cdot {p_2} \cdot {q_2} } } \quad , \mbox{ evaluated in } P^*: \var{epsq2p2} \\ {\epsilon}^{q_1}_{y} & = & {\frac {{\partial}{q_1}} {{\partial} {y}}} \cdot {\frac {y} {q_1}} = {\frac {1} {{\var{n1}} \cdot {p_1}} } \cdot {\frac {y} {q_1}} , \mbox{ evaluated in }P^*: \var{epsq1y}\\ {\epsilon}^{q_2}_{y} & = & {\frac {{\partial}{q_2}} {{\partial} {y}}} \cdot {\frac {y} {q_2}} = {\frac {1} {{\var{n1}}}} \cdot {\frac {1} {{\var{n1min1}} \cdot {p_2}} } \cdot {\frac {y} {q_2}} \quad , \mbox{ evaluated in } P^*: \var{epsq2y}\\ \end{eqnarray}$

Answers:

${\epsilon}^{q_1}_{p_1} {( \var{gegp1} , \var{gegp2} , \var{gegy} )} \approx \var{benepsq1p1}$

${\epsilon}^{q_2}_{p_2} {( \var{gegp1} , \var{gegp2} , \var{gegy} )} \approx \var{benepsq2p2}$

${\epsilon}^{q_1}_{y} {( \var{gegp1} , \var{gegp2} , \var{gegy} )} \approx \var{benepsq1y}$

${\epsilon}^{q_2}_{y} {( \var{gegp1} , \var{gegp2} , \var{gegy} )} \approx \var{benepsq2y}$

${\epsilon}^{q_1}_{p_2} {( \var{gegp1} , \var{gegp2} , \var{gegy} )} \approx \var{benepsq1p2}$

${\epsilon}^{q_2}_{p_1} {( \var{gegp1} , \var{gegp2} , \var{gegy} )} \approx \var{benepsq2p1}$

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Stone-Geary utility and calculation of elasticities

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Paul Verheyen 4 years, 11 months ago

Paul Verheyen 4 years, 11 months ago

Paul Verheyen 4 years, 11 months ago

An individual has as utility function

$U(q_1,q_2)={\var{c1}} \cdot (q_1-{\var{a1}})^{{1}/{\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}}/{\var{n1}}}$

where $q_1$ and $q_2$ denote the quantities consumed of respectively the products 1 and 2.

Let $P^*$ be the point $(p_1 , p_2 , y) = ( \var{gegp1} , \var{gegp2} , \var{gegy} ) .$

Error in variable testing condition

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Generated value: `integer`

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Calculate the own price elasticity of demand for product 1 in $P^*$ :

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Stone-Geary utility and calculation of elasticities

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Checkpoint description

Paul Verheyen 4 years, 11 months ago

Paul Verheyen 4 years, 11 months ago

Paul Verheyen 4 years, 11 months ago

An individual has as utility function

U(q1,q2)={c1}⋅(q1−{a1})1/{n1}⋅(q2−{a2}){n1min1}/{n1} U(q_1,q_2)={\var{c1}} \cdot (q_1-{\var{a1}})^{{1}/{\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}}/{\var{n1}}}

where q1q_1 and q2q_2 denote the quantities consumed of respectively the products 1 and 2.

Let P∗P^* be the point (p1,p2,y)=({gegp1},{gegp2},{gegy}). (p_1 , p_2 , y) = ( \var{gegp1} , \var{gegp2} , \var{gegy} ) .

Error in variable testing condition

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Generated value: integer

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Calculate the own price elasticity of demand for product 1 in P∗P^*:

Advanced settings

$U(q_1,q_2)={\var{c1}} \cdot (q_1-{\var{a1}})^{{1}/{\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}}/{\var{n1}}}$

where $q_1$ and $q_2$ denote the quantities consumed of respectively the products 1 and 2.

Let $P^*$ be the point $(p_1 , p_2 , y) = ( \var{gegp1} , \var{gegp2} , \var{gegy} ) .$

Generated value: `integer`

Calculate the own price elasticity of demand for product 1 in $P^*$ :