In this exercise, we are going to:
- Review how to maximize the utility of an agent under a budget constraint.
- Augment the standard model with the social multiplier.
- Study the fads and fashions model and its unconventional pricing implications.
- Augment the standard model by adding dynamics, and ask whether a social multiplier on savings can amplify interest-rate shocks.
Standard utility maximization program
A representative agent derives utility from consuming two goods, \(x\) and \(y\). Each good comes at price \(p_x\) and \(p_y\), respectively. The agent is also endowed with an income \(I\) to spend on each good.
The utility function is Cobb-Douglas and writes: \[
U(x,y) = x^\alpha y^\beta
\]
with \(\alpha,\beta>0\).
- Write the budget constraint.
- Write the utility maximization program.
To solve the utility maximization program, we can set up a Lagrangian. The function \(\mathcal{L}(.)\) is defined as the objective function minus the product between the budget constraint and a scalar \(\lambda \in \mathbb{R}\).
- Write the Lagrangian and set the first order conditions (partial derivative with respect to \(x\), \(y\) and \(\lambda\) equal to zero) as a system.
- Solve the system for \(x^\star\) and \(y^\star\) and express them as a function of income and prices.
- Compute the indirect utility function \(V = U(x^\star,y^\star)\).
- Show that \(\lambda^\star = \partial V/\partial I\). Interpret.
- Compute the elasticity of demand of \(x\) with respect to \(p_x\), \(p_y\) and \(I\).
The second order condition to reach a local maximum is: \(2p_x p_y U_{xy} - U_{xx}p_y^2 - U_{yy} p_x^2 > 0\) (where \(U_{xx}\) is the second derivative of \(U\) with respect to \(x\)).
- For \(p_x = 2\), \(p_y = 1\), \(I=10\), \(\alpha=\beta=0.5\), did we reach a (local) maximum?
Solution
Q1.
The agent spends her entire income \(I\) on goods \(x\) and \(y\): \[p_x x + p_y y = I\]
Q2. \[\max_{x,y} \; x^\alpha y^\beta \quad \text{subject to} \quad p_x x + p_y y = I, \quad x,y \geq 0\]
Q3.
The Lagrangian is: \[\mathcal{L}(x, y, \lambda) = x^\alpha y^\beta - \lambda(p_x x + p_y y - I)\]
Setting partial derivatives to zero: \[\frac{\partial \mathcal{L}}{\partial x} = \alpha x^{\alpha-1} y^\beta - \lambda p_x = 0\] \[\frac{\partial \mathcal{L}}{\partial y} = \beta x^\alpha y^{\beta-1} - \lambda p_y = 0\] \[\frac{\partial \mathcal{L}}{\partial \lambda} = -(p_x x + p_y y - I) = 0\]
Q4.
Dividing the first FOC by the second: \[\frac{\alpha y}{\beta x} = \frac{p_x}{p_y} \implies p_y y = \frac{\beta}{\alpha} p_x x\]
Substituting into the budget constraint: \[p_x x + \frac{\beta}{\alpha} p_x x = I \implies p_x x \cdot \frac{\alpha + \beta}{\alpha} = I\]
\[\boxed{x^\star = \frac{\alpha}{\alpha+\beta} \cdot \frac{I}{p_x}, \qquad y^\star = \frac{\beta}{\alpha+\beta} \cdot \frac{I}{p_y}}\]
The agent spends fixed shares \(\frac{\alpha}{\alpha+\beta}\) and \(\frac{\beta}{\alpha+\beta}\) of income on \(x\) and \(y\) respectively — a hallmark of Cobb-Douglas preferences.
Q5.
\[V = (x^\star)^\alpha (y^\star)^\beta = \left(\frac{\alpha}{\alpha+\beta}\right)^\alpha \left(\frac{\beta}{\alpha+\beta}\right)^\beta \cdot \frac{I^{\alpha+\beta}}{p_x^\alpha p_y^\beta}\]
Q6.
