### Introduction

Notes are from the 12th edition of *Microeconomic Theory* by Nicholson and Snyder, which you can find here. For notes on Midterm 1, go here.

### Notes

#### Altruism

Utility maximization can account for altruism. Suppose that Kevin and I have utility function \(u(c)\) with \(u'>0, u''<0\). If I am altruistic towards Kevin and donate \(D\), I maximize \(u(c_P) + \alpha u(c_K)\) with \(\alpha > 0\) subject to \(c_P \leq M_P - D\), while Kevin selfishly maximize \(u(c_K)\) subject to \(c_K \leq M_K + D\).

Substituting, we can derive \(\max u(M_P - D) + \alpha u(M_K + D)\) with FOC \(-u'(M_P - D^*) + \alpha u'(M_K + D^*)\) and SOC \(u''(M_P - D^*) + \alpha u''(M_K + D^*) < 0\).

If \(\alpha = 1\), then \(u'(M_P - D^*) = u'(M_K + D^*)\) so \(D* = \frac{M_P - M_K}{2}\)—if Kevin earns less than me, then I donate such that the incomes equate.

We can compute comparative statics to altruism, donor income, and recipient income: \(\frac{\delta D^*}{\delta \alpha} = - \frac{u'(M_K + D^*)}{u''(M_P - D^*) + \alpha u''(M_K + D^*)} > 0\\ \frac{\delta D^*}{\delta M_P} = - \frac{-u''(M_P - D^*)}{u''(M_P - D^*) + \alpha u''(M_K + D^*)} > 0\\ \frac{\delta D^*}{\delta M_K} = - \frac{\alpha u''(M_K + D^*)}{u''(M_P - D^*) + \alpha u''(M_K + D^*)} < 0\)

#### Risk Aversion

For risk averse people, extra resources may provide decreasing marginal utility. Someone who makes $50,000 a year would be unlikely to accept a $20,000 bet on a coin flip because the negative utility of a $30,000 income is worse than the additional utility of a $70,000 income. You can see this graphically below for a starting income of \(W_0\) and earnings of \(h\). The certainty equivalent, \(CE_A\), is the amount that people would be willing to pay to avoid taking the bet.

We can measure **absolute risk aversion** with Pratt’s risk aversion measure \(r(W) = -\frac{U''(W)}{U'(W)}\). The amount a risk-averse individual is willing to pay is proportional to Pratt’s measure so that \(p = kr(W)\) where \(k = E(h^2)/2\) and \(h\) is a random variable representing winnings.

However, it often makes more sense to ask of **relative risk aversion**, since an increase in wealth may reduce absolute risk aversion. If we assume wealth is inversely related to risk aversion, then \(rr(W) = Wr(W)\) is constant.

Risk averse people are willing to purchase unfair insurance to cover losses. We can model \(\alpha\) units of coverage bought at premium \(q\) for an event with probability \(p\) and loss \(L\) as the maximization of: \(\max_\alpha (1-p)u(w-q\alpha) + pu(w-q\alpha - L + \alpha)\) Solving for first order conditions, we get: \(\frac{u'(w-q\alpha)}{u'(w-q\alpha - L + \alpha} = \frac{1-q}{q} \frac{p}{1-p}\) If \(q = p\), we can solve and get \(a^* = L\). Meanwhile, if \(q > p\), then insurance will be partial \(a^* < L\).

Now, suppose that someone with wealth \(w\) and utility function \(u\). Suppose they invested \(S=alpha\) in stock with expected return \(pr_+ + (1-p)r_- > 0\) and the rest in a bond with guaranteed return of 1. The individual maximizes: \(\max_\alpha (1-p)u(w[(1-\alpha) + \alpha(1+r_-)]) + pu(w[(1-\alpha) + \alpha(1+r_+)])\) If we assume risk neutrality, e.g. \(u(x) = bx, b>0\), then this simplifies to: \(\max_\alpha bw + \alpha bw[(1-p)r_- + pr_+]\) Since expected returns are greater than 0, \(\alpha^* = 1\).

If the case of risk aversion, we would still have \(\alpha^* > 0\).

#### Firms

A **production function** \(y=f(z)\) describes the way a firm turns inputs \(z = (z_1, z_2, ..., z_n)\) input a quantity of output \(y\). Often, we use the inputs \(z_1 = L\) (labor) and \(z_2 = K\) (capital). An **isoquant** shows the combinations of \(k\) and \(l\) that produce a given level of output \(q_0\). It represents the curve \(f(k, l) = q_0\).

Marginal physical product of an input is the additional output from one unit of input holding all other inputs constant. So we have \(MP_k = \frac{\delta q}{\delta k} = f_k\) and \(MP_l = \frac{\delta q}{\delta l} = f_l\). Importantly, marginal physical productivity is diminishing, so that \(f_{kk} < 0\) and \(F_{ll} < 0\). Productivity is often a shorthand for **average productivity**. We define the average product of labor to be \(AP_l = \frac{q}{l} = \frac{f(k,l)}{l}\).

Isoquants are convex if \(\frac{d^2 K}{d^2 L} > 0\). Consider an increase in inputs \(f(tz)\) with \(t > 1\). Under decreasing returns to scale, we have \(f(tz) < tf(z)\). For constant returns to scale, we have \(f(tz) = tf(z)\). Under increasing returns to scale, \(f(tz) > tf(z)\).

