Peter Zhang

EC 101A MT1

Introduction

I’m taking these notes for the 101A midterm on Thursday. I did not go to lecture, so these may not reflect the content on the test. The notes are based on the 12th edition of Microeconomic Theory by Nicholson and Snyder. You may or may not find a copy of that textbook here.

Chapter 1

The ceteris paribus assumption restricts the forces at play. In economics, controlled experiments are usually impossible to conduct, so statistical methods are used to control for other factors.

Factors that are outside of a decisionmaker’s control, such as prices, are called exogenous variables. Factors within their control, such as quantity bought, are called endogenous variables. Often, we want to study the effect of changing an exogenous variable.

Economic actors are assumed to be optimizers. It helps to create solvable models and the assumption has some empirical validity.

Economists disagree over the positive-normative distinction. Some economists believe in pure positive analysis, where economics should only describe events. Others believe that economics is inevitably laden with views on ethics, morality, and fairness.

Originally, economists like Adam Smith struggled with the water-diamond paradox. Water, imminently useful, was cheap. Diamonds, which had little practical use, were expensive. The marginal revolution resolved this paradox. Prices reflect the marginal evaluations of demanders and producers. Water is cheap because it has little marginal value and a low cost of production. Diamonds are expensive because they have high marginal value and a high cost of production.

The production possibility frontier is models tradeoffs between two goods. Because of scarcity, producing good A imposes an opportunity cost in terms of good B.

Modern developments include mathematical foundations sophisticated models of markets, incorporation of uncertainty and information, and the study of behavioral economics. Computers now enable researchers to leverage large amounts of real world data to test models.

Chapter 2

To maximize a function \(f\), we must have \(f'(x^*) = 0\) and \(f''(x^*) < 0\).

Elasticity between two goods \(x\) and \(y\) is defined as \(e_{y, x} = \frac{\frac{\delta y}{y}}{\frac{\delta x}{x}} = \frac{dy(x)}{dx} \frac{x}{y}\).

Implicit functions reveal tradeoffs between variables. For example, suppose we have \(f(x_1, x_2) = 0\). We can express \(x_2\) in terms of \(x_1\) so that \(x_2 = g(x_1)\). Differentiating with respect to \(x_1\) yields \(\frac{dg(x_1)}{dx_1} = -\frac{f_1}{f_2}\).

Comparative statics denotes the effect of an exogenous variable \(a\) on an endogenous variable \(x\). In other words, \(\frac{dx(a)}{da} = - \frac{\frac{df}{da}}{\frac{df}{dx}}\).

To maximize a function of multiple variables, the partials with respect to each variable must equal 0 and the second partials must be negative.

To assess the impact of an exogenous variable \(a\) on \(y(x)\), we could solve for \(x^*\) in terms of \(a\) and compute \(y(x^*)\). However, we could also apply the Envelope Theorem and immediately conclude that \(\frac{dy^*}{da} = x^*(a)\). The theorem also holds for the many variable case, so that \(\frac{dy^*}{da} = \frac{df}{da}\|_{x_i = x_i^*(a)}\).

Langrange multipliers are used for constrained optimization. To maximize \(y=f(x_1, x_2,..., x_n)\) subject to constraint \(g(x_1, x_2, ..., x_n) = 0\), set up the Langrangian \(\mathcal{L} = f + \lambda g\). The conditions are \(\frac{\delta \mathcal{L}}{\delta x_i} = f_i + \lambda g_i = 0\) for each good $x_i$ and \(g = 0\).

The multiplier can be interpreted as the benefit-cost ratio, i.e. \(\lambda = \frac{mb}{mc}\) for all \(x_i\). It means that if the constraint were slightly relaxed, it wouldn’t matter which \(x_i\) were increased.

The envelope theorem means \(\frac{dy^*}{da} = \frac{d \mathcal{L}}{da} (x_1^*,...,x_n^*; a)\).

Often, we wish to deploy inequality constraints such a \(g(x_1, x_2) \geq 0\) or \(x_1 \geq 0\). We can use slack variables, such as \(g(x_1, x_2) - a^2 = 0\) and \(x_1 - b^2 = 0\). Substituting, we introduce the additional first-order conditions \(\frac{d\mathcal{L}}{da} = -2a\lambda_1\) and \(\frac{d\mathcal{L}}{d\lambda_1} =g(x_1, x_2) - a^2 = 0\).

Chapter 3

Preference relations have three properties. First, completeness. For two options A and B, it must be the case that either A is preferred to B, B is preferred to A, or that they are equally attractive. Second is transitivity. If A is preferred to B and B is preferred to C, then A is preferred to C. Third is continuity. If A is preferred to B, then options “close enough” to A are also preferred to B.

Utility is only defined as a monotonic transformation. We can’t ask “how much more is A preferred to B?” because there isn’t a unique answer. One consequence is that utility can’t be compared across individuals. Ceteris paribus is in effect: when utility is specified as a function of goods, then it is assumed that all other factors are held constant. A utility function of goods takes the form:

\[U(x_1, x_2, ..., x_n)\]

Indifference curves represent all of the combinations of goods with a given utility \(U_1\). The slope of the indifference curves indicates the trade someone is willing to make between goods. Indifference curves never intersect. Otherwise, the axioms of transitivity suggest produce a contradiction.

Goods face diminishing marginal utility, since people do not want too much of any one good. A diminishing MRS produces a convex indifferent curve.

There are several key utility functions. One common function is Cobb-Douglas, where \(U(x, y) = x^\alpha y^\beta\), often normalized to \(U(x, y) = x^\alpha y^{1-\alpha}\).

Linear indifference curves are generated by \(U(x, y) = \alpha x + \beta y\). These denote perfect substitutes, since MRS is constant. An example would be indifference towards where to buy gasoline.