From the FOCs: \(\lambda = \frac{\alpha x^{\alpha-1}y^\beta}{p_x}\). Substituting \(x^\star\) and \(y^\star\):
\[\lambda^\star = \frac{\alpha+\beta}{I} \cdot V\]
Taking \(\partial V / \partial I\) from the expression in Q5 gives the same result. This confirms that \(\lambda^\star\) is the marginal utility of income: it measures how much utility rises if the budget increases by one unit.
Q7.
From \(x^\star = \frac{\alpha}{\alpha+\beta} \cdot \frac{I}{p_x}\):
- Own-price elasticity: \(\varepsilon_{x,p_x} = \frac{\partial x^\star}{\partial p_x}\frac{p_x}{x^\star} = -1\)
- Cross-price elasticity: \(\varepsilon_{x,p_y} = 0\) (demand for \(x\) does not depend on \(p_y\) with Cobb-Douglas)
- Income elasticity: \(\varepsilon_{x,I} = 1\)
\(x\) is a unit-elastic good: demand falls one-for-one with its own price, is independent of the other good’s price, and rises proportionally with income.
Q8.
Compute the second derivatives at \(\alpha = \beta = 0.5\), so \(U = x^{0.5} y^{0.5}\):
\[U_{xx} = -0.25 x^{-1.5} y^{0.5}, \quad U_{yy} = -0.25 x^{0.5} y^{-1.5}, \quad U_{xy} = 0.25 x^{-0.5} y^{-0.5}\]
With \(p_x=2\), \(p_y=1\): \(x^\star = I/(2(\alpha+\beta)) = 2.5\), \(y^\star = I/(\alpha+\beta) \cdot 0.5 = 5\).
\[2 p_x p_y U_{xy} - U_{xx} p_y^2 - U_{yy} p_x^2\] \[= 2(2)(1)(0.25)(2.5)^{-0.5}(5)^{-0.5} - (-0.25)(2.5)^{-1.5}(5)^{0.5}(1)^2 - (-0.25)(2.5)^{0.5}(5)^{-1.5}(4)\]
All three terms are positive (the first directly, the other two because of the negatives in \(U_{xx}\) and \(U_{yy}\)), so the SOC is satisfied: we are at a local maximum.
Social multiplier
We augment the agent’s program. Utility can also depend on the behavior of others: for example, if you enjoy playing tennis, you may derive additional utility from the fact that many people around you also play tennis. We now assume that \(p_y = 1\). Utility now writes \(U = U(x, y, S)\), where \(S\) is a measure of social capital (e.g., the average number of hours peers spend playing tennis), which each individual takes as given.
- Solve the new program. Explain why including social capital \(S\) does not change the form of the equilibrium condition.
- Show that \(\partial x^\star / \partial S > 0 \iff U_{xS} > p_x U_{yS}\). Interpret. Hint: Differentiate the equilibrium condition and the budget constraint totally with respect to \(S\). Use the second-order condition for a local maximum. Reminder — the Schwarz theorem states: \(f_{xy} = f_{yx}\).
Solution
Q1.
The Lagrangian is: \[\mathcal{L}(x, y, \lambda) = U(x, y, S) - \lambda(p_x x + y - I)\]
Since \(S\) is taken as given by each individual (it is the group average, which no single agent can influence), the FOCs are: \[U_x(x, y, S) = \lambda p_x\] \[U_y(x, y, S) = \lambda\]
Dividing: \(U_x = p_x U_y\), the same equilibrium condition as before. Social capital does not change the optimality condition because it enters utility additively from the individual’s perspective: \(S\) shifts the level of utility but agents cannot adjust \(S\) themselves, so it plays no role in the trade-off between \(x\) and \(y\) at the margin. The budget constraint is also unchanged.
Q2.
