A consumer has two goods: vegetables $v \geq 0$ and salt $s \geq 0$. Preferences satisfy:
(i) More vegetables is always strictly preferred.
(ii) Salt alone has no value.
(iii) When $s < v$, increasing salt (holding $v$ fixed) is strictly preferred.
(iv) When $s \geq v$, additional salt does not change the ranking of the bundle.
(a)Write properties (ii) and (iv) formally using $\succ, \succeq, \sim$.
(b)Draw indifference curves in $(v,s)$ space. Label axes and the line $s=v$. Indicate the higher-utility direction. Explain key features.
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Salt, on its own, provides zero utility — the consumer is indifferent between any $(0, s)$ bundle and having nothing.
Once you're on or above the diagonal $s = v$, additional salt is completely neutral.
The function $u(v, s) = v + \min(v,\, s)$ works. Check:
- (i) More $v$ increases $u$: if $v' > v$, then $u(v',s) = v' + \min(v',s) > v + \min(v,s)$ ✓
- (ii) $u(0, s) = 0 + \min(0,s) = 0$ for all $s \geq 0$ ✓
- (iii) When $s < v$: $u(v,s) = v + s$, which strictly increases in $s$ ✓
- (iv) When $s \geq v$: $u(v,s) = v + v = 2v$, constant in $s$ ✓
Region A — Below diagonal ($s < v$):
$$u(v,s) = v + s = k \implies s = k - v$$ICs are straight lines with slope $-1$.
Region B — On/above diagonal ($s \geq v$):
$$u(v,s) = 2v = k \implies v = k/2$$ICs are vertical lines at $v = k/2$ — $s$ doesn't matter here.
Putting it together: Each IC is an L-shape. The "corner" (kink) falls exactly on the diagonal $s = v$. Below the diagonal, the IC slopes at $-1$ (you trade vegetables and salt 1-for-1). Above the diagonal, the IC is vertical (changing salt doesn't matter).
- Axes: $v$ (vegetables) on horizontal, $s$ (salt) on vertical.
- Diagonal: Draw the line $s = v$. This is where all kinks occur.
- IC shape: L-shaped, kink on diagonal. Right-angle corner points toward the origin from the kink.
- Higher utility: Moving rightward (more $v$) reaches higher ICs. Moving upward above the diagonal leaves utility unchanged.
- Satiation: Salt is partially satiated — fully satiated once $s \geq v$, but never globally satiated (more $v$ always helps).
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A consumer chooses visible good $x \geq 0$ and numeraire $y \geq 0$. Income $m > 0$, price of $x$ is $p > 0$, budget: $px + y = m$.
Preferences: $U(x,y;p) = a\ln x + y + bx\ln p,\quad a > 0,\, b \geq 0.$
Assume $p - b\ln p > 0$ and $m$ large enough for an interior solution.
(a)Derive the Marshallian demand for $x$.
(b)Is demand necessarily downward-sloping? For which $p$ does demand increase with price? Decrease?
(c)What does $bx\ln p$ capture economically? Give an example.
(d)A student claims upward-sloping demand here means $x$ is a Giffen good. Do you agree? Explain.
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From the budget constraint: $y = m - px$. Substitute:
$$U = a\ln x + (m - px) + bx\ln p$$ $$U = a\ln x - px + bx\ln p + m$$Note: $x^*$ does not depend on $m$. This is the hallmark of quasilinear utility — the demand for $x$ has no income effect.
Let $D(p) = p - b\ln p$. Then $x^* = a/D(p)$, so:
$$\frac{\partial x^*}{\partial p} = \frac{-a \cdot D'(p)}{[D(p)]^2}$$ $$D'(p) = 1 - \frac{b}{p}$$ $$\therefore\quad \frac{\partial x^*}{\partial p} = \frac{-a\left(1 - \frac{b}{p}\right)}{(p - b\ln p)^2}$$Sign analysis: The denominator is always positive (given $p - b\ln p > 0$). The sign of $\partial x^*/\partial p$ is thus determined by $-(1 - b/p)$:
- If $p > b$: $D'(p) > 0$, so $\partial x^*/\partial p < 0$ — demand is downward-sloping (normal).
