We start from preferences what people want, and how to translate it into maths so that they can drive demand curves. This should be very similar to the Von Neumann-Morgestern notes. This is the utility function.

We have a different set of assumptions between different Morgestern things.

Axioms of Preferences

  1. Every consumer has a direct preference between the two so and things like that.
  2. Non satiation: this is a strong assumption, it works for money for example. More is better in this case.
  3. Transitivity.

We assume that preferences are given. But you can also start studying that too!

Indifference Curves

it's the rate we are willing to trade-off one good for another

I can exchange two slides of pizza and one cookie, or having two cookies, on my preference curve they are the same. This is a map to how to choose between two or more goods?.

  1. Consumers prefer higher indifference curves. This follows from the non-satiation argument.
  2. Indifference curves are downward sloping. It follows also from non-satiation. If it goes upward sloping then having more would be the same for me, so this is not possible.
  3. Indifference curves cannot cross, if so it would violate transitivity.
  4. Only one curve for one point. this si completeness.
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[>=stealth, line width=0.9pt, scale=1.5]

  % ---- axes ----
  \draw[->] (0,0) -- (0,3.4);
  \draw[->] (0,0) -- (4.3,0);
  \node[below] at (2.2,-0.75) {\textbf{Cookies}};
  \node[rotate=90] at (-1.0,1.7) {\textbf{Slices of Pizza}};
  \node[below left] at (0,0) {\textbf{0}};

  % ---- indifference curves: U = xy, at levels 2 and 4 ----
  \draw[domain=0.75:3.7, smooth, samples=120] plot (\x,{2/\x});
  \draw[domain=1.35:4.0, smooth, samples=120] plot (\x,{4/\x});

  % ---- ticks ----
  \foreach \y in {1,2} {
    \draw (-0.1,\y) -- (0.1,\y);
    \node[left] at (-0.18,\y) {\textbf{\y}};
  }
  \foreach \x in {1,2} {
    \draw (\x,-0.1) -- (\x,0.1);
    \node[below] at (\x,-0.18) {\textbf{\x}};
  }

  % ---- bundles ----
  \fill (1,2) circle (2.5pt) node[above left]  {\textbf{A}};
  \fill (2,1) circle (2.5pt) node[above right] {\textbf{B}};
  \fill (2,2) circle (2.5pt) node[above left]  {\textbf{C}};

\end{tikzpicture}
\end{document}

We can see here the other downward sloping case.

\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[x=1.6cm, y=1.1cm]

% ---- axes ----
\draw[line width=1pt] (0,0) -- (0,3.4);
\draw[line width=1pt] (0,0) -- (3.3,0);
\node[below left, font=\bfseries] at (0,0) {0};
\node[below, font=\bfseries] at (1.65,-0.75) {Cookies};
\node[rotate=90, font=\bfseries] at (-0.75,1.7) {Slices of Pizza};

% ---- ticks ----
\foreach \v in {1,2} {
  \draw[line width=1pt] (-0.14,\v) -- (0.14,\v);
  \node[left=8pt, font=\bfseries] at (0,\v) {\v};
  \draw[line width=1pt] (\v,-0.1) -- (\v,0.1);
  \node[below=4pt, font=\bfseries] at (\v,0) {\v};
}

% ---- flatter curve: y = 2 x^{-0.415}, through A=(1,2) and B=(2,1.5) ----
\draw[line width=0.9pt, smooth, samples=90, domain=0.4:2.45]
      plot (\x, {2*exp(-0.415*ln(\x))});

% ---- steeper curve: y = 2/x, through A=(1,2) and C=(2,1) ----
\draw[line width=0.9pt, smooth, samples=90, domain=0.75:2.9]
      plot (\x, {2/\x});

% ---- bundles ----
\fill (1,2)   circle (3pt) node[above left=1pt]  {\textbf{A}};
\fill (2,1.5) circle (3pt) node[above right=1pt] {\textbf{B}};
\fill (2,1)   circle (3pt) node[above right=1pt] {\textbf{C}};

\end{tikzpicture}
\end{document}

\begin{document}
\begin{tikzpicture}[x=1.6cm, y=1.1cm]

\draw[line width=1pt] (0,0) -- (0,3.4);
\draw[line width=1pt] (0,0) -- (3.1,0);

