Demand_functions_from_ a_ utility_function

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Paul Verheyen

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Paul Verheyen 5 years ago

Created this as a copy of SG_Vraagfuncties_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

Consider the 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.
Find the corresponding demand functions.

Remark: an expression such as $p_1$ must be written down using an underscore in the form p_1 !

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			Name	Type	Generated Value

n1	integer	6
theta	integer	9
c1	integer	3
s	integer	10
gegp2	integer	9
t1	integer	1
v	integer	91
eta	integer	135
n1min1	integer	5
a1	integer	54
a2	integer	8100
t2	integer	5
n2	integer	6
n1min1maala1	integer	270
a1maalt2	integer	270

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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The demand function for product 1 is given by $q_1=$ Unnamed gap

Advice

Step 1: Formulate the corresponding maximum problem using the budget restriction.
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 calulate 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:

$\begin{eqnarray} {p_1} \cdot {q_1} + {\var{a2}} \cdot {p_2} + {\var{n1min1}} \cdot {p_1}\cdot {(q_1-{\var{a1}})} & = & y\\ {p_1} \cdot {q_1} + {\var{a2}} \cdot {p_2} + {\var{n1min1}} \cdot {p_1} \cdot{q_1} - {\var{n1min1}} \cdot {\var{a1}} \cdot {p_1} & = & y\\ {\var{n1}} \cdot {p_1}\cdot {q_1} & = & y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2} \\ {q_1} & = & {\frac {1}{\var{n1} \cdot {p_1}}} \cdot { ( y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2} ) } \\ {q_1} & = & {\frac {y + {\var{n1min1maala1}} \cdot {p_1} - {\var{a2}} \cdot {p_2}}{\var{n1} \cdot {p_1}}} \end{eqnarray}$
Substituting this last formula for

$q_1$ in the equation (4) leads to a formula for

$q_2$ :

$\begin{eqnarray} {q_2} & = & \frac{\var{t2}\cdot y-\var{a1maalt2} \cdot {p_1} + \var{a2}\cdot{p_2}}{\var{n1} \cdot {p_2}} \end{eqnarray}$

Answer.
The demand functions are

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

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Paul Verheyen 5 years ago

Paul Verheyen 5 years ago

Paul Verheyen 5 years ago

Consider the 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.
Find the corresponding demand functions.

Remark: an expression such as $p_1$ must be written down using an underscore in the form p_1 !

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

The demand function for product 1 is given by $q_1=$ Unnamed gap

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Demand_functions_from_ a_ utility_function

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Comment

Checkpoint description

Paul Verheyen 5 years ago

Paul Verheyen 5 years ago

Paul Verheyen 5 years ago

Remark: an expression such as p1p_1 must be written down using an underscore in the form p_1 !

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

The demand function for product 1 is given by q1=q_1= Unnamed gap

Remark: an expression such as $p_1$ must be written down using an underscore in the form p_1 !

Generated value: `integer`

The demand function for product 1 is given by $q_1=$ Unnamed gap