- If $p < b$: $D'(p) < 0$, so $\partial x^*/\partial p > 0$ — demand is upward-sloping (Veblen region).
- If $p = b$: demand is at its maximum ($x^*$ is locally flat in $p$).
The term $bx\ln p$ says: consuming $x$ units of a good priced at $p$ gives extra utility $bx\ln p$. Since $\ln p$ increases in $p$, a higher price makes each unit of $x$ more desirable. This is the Veblen/status effect: the good provides social signaling value precisely because it is expensive.
Example: Designer handbags (Birkin, Chanel), limited-edition sneakers, luxury champagne (Dom Pérignon), high-end watches (Rolex). Owning these goods at high prices is the point — they are consumed partly for the signal, not just the substance.
Disagree with the student. Here is the distinction:
| Feature | Giffen Good | Veblen Good (this problem) |
|---|---|---|
| Source of upward slope | Large negative income effect dominates substitution effect | Price appears directly in utility (status effect) |
| Income effect on $x$? | Yes, large and negative (inferior good) | Zero (quasilinear utility) |
| Good type | Inferior good (budget staple) | Need not be inferior — luxury good |
| Slutsky SE direction | Negative (as always) | N/A — price is in preferences |
In this model, $x^*$ does not depend on $m$ at all. There is zero income effect. The upward slope comes entirely from the direct utility effect of price — not from income-compensated behavior. A Giffen good requires a negative income effect large enough to overwhelm the substitution effect. That mechanism is simply absent here.
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When $b = 0$: $x^* = a/p$. Then $\partial x^*/\partial p = -a/p^2 < 0$ — strictly downward sloping (standard log demand).
When $a=2, b=4$: Demand is maximized where $\partial x^*/\partial p = 0$, which requires $D'(p) = 1 - b/p = 0 \implies p^* = b = 4$. So demand is maximized at price $p = 4$. At prices below 4, demand rises with price (Veblen region); above 4, demand falls normally.
Omar lives for two periods $t = 0, 1$. Each period: consumption $c_t \geq 0$, labor $\ell_t \geq 0$, wages $w_t > 0$. Can borrow/save at gross rate $R > 1$. No initial wealth.
$$U = \ln c_0 - \frac{\ell_0^2}{2} + \beta\!\left(\ln c_1 - \frac{\ell_1^2}{2}\right), \quad \beta \in (0,1)$$Intertemporal budget: $c_0 + \dfrac{c_1}{R} = w_0\ell_0 + \dfrac{w_1\ell_1}{R}$
(a)What does $\ell_t^2/2$ capture? What does $\beta$ represent?
(b)(i)Write the Lagrangian, derive FOCs, and find the consumption Euler equation. When does consumption grow?
(b)(ii)Derive the labor FOCs. Interpret (marginal benefit vs. marginal cost).
(c)Let $V(w_0, w_1, R)$ denote the indirect utility function. Use the Envelope Theorem to find $dV/dw_0$. Interpret.
(d)Without solving fully: does $c_0^*$ increase or decrease when $\beta$ increases? Give intuition.
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$\ell_t^2/2$ (disutility of labor): This is a convex cost of working. The marginal disutility of the $\ell$-th hour of work is $\ell$ itself. Each additional hour of work is more painful than the last — the $10$th hour hurts more than the $1$st. This is the standard "increasing marginal disutility" assumption.
$\beta \in (0,1)$ (discount factor): Omar values future utility less than present utility. A $\beta$ close to 1 means he is very patient (almost indifferent between now and later); a $\beta$ close to 0 means he heavily discounts the future. Formally, $\beta = 1/(1+\rho)$ where $\rho$ is the subjective rate of time preference.