\node[anchor=north east] at (0,0) {\textbf{0}};
\node[anchor=north] at (1.6,-0.6) {\textbf{Cookies}};
\node[anchor=south, rotate=90] at (-0.7,1.7) {\textbf{Slices of Pizza}};
\node[anchor=south] at (1.6,3.6) {\textbf{Upward-sloping indifference curve}};

\draw[line width=1pt] (-0.12,1) -- (0.12,1);
\draw[line width=1pt] (-0.12,2) -- (0.12,2);
\node[anchor=east] at (-0.2,1) {\textbf{1}};
\node[anchor=east] at (-0.2,2) {\textbf{2}};
\draw[line width=1pt] (1,-0.1) -- (1,0.1);
\draw[line width=1pt] (2,-0.1) -- (2,0.1);
\node[anchor=north] at (1,-0.2) {\textbf{1}};
\node[anchor=north] at (2,-0.2) {\textbf{2}};

\draw[line width=0.9pt] plot[smooth] coordinates {
  (0.35,0.64) (0.6,0.76) (0.9,0.93) (1.2,1.15) (1.5,1.41)
  (1.8,1.74) (2.1,2.14) (2.3,2.46) (2.5,2.83) };

\fill (1,1) circle (2.5pt);
\fill (2,2) circle (2.5pt);
\node[anchor=north west] at (1.05,0.85) {\textbf{(1, 1)}};
\node[anchor=west] at (2.1,2) {\textbf{(2, 2)}};

\end{tikzpicture}
\end{document}

Utility Function

The utility creates the indifference curves above. For example one common form is the log utility function. e.g.: pizza and cookies. Such utility function is consistent with the indifference curve above. It's a infinite set. It means only something ordinally, in the sense of ranking choice. Not cardinally!

Diminishing Marginal Utilities

We also have diminishing marginal utilities. This means that as we continue to get more, having more, increments a little bit less that the thing before.

\begin{document}
\begin{tikzpicture}[x=2.2cm, y=1.9cm, line width=0.9pt]

  % ---- axes ----
  \draw[line width=1.4pt] (0,0) -- (0,3.6);
  \draw[line width=1.4pt] (0,0) -- (5.1,0);
  \node[below left, font=\bfseries] at (0,0) {0};
  \node[below, font=\bfseries] at (2.5,-0.42) {Cookies};
  \node[rotate=90, font=\bfseries] at (-0.42,1.8) {Utility};

  % ---- ticks ----
  \foreach \x in {1,2,3,4} {
    \draw[line width=1.2pt] (\x,0) -- (\x,0.09);
    \node[below=6pt] at (\x,0) {\x};
  }
  \foreach \y in {1,2,3} {
    \draw[line width=1.2pt] (0,\y) -- (0.13,\y);
    \node[left=6pt] at (0,\y) {\y};
  }

  % ---- U = sqrt(2C) ----
  \draw[domain=0:4.35, smooth, samples=60]
       plot (\x,{sqrt(2*\x)})
       node[right=14pt] {$U=\sqrt{2\cdot C}$};

  % ---- bundles ----
  \foreach \x/\lab in {1/{(1, 1.41)}, 2/{(2, 2)}, 3/{(3, 2.45)}, 4/{(4, 2.83)}} {
    \fill (\x,{sqrt(2*\x)}) circle (3.2pt);
    \node[above left=1pt, font=\bfseries] at (\x,{sqrt(2*\x)}) {\lab};
  }

  % ---- caption ----
  \node[align=left] at (1.55,3.15) {Holding slices of\\pizza constant at 2};

\end{tikzpicture}
\end{document}

Marginal Rate of Substitution

So if we have two goods, we have:

This also defines the indifference curve. And one base idea of the trading, since here we are trading with .

Marginal Utility

We can also have that the slope of the indifference curve is the negative slope of the ratio of the marginal utilities. We have a negative sign because MU is a negative of quantity, the more we get, the less marginal utility we get.