Perfect complements denote pairs of goods that “go together,” like coffee and cream. In the extreme case, \(U(x, y) = \min (\alpha x, \beta y)\). The ratio of goods consumed is given by \(\frac{y}{x} = \frac{\alpha}{\beta}\).

The Constant Elasticity of Substitution (CES) function takes the form of \(U(x, y) = [x^\alpha + y^\alpha]^{\frac{1}{\alpha}}\), and it incorporates each of the previous utility functions. As \(\alpha \to 0\), it approaches Cobb-Douglas. As \(\alpha \to -\infty\), it approach perfect complements.

When the utility function takes in many goods, the MRS between two goods is

\[MRS = -\frac{dx_2}{dx_1}|_{U(x_1 x_2, ..., x_n) = k} = \frac{U_{x_1}(x_1, x_2,..., x_n)}{U_{x_2}(x_1, x_2, ..., x_n)}\]

Chapter 4

If someone faces a budget constraint \(p_xx+p_yy \leq I\), then their possible combinations of goods is bounded by a triangle with slope \(-\frac{p_x}{p_y}\). The first order conditions is that the slope of the indifference curves equals the slope of the indifference curve, i.e. that:

\[\frac{-p_x}{p_y} = \frac{dy}{dx}|_{U = constant}\]

Tangency is only a sufficient condition when the MRS is diminishing, which guarantees that the utility function is quasi-concave. Often, there can be a corner solution, e.g. \(x = x^*\) and \(y = 0\), which we must check separately.

In the \(n\) good case, we have \(\mathcal{L} = U(x_1, x_2,..., x_n) + \lambda(I - p_1x_1 - p_2x_2 - ... -p_nx_n)\). The first order conditions dictate that \(MRS(x_i for x_j) = \frac{dU/dx_i}{dU/dx_j} \frac{p_i}{p_j}\). Here, \(\lambda\) represents the marginal utility per extra dollar of income, which is constant across goods.

To account for corner solutions, we consider the case where \(\frac{d\mathcal{L}}{dx_i} = \frac{dU}{dx_i} - \lambda p_i \leq 0\), where if the the marginal utility is less than the basket-wide marginal utility per income, then none of that good is consumed, i.e. \(x_i = 0\).

Since demand functions dictate the consumption of goods such that \(x^* = x(p_1, p_2, ..., p_n, I)\), utility can be rewritten as a function of exogenous variables such that \(U = V(p_1, p_2, ..., p_n, I)\). Indirect utility allows us to assess the effect of a tax on one good. The lump sum principle states that for the same amount of revenue, an income tax will depress utility less than a tax imposed on one good.

Expenditures minimization involves an expenditure function \(E = p_1x_1 + p_2x_2 +...+p_nx_n\) with constraint \(\bar{U} = U(x_1, x_2, ..., x_n)\). Utility maximization with a budget constraint and expenditure minimization with a target utility constraint are equivalent problems. Notice that this is the inverse of the utility function. In practice, the expenditure function is more useful because the envelope theorem helps reveal key elements of demand.

Expenditure functions share several properties. First, homogeneity. A doubling of all prices will double the required expenditure. Second, expenditure functions are nondecreasing in prices. That is to say, \(\frac{dE}{dp_i} \geq 0\) for all goods. Third, expenditure functions are concave in prices. If the price of a good decreases, expenditure decreases even faster because it can efficient reallocate goods.

Chapter 5

A demand function for a good \(x_i\) takes the form \(x_i ^* = x(p_{x_1}, p_{x_2}, ..., I)\). One property is homogeneity: \(x_i^* = x_i(tp_1, tp_2, ..., tp_n, tI)\). Changing all prices and incomes with the same proportion will not change quantity of goods demanded.

Consumption of normal goods increases and income increases. For inferior goods, quantity purchased decreases as income rises. If there are only two goods, if one good is inferior then the other must be normal.

When a good’s price changes, there are two forces at play that affect consumption of that good. First, there is the substitution effect, as the MRS changes to reflect the new ratio of prices. Second, there is the income effect, as a decrease in price increases real income.

For normal goods, these effects reinforce each other. For inferior goods, the substitution effect still increases consumption, but the increase in income works in the opposite direction. When the income effect is strong enough, an increase in the price of an inferior good can paradoxically increase consumption.

Utility varies along an uncompensated demand curve \(x = x(p_x, p_y, I)\), since income and other prices are constant. A decline in \(p_x\) therefore increases purchasing power. We could also hold real income constant, forming a Hicksian or compensated demand curve \(x^c = x^c(p_x, p_y, U)\). Notice that the Hicksian of a good \(x\) can be obtained by differentiating the expenditure function with respect to \(p_x\). The compensated demand curve must have a negative slope: an increase in price will always decrease the amount consumed. Because the income effect doesn’t play a role for compensated demand curves, the uncompensated curve is more price-sensitive.

Now, notice that \(x^c(p_x, p_y, U) = x[p_x, p_y, E(p_x, p_y, U)]\). By differentiating with respect to \(x\), we obtain

\[\frac{dx^c}{dp_x} = \frac{dx}{dp_x} + \frac{dx}{dE} * \frac{dE}{dp_x}\]

and rearrange to yield

\[\frac{dx}{dp_x} = \frac{dx^c}{dp_x} - \frac{dx}{dE}*x^c\]

We can interpret this to mean that the change in quantity consumed to price is equal to the compensated change in consumption (the substitution effect) minus the sensitivity of consumption to expenditure multiplied by the sensitivity of expenditure to a change in a good’s price (the income effect). The consequence is that we can decompose a good’s price sensitive into the income effect and substitution effect.

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