Euler Equation: Divide (1) by (2):
$$\frac{1/c_0}{\beta/c_1} = \frac{\lambda}{\lambda/R} \implies \frac{c_1}{\beta c_0} = R \implies \boxed{c_1 = \beta R \cdot c_0}$$Consumption grows ($c_1 > c_0$) when $\beta R > 1$, i.e., when the return on savings outweighs impatience. If $\beta R < 1$, consumption falls over time.
Interpretation: FOC (3) says: the marginal disutility of working ($\ell_0$) equals the shadow value of the income earned ($\lambda w_0$). Using $\lambda = 1/c_0$ from (1):
$$\ell_0 = \frac{w_0}{c_0}, \qquad \ell_1 = \frac{w_1}{c_1}$$In each period, Omar works until the marginal disutility of labor equals the real wage measured in consumption units.
Using $\lambda^* = 1/c_0^*$:
$$\boxed{\frac{dV}{dw_0} = \frac{\ell_0^*}{c_0^*}}$$Interpretation: A marginal increase in $w_0$ raises lifetime income by $\ell_0^*$ units (the number of hours Omar is working times the wage increase). Each extra unit of income is worth $\lambda^* = 1/c_0^*$ utils. So the total utility gain is $\ell_0^*/c_0^*$. Higher wages are better if you're working — but the benefit scales with how much you're working.
$c_0^*$ decreases when $\beta$ increases.
When $\beta$ rises, Omar places more weight on future utility. The Euler equation $c_1 = \beta R c_0$ shows that a higher $\beta$ increases $c_1$ relative to $c_0$ — Omar wants a steeper consumption profile. Given a fixed lifetime budget, consuming more in period 1 means consuming less in period 0. So $c_0^*$ falls.
Intuition: a more patient consumer saves more today to fund higher future consumption. Patience (high $\beta$) is synonymous with sacrificing present consumption.
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Envelope Theorem: $\dfrac{dV}{dw_1} = \dfrac{\partial\mathcal{L}}{\partial w_1}\bigg|_{\text{opt}} = \dfrac{\lambda \ell_1^*}{R} = \dfrac{\ell_1^*}{R c_0^*}$.
Labor in period 1: A higher $w_1$ raises the return to working in period 1, making leisure more expensive (substitution effect). In this model, there's no "wealth effect" that would reduce labor (since labor here is the only income source). So Omar works more in period 1 — $\ell_1^*$ increases with $w_1$. Formally: $\ell_1^* = w_1/c_1^*$, and a higher $w_1$ — at least partially — feeds back into higher $\ell_1^*$.
A consumer has utility $u(x,y) = x^{1/2}y^{1/2}$ and income $m = 36$. The price of $y$ is $p_y = 1$ throughout. Initially $p_x = 4$. The price then rises to $p_x' = 9$.
(a)Find the initial and final Marshallian demands $(x_0^*, y_0^*)$ and $(x_1^*, y_1^*)$.
(b)Decompose the total effect on $x$ into substitution effect (SE) and income effect (IE) using the Slutsky method. State the compensated income $m'$.
(c)Is $x$ a normal or inferior good? Explain using the sign of IE.
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Initial ($p_x = 4$):
$$x_0^* = \frac{36}{2 \times 4} = \frac{36}{8} = 4.5, \qquad y_0^* = \frac{36}{2 \times 1} = 18$$Final ($p_x' = 9$):
$$x_1^* = \frac{36}{2 \times 9} = \frac{36}{18} = 2, \qquad y_1^* = \frac{36}{2 \times 1} = 18$$Note: $y^*$ doesn't change because Cobb-Douglas demand for $y$ depends only on $m$ and $p_y$ — both are fixed. This is a useful check.
The Slutsky compensated income holds the consumer on the same budget set containing the original bundle at new prices:
$$m' = p_x' x_0^* + p_y y_0^* = 9 \times 4.5 + 1 \times 18 = 40.5 + 18 = 58.5$$Demand at new price $p_x' = 9$ with compensated income $m' = 58.5$:
$$x^H = \frac{m'}{2p_x'} = \frac{58.5}{18} = 3.25$$Verify: $\text{SE} + \text{IE} = -1.25 + (-1.25) = -2.5 = \text{TE}$ ✓
$x$ is a normal good.
