Handout 2APEC5152xxxx.pdf - APEC 5152 Utility Indirect Utility and Expenditure Consumption Well use Cobb-Douglas preferences u = q11 q22 q33 where 1 2 3

Handout 2APEC5152xxxx.pdf - APEC 5152 Utility Indirect...

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APEC 5152 Utility, Indirect Utility and Expenditure Consumption We’ll use Cobb-Douglas preferences, u = q1 γ 1 q2 γ 2 q3 γ 3 , where γ 1 + γ 2 + γ 3 = 1. It turns out that using the Cobb-Douglas function doesn’t work with the Solve function. Taking a monotonic transforma- tion of the function - a log transformation - however, yields a solution. The Indirect Utility Function The lagrangian In[62]:= U = Log q1 γ 1 q2 γ 2 q3 γ 3 // PowerExpand L0 = U + λ  wv - p1 q1 - p2 q2 - p3 q3 Out[62]= γ 1 Log [ q1 ] + γ 2 Log [ q2 ] + γ 3 Log [ q3 ] Out[63]= - p1 q1 - p2 q2 - p3 q3 + wv  λ + γ 1 Log [ q1 ] + γ 2 Log [ q2 ] + γ 3 Log [ q3 ] The first-order conditions In[3]:= FOC10 = q1 L0 FOC20 = q2 L0 FOC30 = q3 L0 FOC λ 0 = λ L0 Out[3]= γ 1 q1 - p1 λ Out[4]= γ 2 q2 - p2 λ Out[5]= γ 3 q3 - p3 λ Out[6]= - p1 q1 - p2 q2 - p3 q3 + wv
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In[7]:= MarshSol = Solve [{ FOC10 0, FOC20 0, FOC30 0, FOC λ 0 0 } , { q1, q2, q3, λ }] / . γ 1 + γ 2 + γ 3 1 // ExpandAll // FullSimplify Out[7]=  q1 wv γ 1 p1 , q2 wv γ 2 p2 , q3 wv γ 3 p3 , λ → 1 wv  Substitute the above demand functions into the Cobb-Douglas utility function. In[9]:= V [ p1 _ , p2 _ , p3 _ , wv _] = Exp [ U ] / . MarshSol [[ 1 ]] // ExpandAll // FullSimplify Out[9]= wv γ 1 p1 γ 1 wv γ 2 p2 γ 2 wv γ 3 p3 γ 3 Further simplifying yields: V [ p1, p2, p3, wv ] = γ 1 p1 γ 1 γ 2 p2 γ 2 γ 3 p3 γ 3 wv Roy’s identity in action In[57]:= - p1 V [ p1, p2, p3, wv ] wv V [ p1, p2, p3, wv ] // ExpandAll // FullSimplify - p2 V [ p1, p2, p3, wv ] wv V [ p1, p2, p3, wv ] // ExpandAll // FullSimplify - p3 V [ p1, p2, p3, wv ] wv V [ p1, p2, p3, wv ] // ExpandAll // FullSimplify Out[57]= wv γ 1 p1 γ 1 + γ 2 + γ 3 Out[58]= wv γ 2 p2 γ 1 + γ 2 + γ 3 Out[59]= wv γ 3 p3 γ 1 + γ 2 + γ 3 Expenditure Function The optimization problem is L = p1 q1 + p2 q2 + p3 q3 + λ (u - q1 γ 1 q2 γ 2 q3 γ 3 ) In[15]:= L = p1 q1 + p2 q2 + p3 q3 + λ  u - q1 γ 1 q2 γ 2 q3 γ 3 ; In[16]:= foc1 = q1 L foc2 = q2 L foc3 = q3 L foc λ = λ L Out[16]= p1 - q1 - 1 + γ 1 q2 γ 2 q3 γ 3 γ 1 λ Out[17]= p2 - q1 γ 1 q2 - 1 + γ 2 q3 γ 3 γ 2 λ Out[18]= p3 - q1 γ 1 q2 γ 2 q3 - 1 + γ 3 γ 3 λ Out[19]= - q1 γ 1 q2 γ 2 q3 γ 3 + u The solution gives the system of demands 2 Consumption_170124.nb
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In[29]:= Solve [{ foc1 0, foc2 0, foc3 0, foc λ ⩵ 0 } , { q1, q2, q3, λ }][[ 1 ]] // ExpandAll // FullSimplify HickSol = % / . γ 1 + γ 2 + γ 3 1 Out[29]= q1 p1 - 1 + γ 1 γ 1 + γ 2 + γ 3 p2 γ 2 γ 1 + γ 2 + γ 3 p3 γ 3 γ 1 + γ 2 + γ 3 u 1 γ 1 + γ 2 + γ 3 γ 1 γ 2 + γ 3 γ 1 + γ 2 + γ 3 γ 2 - γ
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