By definition of optimality, the condition \(U_x(x^\star, y^\star, S) = p_x U_y(x^\star, y^\star, S)\) holds as an identity in \(S\). Total differentiation with respect to \(S\) gives:
\[U_{xx}\frac{\partial x^\star}{\partial S} + U_{xy}\frac{\partial y^\star}{\partial S} + U_{xS} = p_x\left(U_{yx}\frac{\partial x^\star}{\partial S} + U_{yy}\frac{\partial y^\star}{\partial S} + U_{yS}\right)\]
The budget constraint \(p_x x^\star + y^\star = I\) is also an identity, so differentiating: \[p_x \frac{\partial x^\star}{\partial S} + \frac{\partial y^\star}{\partial S} = 0 \implies \frac{\partial y^\star}{\partial S} = -p_x \frac{\partial x^\star}{\partial S}\]
Substituting and using the Schwarz theorem (\(U_{xy} = U_{yx}\)):
\[\frac{\partial x^\star}{\partial S}\left(U_{xx} - p_x U_{yx} - p_x U_{xy} + p_x^2 U_{yy}\right) = p_x U_{yS} - U_{xS}\]
\[\frac{\partial x^\star}{\partial S}\left(U_{xx} - 2p_x U_{xy} + p_x^2 U_{yy}\right) = p_x U_{yS} - U_{xS}\]
The second-order condition tells us that \(2p_x p_y U_{xy} - U_{xx}p_y^2 - U_{yy}p_x^2 > 0\) with \(p_y = 1\), which rearranges to \(-(U_{xx} - 2p_x U_{xy} + p_x^2 U_{yy}) > 0\), i.e. the bracket on the left is negative. Therefore:
\[\frac{\partial x^\star}{\partial S} > 0 \iff p_x U_{yS} - U_{xS} < 0 \iff U_{xS} > p_x U_{yS}\]
Interpretation: Social capital raises demand for \(x\) if and only if it raises the marginal utility of \(x\) more than \(p_x\) times it raises the marginal utility of income (the numeraire \(y\)). In other words, \(S\) must increase the attractiveness of \(x\) relative to \(y\). If \(S\) raises the utility of both goods equally (at the margin), it has no effect on the composition of spending. Note that this does not require \(S\) to raise utility overall (\(U_S > 0\) is not assumed): even if being surrounded by drug users is unpleasant overall, it can still increase one’s own drug consumption if it makes drugs more appealing relative to other goods.
Fads and fashions
Recall from the lecture the Becker (1991) model of fads. Aggregate demand for a good is \(Q = D(p, Z, Q)\), where \(p\) is the price, \(Z\) is a vector of exogenous demand shifters, and \(Q\) also appears on the right-hand side because popularity itself raises demand (\(D_Q > 0\)). At equilibrium, quantity demanded must equal quantity supplied — so \(Q\) is determined jointly by price and the self-referential popularity effect.
The supply curve is \(Q = S(p)\) with \(S'(p) > 0\) as usual. Denote by \(m = D_Q \in [0,1)\) the social multiplier (we focus on the stable case for now).
The implicit equation \(Q = D(p, Z, Q)\) defines \(Q\) as a function of \(p\) (and \(Z\)). Using total differentiation, derive \(\frac{dQ}{dp}\) and show that it equals \(\frac{D_p}{1-m}\). What is the sign of \(\frac{dQ}{dp}\), and under what condition is demand more elastic than it would be without social interactions?
Explain intuitively why the social multiplier makes the demand curve steeper when \(m\) is close to 1-.
Now suppose \(m > 1\) locally (around some popularity threshold \(Q^\star\)). Show that \(\frac{dQ}{dp} > 0\) in this region, i.e. the demand curve is upward-sloping. Give an intuition for why this is not as paradoxical as it sounds.
A producer faces upward-sloping demand in the range \(Q < Q^\star\) and is currently at a low-popularity equilibrium \(Q_\ell < Q^\star\). Explain why she could be interested in lowering her price?
Solution
Q1.
At equilibrium, \(Q = D(p, Z, Q)\) holds as an identity in \(p\). Total differentiation with respect to \(p\):
\[\frac{dQ}{dp} = D_p + D_Q \cdot \frac{dQ}{dp} = D_p + m \cdot \frac{dQ}{dp}\]
Solving:
\[\frac{dQ}{dp}(1 - m) = D_p \implies \boxed{\frac{dQ}{dp} = \frac{D_p}{1-m}}\]
Since \(D_p < 0\) (demand falls with price) and \(1 - m > 0\) (in the stable case), we have \(dQ/dp < 0\): demand is downward-sloping as usual. However, \(|dQ/dp| = |D_p|/(1-m) > |D_p|\) — demand is more elastic than it would be without social interactions (\(m = 0\)). A price increase triggers a direct reduction in quantity (\(D_p\)), which then reduces popularity, which reduces demand further, amplifying the initial effect. The closer \(m\) is to 1, the larger this amplification.