The income effect is $-1.25 < 0$. When the price of $x$ rises, real income falls, and the consumer buys less $x$. A normal good is one where demand increases with income — so a fall in real income reduces demand. This is exactly what we see: $\text{IE} < 0$ confirms normality.
Sanity check: Cobb-Douglas goods always have positive income elasticity — they are always normal goods. The Marshallian demand $x^* = m/(2p_x)$ is linear in $m$, so income elasticity equals 1.
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Hicksian approach: $e(p_x', p_y, u_0) = 2 \times 9 \times \sqrt{9 \times 1} = 18 \times 3 = 54$.
Hicksian demand: $x^H = e/(2p_x') = 54/18 = 3$.
Hicks SE $= 3 - 4.5 = -1.5$. Hicks IE $= 2 - 3 = -1$.
Comparison: The Slutsky method gave SE = $-1.25$, IE = $-1.25$. The Hicksian method gives SE = $-1.5$, IE = $-1$. The total effect is the same ($-2.5$), but the decomposition differs. Hicksian compensation (utility-constant) gives a slightly larger substitution effect because it holds utility constant rather than just affording the old bundle.
Two consumers (A, B) with Cobb-Douglas preferences: $u^A(x,y) = x^{2/3}y^{1/3}$ and $u^B(x,y) = x^{1/3}y^{2/3}$. Endowments: $\omega^A = (6, 0)$ and $\omega^B = (0, 6)$ (A owns only $x$, B owns only $y$).
(a)Find the Walrasian equilibrium price $p_x^*$ (normalize $p_y = 1$) and equilibrium allocations $(x_A^*, y_A^*)$, $(x_B^*, y_B^*)$.
(b)Verify market clearing.
(c)Is the equilibrium allocation Pareto optimal? How do you know?
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Consumer A ($\alpha_A = 2/3$):
$$x_A^* = \frac{2}{3} \cdot \frac{m_A}{p_x} = \frac{2}{3} \cdot \frac{6p_x}{p_x} = 4, \qquad y_A^* = \frac{1}{3} \cdot m_A = \frac{1}{3} \cdot 6p_x = 2p_x$$Consumer B ($\alpha_B = 1/3$):
$$x_B^* = \frac{1}{3} \cdot \frac{m_B}{p_x} = \frac{1}{3} \cdot \frac{6}{p_x} = \frac{2}{p_x}, \qquad y_B^* = \frac{2}{3} \cdot m_B = \frac{2}{3} \cdot 6 = 4$$Market clearing for $x$: Total endowment of $x$ = 6.
$$x_A^* + x_B^* = 6$$ $$4 + \frac{2}{p_x} = 6$$ $$\frac{2}{p_x} = 2 \implies \boxed{p_x^* = 1}$$Notice the symmetry: A consumes 4 of $x$ and 2 of $y$ (mostly $x$, matching her preferences), and B consumes 2 of $x$ and 4 of $y$ (mostly $y$). Exactly right given their tastes.
Both markets clear. The equilibrium is valid.
First Welfare Theorem: This is a competitive equilibrium with complete markets and no externalities. By the First Welfare Theorem, it is Pareto optimal.
Verify directly: At a Pareto optimum, $MRS^A = MRS^B$. For Cobb-Douglas $u = x^\alpha y^{1-\alpha}$: $MRS = \frac{\alpha}{1-\alpha} \cdot \frac{y}{x}$.
$$MRS^A = \frac{2/3}{1/3} \cdot \frac{y_A^*}{x_A^*} = 2 \cdot \frac{2}{4} = 1$$ $$MRS^B = \frac{1/3}{2/3} \cdot \frac{y_B^*}{x_B^*} = \frac{1}{2} \cdot \frac{4}{2} = 1$$ $$MRS^A = MRS^B = 1 = \frac{p_x^*}{p_y} \checkmark$$Both consumers are tangent to the same price line, confirming Pareto optimality. There is no room to make one better off without hurting the other.