The concepts seem intuitive enough, probably.

\begin{document}
\begin{tikzpicture}[x=1.5cm, y=1.1cm, line width=0.9pt]

  % ---- axes ----
  \draw[line width=1.4pt] (0,0) -- (0,4.7);
  \draw[line width=1.4pt] (0,0) -- (5.6,0);
  \node[below left, font=\bfseries] at (0,0) {0};
  \node[below, font=\bfseries] at (2.8,-0.85) {Cookies};
  \node[rotate=90, align=center, font=\bfseries] at (-0.85,2.3)
       {Slices of\\Pizza};

  % ---- ticks ----
  \foreach \x in {1,2,3,4} {
    \draw[line width=1.2pt] (\x,0) -- (\x,0.12);
    \node[below=5pt] at (\x,0) {\x};
  }
  \foreach \y in {1,2,3,4} {
    \draw[line width=1.2pt] (0,\y) -- (0.16,\y);
    \node[left=5pt] at (0,\y) {\y};
  }

  % ---- indifference curve: P*C = 4 ----
  \draw[domain=0.9:5.3, smooth, samples=80] plot (\x,{4/\x});

  % ---- step A -> B ----
  \draw[dotted, line width=1pt] (1,4) -- (1,2) -- (2,2);
  \node[right=2pt] at (1,3) {-2};
  \node[above=2pt] at (1.5,2) {1};
  \node[below left=1pt] at (1,2) {$MRS=-2$};

  % ---- step B -> C ----
  \draw[dotted, line width=1pt] (2,2) -- (2,1) -- (4,1);
  \node[right=2pt] at (2,1.5) {-1};
  \node[above=2pt] at (3,1) {2};
  \node[below left=1pt] at (2,1) {$MRS=-1/2$};

  % ---- bundles ----
  \foreach \x/\y/\lab in {1/4/A, 2/2/B, 4/1/C} {
    \fill (\x,\y) circle (3.5pt);
    \node[above=4pt, font=\bfseries] at (\x,\y) {\lab};
    \node[right=6pt, font=\bfseries] at (\x,\y) {(\x,\y)};
  }

\end{tikzpicture}
\end{document}

Think about the case where McDonald's have not double the cost when you double the content for like drinks. This can be explained by this simple model, but can also be explained by other pricing strategies etc...

Concave Utilities

If it were concave, for example: Doesn't satisfy a common principle, which is when we have more of one resource, we are not likely to give up more of that. And when you have less of that you are willing to give up even more. The utility maths works, some cases might be possible, but doesn't make sense. The professor also doesn't believe economics can be a real strong science in the sense of physics.

\begin{document}
\begin{tikzpicture}[x=0.85cm, y=0.62cm, line width=0.9pt]

  % ---- axes ----
  \draw[line width=1.4pt] (0,0) -- (0,9.4);
  \draw[line width=1.4pt] (0,0) -- (9.6,0);
  \node[below left, font=\bfseries] at (0,0) {0};
  \node[below, font=\bfseries] at (4.8,-1.5) {Cookies};
  \node[rotate=90, align=center, font=\bfseries] at (-1.5,4.7)
       {Slices of\\Pizza};

  % ---- ticks ----
  \foreach \x in {1,4,7} {
    \draw[line width=1.2pt] (\x,0) -- (\x,0.22);
    \node[below=5pt, font=\bfseries] at (\x,0) {\x};
  }
  \foreach \y in {4,7,8} {
    \draw[line width=1.2pt] (0,\y) -- (0.28,\y);
    \node[left=5pt, font=\bfseries] at (0,\y) {\y};
  }

  % ---- indifference curve: C^2 + P^2 = 65 ----
  \draw[line width=1.2pt, domain=0.85:7.45, smooth, samples=80]
       plot (\x,{sqrt(65 - \x*\x)});

  % ---- step A -> B ----
  \draw[dotted, line width=1pt] (1,8) -- (1,7) -- (4,7);
  \node[right=2pt] at (1,7.5) {-1};
  \node[above=2pt] at (2.5,7) {3};
  \node[below left=1pt, font=\bfseries] at (1,7)
       {MRS$=\frac{-1}{3}$};

  % ---- step B -> C ----
  \draw[dotted, line width=1pt] (4,7) -- (4,4) -- (7,4);
  \node[right=2pt] at (4,5.5) {-3};
  \node[above=2pt] at (5.5,4) {3};
  \node[below left=1pt, font=\bfseries] at (4,4) {MRS$=-1$};

  % ---- bundles ----
  \foreach \x/\y/\lab in {1/8/A, 4/7/B, 7/4/C} {
    \fill (\x,\y) circle (3.5pt);
    \node[above left=2pt, font=\bfseries] at (\x,\y) {\lab};
    \node[right=6pt] at (\x,\y) {(\x,\y)};
  }

\end{tikzpicture}

\end{document}

Consumer Budgets

Every penny you have you spend it. This is another assumption that we have it here.

