Budget minimization for a given Stone-Geary utility function

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Created this as a copy of Budget minimization for a given Stone-Geary utility function old.

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.
The unit prices for this products are respectively $p_1 = \var{p1}$ and $p_2 = \var{p2}$ .
Minimize the budget which is completely spent by this individual on these products when the utility level equals {u}.

Give any introductory information the student needs.

			Name	Type	Generated Value

n1	integer	3
v	integer	4
c1	integer	3
u	integer	12
a1	integer	6
a2	integer	6
t1	integer	1
t2	integer	2
w	integer	4
q1	integer	10
q2	integer	10
gegp1	integer	3
ybud	integer	90
n1min1	integer	2
n2	integer	3
p1	integer	3
p2	integer	6
nutfinaal	number	12
ybmin	rational	45
ybmax	rational	135
cu	decimal	dec("8")
ydeelp2	rational	15
ydeelp1	rational	30
s	rational	1/2
ymindeelp2	rational	15/2
ymaxdeelp2	rational	45/2
ymindeelp1	rational	15
ymaxdeelp1	rational	45
yb1	rational	30
yb2	rational	15

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Description

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Generated value: `integer`

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cu
n1min1
n2
nutfinaal
p2
s
t2

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Step 1: Formulate the corresponding minimum problem using the utility restriction.
\begin{eqnarray}
\min_{q_1,q_2} \var{p1} \cdot q_1 + \var{p2} \cdot q_2 \\
\mbox{ if } \quad U(q_1,q_2) &=& {\var{u}} \\
\mbox{i.e. if } \quad {\var{u}}&=& {\var{c1}} \cdot (q_1-{\var{a1}})^{1 / {\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}} / {\var{n1}}}
\end{eqnarray}
which is equivaltent with
\begin{eqnarray}
\min_{q_1,q_2} \var{p1} \cdot q_1 + \var{p2} \cdot q_2 \\
\mbox{ if } \quad \ln \, U(q_1,q_2) &=& \ln {\var{u}}\\
\mbox{i.e. if } \quad {\ln {\var{u}}} &=& \ln {\var{c1}} + {\frac{1}{\var{n1}}} \cdot \ln {(q_1-{\var{a1}})} + {\frac{\var{n1min1}}{\var{n1}}} \cdot \ln {(q_2-{\var{a2}})}
\end{eqnarray}

Step 2: Write down the corresponding Lagrange function for this constrained minimum problem.
\begin{eqnarray}
L(q_1, q_2, \lambda) &=& \var{p1} \cdot q_1 + \var{p2} \cdot q_2 + \lambda \cdot \left( {\ln {\var{u}}} - \ln U(q_1,q_2)\right) \\
&=& \var{p1} \cdot q_1 + \var{p2} \cdot q_2 + \lambda \cdot \left( {\ln {\var{u}}} - \ln {\var{c1}} - {\frac{1}{\var{n1}}} \cdot \ln {(q_1-{\var{a1}})} - {\frac{\var{n1min1}}{\var{n1}}} \cdot \ln {(q_2-{\var{a2}})} \right)
\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} & = & \var{p1} - \lambda \cdot {\frac{1}{\var{n1}}} \cdot {\frac {1} {q_1-{\var{a1}}} } = 0 \\
\mbox{(2) } \quad \frac{\partial {L}}{\partial q_2} & = & \var{p2} - \lambda \cdot {\frac{\var{n1min1}}{\var{n1}}} \cdot {\frac {1} {q_2-{\var{a2}}}} = 0 \\
\mbox{(3) } \quad \frac{\partial {L}}{\partial \lambda} & = & {\ln {\var{u}}} - \ln {\var{c1}} - {\frac{1}{\var{n1}}} \cdot \ln {(q_1-{\var{a1}})} - {\frac{\var{n1min1}}{\var{n1}}} \cdot \ln {(q_2-{\var{a2}})} = 0
\end{eqnarray}

Step 4: Find the solution of this system of equations in $q_1$, $q_2$ and $\lambda$.
Solving $\lambda$ from the equations (1) and (2) leads to
\begin{eqnarray}
\frac{\var{p1}}{{\frac{1}{\var{n1}}} \cdot {\frac {1} {q_1-{\var{a1}}} } } & = & \frac{\var{p2}}{{\frac{\var{n1min1}}{\var{n1}}} \cdot {\frac {1} {q_2-\var{a2}}}} \\
\var{p1} \cdot \var{n1min1} \cdot {(q_1-{\var{a1}})} &=& \var{p2} \cdot {(q_2-{\var{a2}})} \\
\mbox{(4)} \quad \quad \quad \quad \quad \quad \quad \quad \quad q_2-\var{a2} &=& q_1-\var{a1}
\end{eqnarray}
i.e.
\[ \simplify{q_2 = q_1 + {a2-a1}} \]
Since (3) is equivalent to
\[ {\var{u}} = {\var{c1}} \cdot (q_1-{\var{a1}})^{1 / {\var{n1}}} \cdot (q_2-{\var{a2}})^{{\var{n1min1}} / {\var{n1}}} \]
we can substitute equation (4) in the given utility restricion:
\begin{eqnarray}
\var{u} &=& \var{c1} \cdot (q_1-\var{a1}) \\
q^*_1 &=& \var{a1+v}
\end{eqnarray}
Substitution of this $q_1-$ value in (4) gives us the optimal value of $q_2$ :
\begin{eqnarray}
{q^*_2} & = & \var{q2}
\end{eqnarray}

Answer.

For
\begin{eqnarray}
q^*_1&=&\var{q1} \\
q^*_2&=&\var{q2}
\end{eqnarray}
we find the minimal value of the budget:
\[ \var{p1} \cdot q^*_1 + \var{p2} \cdot q^*_2 = \var{ybud}. \]

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Budget minimization for a given Stone-Geary utility function

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Paul Verheyen 4 years, 7 months ago

Paul Verheyen 4 years, 7 months ago

Paul Verheyen 4 years, 7 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.
The unit prices for this products are respectively $p_1 = \var{p1}$ and $p_2 = \var{p2}$ .
Minimize the budget which is completely spent by this individual on these products when the utility level equals {u}.

Error in variable testing condition

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Generated value: `integer`

Parts

Number entry

Calculate the optimal value of $q_1$:

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Budget minimization for a given Stone-Geary utility function

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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History

Comment

Checkpoint description

Paul Verheyen 4 years, 7 months ago

Paul Verheyen 4 years, 7 months ago

Paul Verheyen 4 years, 7 months ago

An individual has as utility functionU(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}}}

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Number entry

Calculate the optimal value of $q_1$:

Advanced settings

An individual has as utility function
$U(q_1,q_2)={\var{c1}} \cdot(q_1-{\var{a1}})^{{1}/{\var{n1}}} \cdot (q_2-{\var{a2}})^ {{\var{n1min1}}/{\var{n1}}}$

Generated value: `integer`