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New incomes: $m_A = 3p_x$, $m_B = 9$.
New demands for $x$: $x_A^* = (2/3)(3p_x/p_x) = 2$. $x_B^* = (1/3)(9/p_x) = 3/p_x$.
Market clearing: $2 + 3/p_x = 3 \implies 3/p_x = 1 \implies p_x^* = 3$.
Intuition: $x$ became relatively scarcer (endowment fell from 6 to 3 while $y$ stayed the same in relative terms). Scarcer $x$ means higher relative price of $x$ — $p_x^*$ rose from 1 to 3. This makes sense.
An investor has CARA utility $u(W) = -e^{-\gamma W}$ with risk aversion $\gamma = 0.02$. Current wealth $W_0 = 100$. A risky asset pays $\tilde{X} \sim N(\mu, \sigma^2)$ with $\mu = 8$ (expected gain) and $\sigma^2 = 400$ (variance). A safe asset pays 0.
(a)Compute the certainty equivalent $CE$ of the risky asset.
(b)The investor can buy the risky asset for a cost $I = 4$. Should she invest?
(c)How does your answer change if $\gamma = 0.05$?
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For $u(W) = -e^{-\gamma W}$ and $\tilde{W} = W_0 + \tilde{X}$ where $\tilde{X} \sim N(\mu, \sigma^2)$:
$$CE = W_0 + \mu - \frac{\gamma}{2}\sigma^2$$The term $\frac{\gamma}{2}\sigma^2$ is called the risk premium — what the investor gives up to avoid risk. The $CE$ is the mean return minus this risk penalty.
The $CE$ satisfies $u(CE) = E[u(\tilde{W})]$, i.e., $-e^{-\gamma CE} = E[-e^{-\gamma(W_0+\tilde{X})}]$.
Using the moment-generating function of a normal: $E[e^{-\gamma\tilde{X}}] = e^{-\gamma\mu + \gamma^2\sigma^2/2}$.
$$-e^{-\gamma CE} = -e^{-\gamma W_0} \cdot e^{-\gamma\mu + \gamma^2\sigma^2/2}$$ $$\gamma CE = \gamma W_0 + \gamma\mu - \frac{\gamma^2\sigma^2}{2}$$ $$CE = W_0 + \mu - \frac{\gamma\sigma^2}{2} \checkmark$$Risk premium = $\frac{\gamma}{2}\sigma^2 = \frac{0.02}{2}(400) = 4$. The investor requires a risk premium of 4 to hold this asset.
Without investing: wealth = $W_0 = 100$. With investment at cost $I = 4$: expected wealth = $W_0 - I + E[\tilde{X}] = 100 - 4 + 8 = 104$. But she's risk averse, so we compare $CE$ to $W_0 + I = 104$.
Invest if: $CE \geq W_0 + I$
$$104 \geq 100 + 4 = 104$$Result: $CE = W_0 + I = 104$. The investor is exactly indifferent. She is willing to pay exactly $I = 4$ for this asset — the risk premium exactly equals the investment cost. She should (weakly) invest.
Now: $CE = 98 < 104 = W_0 + I$. Do not invest.
The risk premium is now $\frac{0.05}{2}(400) = 10$, which exceeds the expected gain of $\mu = 8$. The variance is so penalizing that even the positive expected return isn't enough to justify the investment at cost $I = 4$.
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Invest if $CE \geq W_0 + I$, i.e., $100 + 8 - \frac{0.02}{2}\sigma^2 \geq 104$.
$108 - 0.01\sigma^2 \geq 104$
$0.01\sigma^2 \leq 4 \implies \sigma^2 \leq 400$.
Maximum tolerable variance is $\bar{\sigma}^2 = 400$. The original problem sits exactly at this threshold — confirming the investor is indifferent at $\sigma^2 = 400$. Any higher variance and she declines.