To produce an output that generates maximal profit, we go through two steps. First, we pick a cost-minimizing choice of inputs: \(\min_{L, K} wL + rK\\ \text{s.t.}\ f(L, K) \geq y\) Deriving the FOCs, we can rewrite: \(\frac{f_L'(L^*, K^*)}{f_K'(L^*, K^*)} = \frac{w}{r}\) yielding a cost function at the optimum: \(c(w, r, y) = wL^*(r, y, w) + rK^*(r, w, y)\) Next, we choose an optimal quantity of \(y\) to produce: \(\max_y py - c(w, r, y)\) The FOC gives us \(p - c'_y(w, r, y) = 0\).

The **elasticity of substitution** \(\sigma\) measures the proportionate change in \(k/l\) relative to the change in RTS.
\(\sigma = \frac{d\ (K/L)}{d\ RTS} \cdot \frac{RTS}{K/L} = \frac{d \ln(K/L)}{d \ln(f_L/f_K)}\)
At \(\sigma = \infty\), we have a linear production function:
\(y = f(K,L) = \alpha K + \beta L\)
At \(\sigma = 0\), we have a fixed-proportion production function:
\(y = f(K, L) = \min (\alpha K, \beta L)\)

At \(\sigma = 1\), we have the Cobbs-Douglas production function: \(y = f(K, L) = AK^\alpha L^\beta\) The constant elasticity of substitution function that incorporates each of the above is given by: \(y = f(K, L) = (K^p + L^p)^{\gamma / p}\) Underneath perfect competition, price \(p\) is set by the market. Solving the profit function for FOCs: \(pf(L, K) - wL - rK\\ pf'_L (L, K) - w = 0\\ pf'_K (L, K) - r = 0\)

#### Equilibrium

If all \(J\) companies \(j=1, 2, ..., J\) are producing some good \(i\), then a given company \(j\) has the supply function \(y_i^j = y_i^{j*} (p_i, w, r)\) with an industry supply function \(Y_i (p_i, w, r) = \sum_{j=1}^J y_i^{j*}(p_i, w, r)\) Similarly, if each of the \(J\) customers had a demand for good \(i\) \(x_i^{j*} = x_i^{j*} (p_1, ..., p_n, M^j)\) then the market demand would be \(X_i(p_1, ..., p_n, M^1, ..., M^J) = \sum_{j=1}^J x_i^{j*}(p_1, ..., p_n, M^j)\) At the equilibrium price \(p_i\), the demand and supply are equal so taht \(Y_i = X_i\). Using \(Y_i - X_i = 0\), we can assess the effect of an exogenous variable \(\alpha\): \(\frac{\delta p^*}{\delta \alpha} = - \frac{\frac{\delta Y^S}{\delta \alpha} - \frac{\delta X^D}{\delta \alpha}}{\frac{\delta Y^S}{\delta p} - \frac{\delta X^D}{\delta p}}\) We can interpret \(\delta p^*/\delta \alpha\) using elasticities, writing \(\epsilon_{x, p} = \frac{\delta x}{\delta p}\frac{p}{x} = \frac{\delta \ln x}{\delta \ln p}\) which measures the effect of a percent change in \(p\) on the percent change in \(x\).

#### Welfare

Taxes decrease welfare by shifting consumption to the left. Although the government collects some revenue, it also imposes deadweight loss. Trade decreases the price and increases the quantity consumed, decreasing the profits of suppliers and increasing consumer surplus.

If supply is completely inelastic, there is no deadweight loss. If supply is completely elastic, there is the usual deadweight loss.

Under perfect competition, small firms accept prices from the market and receive no profits. In a monopoly, one large firm sets the price \(p\) at the point where marginal revenue equals marginal cost. Price discrimination allows monopolies to capture all of consumer surplus.

#### Oligopoly

Game theory is the study of decisions where the payoff of a player \(i\) depends on the actions of another player \(j\). A set of strategies is in equilibrium if there are no positive utility deviations. A Nash Equilibrium always exists in mixed strategies \(\sigma\).

In **Cournot oligopy**, firms simultaneously choose quantities, leading to pricing above marginal cost. In **Bertrand oligopy**, firms choose prices and then produce quantities; it suggests that firms should be in perfect competition. In the case of second-price auctions, it is weakly dominant for everyone to bid their value \(v_i\)

#### Intertemporal

Thus far, we assume time consistency in intertemporal decisions. Suppose that across each period \(t = 0, 1, 2\), agents have income \(M'_i = M_i + \text{savings/debts}\) and choose consumption \(c_i\). Plans for the future coincide with future actions, since we can rewrite the utility function: \(u(c_0, c_1, c_2) = U(c_0) + \frac{1}{1+\delta}[U(c_1) + \frac{1}{1+\delta}U(c_2)] = u(c_1) + \frac{1}{1+\delta}u(c_2)\) But the time consistency doesn’t seem to model addictive products, good actions, or immediate gratification. Instead, we can apply a constant discount rate to all non-immediate utility so that: \(u(c_t, c_{t+1}, ...) = u(c_t) + \frac{\beta}{1+\delta} u(c_{t+1}) + \frac{\beta}{(1+\delta)^2} u(c_{t+2}) + ...\) The FOC is now inconsistent over periods. For \(t=1\), we have \(\frac{U'(c_1^*)}{U'(c_2^*)} = \beta \frac{1+r}{1+\delta}\). But for \(t=0\), we still have \(\frac{U'(c_1)}{U'(c_2)} = \frac{1+r}{1+\delta}\).

We see this in practice for health clubs. People buy long-term contracts as commitment devices, but they overestimate future attendance and delay cancellation.