Q2.
In the \((Q, p)\) space (with \(Q\) on the horizontal axis), the demand curve being “steeper” means a given change in \(Q\) is associated with a large change in \(p\) — equivalently, the quantity response to a price change is small. This might seem to contradict Q1, but the two statements are consistent: more elastic demand in the \((p, Q)\) sense means a given price cut triggers a large quantity increase — which when drawn with \(Q\) on the x-axis corresponds to a flatter curve, not a steeper one.
Let us restate carefully: when \(m \to 1^-\), \(dQ/dp \to -\infty\): a tiny price increase causes a catastrophic collapse in demand. In the standard \((Q,p)\) diagram, this manifests as a demand curve that is nearly horizontal (infinitely elastic) around the equilibrium. The intuition: raise price slightly → fewer buyers → good becomes less popular → even fewer buyers → … the cascade is almost self-sustaining near \(m = 1\).
Q3.
When \(m > 1\) locally, the denominator \(1 - m < 0\), so:
\[\frac{dQ}{dp} = \frac{D_p}{1-m} = \frac{(-)}{(-)} > 0\]
Demand is upward-sloping: a higher price is associated with higher quantity demanded. This is not paradoxical once we understand the mechanism: in this range, the social feedback is so strong that it overwhelms the direct price effect. A small price increase causes a small direct drop in buyers; but if somehow quantity stays high (e.g. due to rationing), popularity remains high, which pulls demand up. The upward-sloping segment is therefore unstable: no equilibrium can persist there, because any perturbation sends demand cascading either up (toward the high-popularity stable equilibrium \(Q_h\)) or down (toward the low-popularity stable equilibrium \(Q_\ell\)).
Q4.
This strategy is a way to push demand towards the upward-sloping part, where price increases increase demand. If it can sustain this short-run, decrease in profitability, it is a way to reach the \(Q_h\) equilibrium.
Intertemporal consumption and the social multiplier on savings
So far, we have studied how social interactions affect the composition of spending (Exercise 2) and the level and stability of demand for a single good (Exercise 3). This exercise asks whether the same logic applies to the **intertemporal* allocation of income — i.e., to saving.
Empirically, savings rates are strongly correlated within peer groups and neighborhoods, even after controlling for income. Is this because similar people sort into similar places, or because people genuinely imitate each other’s saving behavior? The framework below lets you think through this question formally.
Part A — Two-period consumption
An agent lives for two periods. She earns income \(I_1\) in period 1 and \(I_2 = 0\) in period 2 (retirement), so she must save in period 1 to fund retirement consumption. The interest rate is \(r\) and the discount factor is \(0 < \beta < 1\). Prices equal unity in both periods.
Her utility is: \[V = \log c_1 + \beta \log c_2\]
Write the period-1 and period-2 budget constraints, then derive the intertemporal budget constraint.
Solve the program using the Lagrange method. Derive the Euler equation and express optimal \(c_1^\star\), \(c_2^\star\), and savings \(s^\star = I_1 - c_1^\star\) as functions of \(I_1\), \(\beta\), and \(r\).
How does \(s^\star\) respond to an increase in \(r\)? Decompose your answer into a substitution effect and an income effect, and explain why, with \(I_2 = 0\) and log utility, only one of them survives.
How does \(s^\star\) respond to an increase in \(\beta\)? Give the economic intuition.
Solution — Part A
Q1.
Period 1: \(c_1 + s = I_1\), so \(s = I_1 - c_1\).
Period 2: \(c_2 = (1+r)s = (1+r)(I_1 - c_1)\).
Combining (eliminating \(s\)): \[c_1 + \frac{c_2}{1+r} = I_1\]
Since \(I_2 = 0\), lifetime wealth is simply \(W = I_1\).