The total spending would then be and this is what is spend on this two goods market. In consumer theory the are kept constant and given.

\begin{document}
\begin{tikzpicture}[x=1.5cm, y=1.1cm, line width=0.9pt]

  % ---- axes ----
  \draw[line width=1.4pt] (0,0) -- (0,5.2);
  \draw[line width=1.4pt] (0,0) -- (7.4,0);
  \node[below left, font=\bfseries] at (0,0) {0};
  \node[below, font=\bfseries] at (3.7,-0.85) {Cookies (C)};
  \node[rotate=90, align=center, font=\bfseries] at (-1.1,2.6)
       {Slices of\\Pizza (S)};

  % ---- intercept labels ----
  \node[left=4pt] at (0,4.6) {$Y/p_s$};
  \node[below right=4pt] at (7,0) {$Y/p_c$};

  % ---- budget line ----
  \draw[line width=1.2pt] (0,4.6) -- (7,0);

  % ---- equation ----
  \node[align=left, font=\bfseries] at (2.1,4.35)
       {Budget line:};
  \node[align=left] at (2.7,3.9) {$Y = p_s S + p_c C$};

  % ---- slope annotation ----
  \draw (4.05,1.94) -- (4.55,2.45);
  \node[right=1pt] at (4.55,2.45) {Slope: $-p_c/p_s$};

\end{tikzpicture}
\end{document}

Marginal Rate of Transformation

This is the slope of the price part, how much we can transform the cookies into pizzas in terms of income usage. We transform one into the other using the market.

Weight watchers set a point, basically a budget, and saying that you cannot get above that budget of points for eating. This was one way of working that out for obesity in the US. The idea is this: make your self as well-off as possible subject to a constraint, in this sense of a budget.

Change of Cost

mostly changes the MRT. The area of possible consumption bundles changes in this sense. Fewer opportunities, so we are worse off here. So the best well-off point is exactly going to the tangency point of the indifference curves, and that should be the consumption bundle point.

So to optimize we want this: .

\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[x=0.62cm, y=0.78cm, line width=0.9pt]

  % ---- shaded regions (drawn first, under everything) ----
  \fill[pattern=dots, pattern color=red!45]
       (0,6) -- (12,0) -- (0,4) -- cycle;
  \fill[pattern=dots, pattern color=black!70]
       (0,0) -- (0,4) -- (12,0) -- cycle;

  % ---- axes ----
  \draw[line width=1.4pt] (0,0) -- (0,8.6);
  \draw[line width=1.4pt] (0,0) -- (13.4,0);
  \node[below left, font=\bfseries] at (0,0) {0};
  \node[below, font=\bfseries] at (6.7,-1.05) {Cookies (C)};
  \node[rotate=90, align=center, font=\bfseries] at (-1.9,4.3)
       {Slices of\\Pizza (S)};

  % ---- ticks ----
  \foreach \x in {2,4,6,8,10,12} {
    \draw[line width=1.2pt] (\x,0) -- (\x,0.22);
    \node[below=5pt, font=\bfseries] at (\x,0) {\x};
  }
  \foreach \y in {2,4,6,8} {
    \draw[line width=1.2pt] (0,\y) -- (1.1,\y);
    \node[left=5pt, font=\bfseries] at (0,\y) {\y};
  }

  % ---- budget lines ----
  \draw[line width=1.1pt] (0,6) -- (12,0);
  \draw[line width=1.1pt] (0,4) -- (12,0);

  % ---- labels ----
  \node[right, font=\bfseries] at (1.4,5.7) {BC$_1$: 4S + 2C = 24};
  \node[right, font=\bfseries] at (1.4,2.6) {BC$_2$: 6S + 2C = 24};

  % ---- shift arrow ----
  \draw[line width=1.1pt, ->] (3.4,4.3) -- (2.0,3.35);

\end{tikzpicture}
\end{document}