\begin{document} \begin{titlepage} \begin{center} \vspace*{2cm} \textbf{\huge{~~~Modeling Project I}} \vspace*{0.5cm} \newline \nlindent \textbf{\huge{$~~~~~~~$Gas or Pass}} \large{\textit{{{A first exploration into modeling utility functions}}}} \vspace*{\fill} \noindent \large{Alex Karbowski Arjun Nageswaran Ryan Xia Michael Zhao} \end{center} \end{titlepage} \section{The Function} The goal of this section is to create a model for the effect gas prices on utility. \subsection{Setting up Problem} First let us define our variables: \begin{itemize} \item Let $\bar{g}$ be an exogenous constant (in short term) representing amount used required transportation $g$ other purposes $c$ consumption goods $w$ income which we will also consider exogenous. \end{itemize} \smallskip Thus household trying find: $$\max_{g c}U(g c \bar{g})$$ subject budget constraint: $$pg + p \bar{g}+ c \leq w.$$ But know that since g and both increase utility optimal at edge constraint. Thus have: $max_{g \bar{g})$ s.t. $pg = w.$ Giving Lagrangian: $$\mathcal{L}(g \lambda) U(g c) \lambda(w - pg \bar{g} c)$$ And First Order Conditions: $$\frac{d L}{d g} U_g(g \bar{g}) -\lambda 0$$ c} U_c(g $$w 0.$$ \subsection{A Functional Form} To continue further must assume functional form $U(g These are assumptions satisfy: can defined only in terms choice variables $c g$. This forced consume $\bar{g}$ so it follows derive from consumption. We derived has no cross effects with purely spent driving work. there monotonic transformation $U(c g $ g)$ staying solely $\frac{dU}{dc} \frac{dU}{dg} > 0$. Consuming more increases your $\frac{d^2U}{dg^2} \frac{d^U}{dc^2} < There decreasing marginal returns all goods. In general Hessian matrix negative definite such function globally concave interior solution. base you (e.g. necessary food housing) (for grocery store). $\frac{dU}{dc}|_{c=0} \infty$ $\frac{dU}{dg}|_{g=0} \infty$. \bigskip A Cobb Douglas fits these assumptions. believe best use scenario because property consumers always spend fixed proportion their free each product. It reasonable real world fun trips scales proportionally income. those higher incomes tend travel (and different methods) than lower incomes. form: $$\boxed{U(g c^\alpha g^\beta}$$ Which apply use: \alpha\ln(c) +\beta\ln(g)}$$ Where $0 \alpha \beta 1$. logarithm Cobb-Douglass Utility Function. properties Douglass restricted domain \neq simplify calculations. plug second Conditions find system $$\frac{dL}{dg} \frac{\beta}{g} \lambda $$\frac{dL}{dc} \frac{\alpha}{c} p\bar{g}.$$ Solving obtain: $\lambda \frac{\alpha}{w p\bar{g}}$ $$g^{*} \frac{\beta (w \bar{g})}{p(\alpha \beta)}$$ $$c^{*} \frac{\alpha(w-p\bar{g})}{(\alpha solutions numeraire. Demand Function Gas} \subsection{Total total demand equivalent value $g^{*} \bar{g}$ equals $$\frac{\beta \beta)} w \alpha p\bar{g}}{p(\alpha \subsection{Share Income Spent share income S $$S p\bar{g}}{wp(\alpha \subsection{Change With Wages?