Q2.
\[\mathcal{L} = \log c_1 + \beta \log c_2 - \lambda\left(c_1 + \frac{c_2}{1+r} - I_1\right)\]
FOCs: \[\frac{1}{c_1} = \lambda, \qquad \frac{\beta}{c_2} = \frac{\lambda}{1+r}\]
Combining: \(\boxed{c_2 = \beta(1+r)c_1}\) — the Euler equation. It equates the marginal utility of consuming today to the discounted marginal utility of consuming tomorrow, scaled by the gross return on savings.
Substituting into the intertemporal constraint: \[c_1 + \beta c_1 = I_1 \implies c_1^\star = \frac{I_1}{1+\beta}\] \[c_2^\star = \frac{\beta(1+r)I_1}{1+\beta}, \qquad s^\star = I_1 - c_1^\star = \frac{\beta I_1}{1+\beta}\]
Q3.
\[\frac{\partial s^\star}{\partial r} = 0\]
Savings are independent of \(r\) when \(I_2 = 0\) and utility is logarithmic. Here is why the two effects exactly cancel:
- Substitution effect: A higher \(r\) makes future consumption cheaper → agent wants to save more → \(s^\star \uparrow\).
- Income effect: A higher \(r\) makes each dollar saved more productive → to reach the same \(c_2\), the agent needs to save less → \(s^\star \downarrow\).
With log utility and \(I_2 = 0\), lifetime wealth \(W = I_1\) is fixed (it does not depend on \(r\)), so the income effect arises only through the productivity of savings. These two forces exactly offset: the optimal savings share \(\beta/(1+\beta)\) of income is invariant to \(r\). This is a property of log utility — the elasticity of intertemporal substitution equals 1, which makes the two effects cancel out.
Q4.
\[\frac{\partial s^\star}{\partial \beta} = \frac{I_1}{(1+\beta)^2} > 0\]
A more patient agent (higher \(\beta\)) saves more. Intuitively, \(\beta\) is the weight placed on future utility: as \(\beta \to 1\), the agent treats the future almost like the present and saves a large fraction of income; as \(\beta \to 0\), she spends everything today.
Part B — A social multiplier on savings
Now suppose the agent’s utility also depends on the average savings rate of her peer group, \(\bar{s}\). Specifically, saving more when peers save a lot is intrinsically rewarding — perhaps because thrift is a local norm, or because savings decisions are made jointly (e.g., peer pressure from family). The utility function becomes:
\[U = \log c_1 + \beta \log c_2 + \gamma \cdot s/ \bar{s}\]
where \(-\beta <\gamma < 0\) and \(s = I_1 - c_1\) is own savings. Each agent takes \(\bar{s}\) as given when optimizing.
Derive the new optimal savings \(s^\star(\bar{s})\) as a function of \(\bar{s}\) (and the parameters \(I_1\), \(\beta\), \(r\), \(\gamma\)). Show that \(\partial s^\star / \partial \bar{s} > 0\): savings are strategic complements.
In a symmetric equilibrium all agents are identical, so \(s^\star = \bar{s} \equiv S\). Find the equilibrium savings level \(S^\star\).
Solution — Part B
Q5.
Rewrite utility in terms of \(c_1\) (since \(s = I_1 - c_1\) and \(c_2 = (1+r)(I_1-c_1)\)):
\[U = \log c_1 + \beta \log[(1+r)(I_1 - c_1)] + \gamma(I_1 - c_1)/\bar{s}\]
Taking the FOC with respect to \(c_1\):
\[\frac{1}{c_1} - \frac{\beta}{I_1 - c_1} - \gamma / \bar{s} = 0\]
Let \(s = I_1 - c_1\), so \(c_1 = I_1 - s\):
\[\frac{1}{I_1 - s} - \frac{\beta}{s} = \frac{\gamma}{ \bar{s}}\]
This implicitly defines \(s^\star(\bar{s})\). To find \(\partial s^\star/\partial \bar{s}\), differentiate implicitly:
\[\left(\frac{1}{(I_1-s)^2} + \frac{\beta}{s^2}\right)\frac{\partial s^\star}{\partial \bar{s}} = -\frac{\gamma}{\bar{s}^2}\]
The bracket on the left and the right hand side of the equality are strictly positive, so:
\[\frac{\partial s^\star}{\partial \bar{s}} > 0\]
Savings are strategic complements: when peers save more, each individual also saves more. This is the direct analogue of the social multiplier on consumption from Exercise 2.
Q6.
In a symmetric equilibrium, \(s^\star = \bar{s} = S^\star\). The FOC at equilibrium becomes:
\[\frac{1}{I_1 - S^\star} - \frac{\beta}{S^\star} = \frac{\gamma}{ S^\star}\]
which implies
\(S^\star = \frac{\gamma + \beta}{1+ \gamma + \beta} I_1 >0\), an increasing function of \(\gamma\) and \(\beta\)
Social multiplier
We augment the agent’s program. Utility can also depend on the behavior of others: for example, if you enjoy playing tennis, you may derive additional utility from the fact that many people around you also play tennis. We now assume that \(p_y = 1\). Utility now writes \(U = U(x, y, S)\), where \(S\) is a measure of social capital (e.g., the average number of hours peers spend playing tennis), which each individual takes as given.
Solution
Q1.
The Lagrangian is: \[\mathcal{L}(x, y, \lambda) = U(x, y, S) - \lambda(p_x x + y - I)\]
Since \(S\) is taken as given by each individual (it is the group average, which no single agent can influence), the FOCs are: \[U_x(x, y, S) = \lambda p_x\] \[U_y(x, y, S) = \lambda\]
Dividing: \(U_x = p_x U_y\), the same equilibrium condition as before. Social capital does not change the optimality condition because it enters utility additively from the individual’s perspective: \(S\) shifts the level of utility but agents cannot adjust \(S\) themselves, so it plays no role in the trade-off between \(x\) and \(y\) at the margin. The budget constraint is also unchanged.
Q2.
By definition of optimality, the condition \(U_x(x^\star, y^\star, S) = p_x U_y(x^\star, y^\star, S)\) holds as an identity in \(S\). Total differentiation with respect to \(S\) gives:
\[U_{xx}\frac{\partial x^\star}{\partial S} + U_{xy}\frac{\partial y^\star}{\partial S} + U_{xS} = p_x\left(U_{yx}\frac{\partial x^\star}{\partial S} + U_{yy}\frac{\partial y^\star}{\partial S} + U_{yS}\right)\]
The budget constraint \(p_x x^\star + y^\star = I\) is also an identity, so differentiating: \[p_x \frac{\partial x^\star}{\partial S} + \frac{\partial y^\star}{\partial S} = 0 \implies \frac{\partial y^\star}{\partial S} = -p_x \frac{\partial x^\star}{\partial S}\]
Substituting and using the Schwarz theorem (\(U_{xy} = U_{yx}\)):
\[\frac{\partial x^\star}{\partial S}\left(U_{xx} - p_x U_{yx} - p_x U_{xy} + p_x^2 U_{yy}\right) = p_x U_{yS} - U_{xS}\]
\[\frac{\partial x^\star}{\partial S}\left(U_{xx} - 2p_x U_{xy} + p_x^2 U_{yy}\right) = p_x U_{yS} - U_{xS}\]
The second-order condition tells us that \(2p_x p_y U_{xy} - U_{xx}p_y^2 - U_{yy}p_x^2 > 0\) with \(p_y = 1\), which rearranges to \(-(U_{xx} - 2p_x U_{xy} + p_x^2 U_{yy}) > 0\), i.e. the bracket on the left is negative. Therefore:
\[\frac{\partial x^\star}{\partial S} > 0 \iff p_x U_{yS} - U_{xS} < 0 \iff U_{xS} > p_x U_{yS}\]
Interpretation: Social capital raises demand for \(x\) if and only if it raises the marginal utility of \(x\) more than \(p_x\) times it raises the marginal utility of income (the numeraire \(y\)). In other words, \(S\) must increase the attractiveness of \(x\) relative to \(y\). If \(S\) raises the utility of both goods equally (at the margin), it has no effect on the composition of spending. Note that this does not require \(S\) to raise utility overall (\(U_S > 0\) is not assumed): even if being surrounded by drug users is unpleasant overall, it can still increase one’s own drug consumption if it makes drugs more appealing relative to other goods.