} whether $S$ decreases calculate: $$\frac{\partial S }{\partial w} \frac{\alpha \bar{g}}{w^2(\alpha $$ From sign partial derivative get as wages go up decreases result poor families gas. \subsection{If $\bar{g} 0$} If 0$ would $\frac{\partial tells if people trips stay (This Function). But when $p\bar{g}$ matter richer individuals smaller trips. Adding two together have same meaning they overall. \subsection{Price Gas Increases} what happens numeraire increases (g^{*} +\bar{g})}{\partial p} \frac{\partial g^{*}}{\partial \frac{-\beta w}{p^2} c^{*}}{\partial -\frac{\alpha \bar{g}}{\alpha \beta}$$ decrease price increases. makes sense $p$ everyone now less disposable pay everything decreases. elasticity $g^*$ $$e_{(g \bar{g})} \frac{d(g^*+\bar{g})}{dp} * \frac{p}{g^* \bar{g}} $$\frac{-\beta \frac{p^2(\alpha \beta)}{\beta p\bar{g}} w(\alpha previously thought. \subsection{Elasticity Demand} how changes depending e_{(g \bar{g})}}{\partial \frac{- \beta^2 a^2 }{(\beta p\bar{g})^2}$$ derivative respect wage negative result goes poorer people. interpretation sensitive adjust people intuitive sense. \subsection{Indirect Plugging $c^*$ indirect function: $$U^*(pw) \alpha\ln\left(\frac{\alpha(w-p\bar{g})}{(\alpha \beta)}\right) +\beta\ln\left(\frac{\beta re-express Cobb-Douglas by conducting raising entire exponent e obtaining: e^{\alpha\ln(\frac{\alpha(w-p\bar{g})}{(\alpha \beta)}) +\beta\ln(\frac{\beta \beta)})} $$\left( p\bar{g})}{\alpha \beta} \right)^\alpha \left( p\bar{g})}{p(\alpha \right)^\beta$$ \subsection{Elasticity} function calculate p. U^*}{\partial \left(\frac{-p\alpha\bar{g} w}{p(w p\bar{g})}\right) \right)^\alpha\left( $w-p\bar{g} rise should decrease. Now as: $$e_{(U^*)} \frac{p}{U^*} \frac{-p\alpha\bar{g} w}{w p\bar{g}}$$ strictly negative expected. \subsection{Regressive} order reduces opposed rich consumers take wage. e_{(U^*)}}{\partial p\alpha\bar{g}}{(w-p\bar{g})^2}$$ positive. Meanwhile itself negative. increase (gets closer 0). \medskip represents percentage change prices. Our findings above suggest experience greater magnitude individuals. lose larger rise. Since increasing thus becomes individuals rather making regressive. \section{Taxes Subsidies} \subsection{Subsidies} Recall Marshallian g^* \frac{\beta(w p\overline{g})}{p(\alpha \beta)}. At initial $p_0$ g^*(p_0) \frac{\beta(w_0 p_0\overline{g})}{p_0(\alpha government subsidize cost $p_1 p_0$ unit consumed \boxed{C_s(p_1) (p_1 p_0)\left(\frac{\beta(w_0 \overline{g}\right)}. \subsection{Checks} 2.7 \left(\frac{\alpha(w-p\bar{g})}{(\alpha \beta)}\right)^\alpha\left(\frac{\beta \beta)}\right)^\beta seek $w_1$ $U^*(p_0w_0) U^*(p_1w_1)$. Here equalized w_1 \left(\frac{p_1}{p_0}\right)^\frac{\beta}{\alpha (w_0 p_0 \overline{g}) p_1\overline{g}. $w_0 p_1g$ \subsection{Cost Checks} 3.2 keep constant then consumer constant. Then doing $w_1-w_0$. As $p_1$ express \boxed{C_c(p_1) p_1\overline{g})}. \subsection{Difference Costs} $\Delta(p_1)$ difference costs between subsidy check. So \begin{aligned} \Delta(p_1) &= C_s(p_1) C_c(p_1) \\ \overline{g}\right) p_1\overline{g}) \end{aligned} \subsection{Determining Magnitudes wish determine $\Delta(p_1)$. $\Delta$ ambiguous compute its see decreasing. When p_0$ \Delta(p_0) (p_0 p_0\overline{g})}{p_1(\alpha \left(\frac{p_0}{p_0}\right)^\frac{\beta}{\alpha p_0\overline{g}) 0. \Delta(p_1)}{\partial p_1} \frac{\beta}{p_0(\alpha \beta)}\left(1 \left(\frac{p_1}{p_0}\right)^{\frac{\beta}{\alpha 1}\right)(w_0 mean theorem some $c \in p_0 p_1$ \frac{\partial\Delta(c)}{\partial p_1}(p_1 p_0) 0 $\frac{\partial\Delta(c)}{\partial 0$ any $c$ Therefore positive