问题 1.

Consider a person that spends all his fixed income $\mathrm{M}$ on two goods. Currently he spends one third of the income on good 2 . If the price of good one rises by $50 \%$ and consumer’s income by one third, what is the change in the consumer’s welfare?

Not worse off. Initial bundle is just affordable (proof)
Initial bundle: $x_1=M / 3 p_1$ and $x_2=2 M / 3 p_2$.
Initial bundle is just affordable under new prices and income
M / 3+1,5^* 2 M / 3=4 M / 3=M^{n e w}

It means that consumer is not worse off (as he can choose initial bundle).
But as relative price of good 1 is different he can substitute away from good 1 that is relatively more expensive. Thus in case of smooth IC he is better off.
Example (graphical or analytical).

问题 2.

Bob uses his monthly income (M) to pay for water services and all other goods (“other goods” represents a composite commodity of all other goods. The price of the water services is p per $\mathrm{m}^3$ (assume that water meters are already installed), and the price of composite commodity is 1 . Bob’s preferences are represented by differentiable utility function. The local water company cannot cover its cost and considers two options to solve the problem. It could raise the price by $10 \%$. In this case Bob’s utility level is reduced from $u^0$ to $u^1$. Alternatively, the water company may keep per unit price constant but in addition introduce fixed per month charge that results for Bob exactly in the same utility loss. Which scheme brings more revenue to the water company? Which scheme results in greater water conservation? Provide graphical and analytical solution.

Let $\mathrm{x}$ stays for water consumption and $\mathrm{y}$-for $\mathrm{AOG}$. Revenue of water company is given by the sum of revenue from sales (price multiplied by quantity) and fixed charge. As his income is the same, then water expenditures equals M-y. Graphically we compare $T R_1=p^1 x\left(p^1, M\right)=M-y\left(p^1, M\right)$ and $T R_2=F+p^0 x\left(p^0, M-F\right)=M-y\left(p^0, M-F\right)$. Thus from graph we get $T R_1x^0-x^2$. Thus the second scheme brings more revenue but the first scheme provides greater water conservation.

Proof. As bundles $\left(x^1, y^1\right)$ and $\left(x^2, y^2\right)$ provide the same utility, then the change in quantity demanded is due to Hicksian SE only. We know that own SE is nonpositive. As relative price goes up when we proceed from $x^2$ to $x^1$ and ICs are smooth (due to differentiability of utility function) then $\Delta x^{S E}=x^1-x^2<0$. Thus $x^1y^2$. Thus $T R_1=M-y^1<M-y^2=T R_2$

问题 3.

Dan has utility function $u(w)=\sqrt{w}$, where $w$ is his wealth. All his initial wealth, equal to $\$ 36$, is deposited at bank M. With probability of 0.5 this bank can become bankrupt. Had this happened, Dan would get only $\$ 4$ guaranteed by the government. A risk neutral firm $\mathrm{N}$ proposes Dan to purchase his problem deposit (before the uncertainty is resolved) for $\$ X$.
(a) Find all values of $\mathrm{X}$ that are mutually beneficial for Dan and firm $\mathrm{N}$, provide graphical solution.
(b) Suppose that $\mathrm{X}=20$. A corrupted manager from bank $\mathrm{M}$ possesses information about the bank’s position and can say with certainty whether bankruptcy will take place. He offers Dan to sell this information. What is the maximum amount that Dan is willing to pay for this information? Provide algebraic solution and illustrate your solution on a diagram with contingent commodities.
(c) Suppose that Zara faces exactly the same problem as Dan but she is risk neutral. Find the maximum sum that Zara is willing to pay for the information offered by the corrupted manager described in (b) and compare with the maximum sum that Dan is willing to pay. Illustrate on the same graph.
(d) Compare the maximum prices found in (b) and (c). Would the result of this comparison be different if Dan had different preferences but the same type of risk attitude?

(a) Dan doesn’t reject iff his EU does not go down as a result of this sale: $\sqrt{x} \geq 0.5 \sqrt{4}+0.5 \sqrt{36}=4, x \geq 16$

Firm accepts iff its expected utility (equal to the expected profit due to risk neutrality) is not reduced as a result of this transaction: $0.5 \times 4+0.5 \times 36-x=20-x \geq 0$
Mutually beneficial $20 \geq x \geq 16$
Graph should be provided

(b) With information Dan sells his deposit in case of bankruptcy and gets $20-Q$ ( $Q$ – price of information) and keeps deposit otherwise (in this case his wealth is $36-Q$ ). The resulting expected utility is $E U^{\text {inf }}=0.5 \sqrt{20-Q}+0.5 \sqrt{36-Q}$. Without information he is better off by selling this deposit as price exceeds 16 and his utility is $\sqrt{x}=\sqrt{20}$. Thus he will purchase information iff $E U^{\mathrm{inf}}=0.5 \sqrt{20-Q}+0.5 \sqrt{36-Q} \geq \sqrt{20}$. The maximum price makes Dan indifferent:
\sqrt{36-Q}=2 \sqrt{20}-\sqrt{20-Q} .
Then $36-Q=4 \times 20+20-Q-4 \sqrt{(20-Q) 20}$, which can be rewritten as $4 \sqrt{(20-Q) 20}=100-36=64$. Thus $\sqrt{(20-Q) 5}=8$, which implies $(20-Q) 5=64$. Solving equation we get $Q=20-\frac{64}{5}=7.2$
Graph should be provided

(c) Zara has the same utility function as bank $\mathrm{N}$, so without information she is indifferent b/w selling deposit at $\mathrm{X}=20$ or keeping it. $E U_B{ }^{\text {inf }}=0.5(20-Q)+0.5(36-Q)=28-Q=20$, which gives $Q=8$
Graph should be provided

(d) $Q_{\text {Zara }}=8>7.2=Q_{\text {Dan }}$
General case:
$0.5 u(4)+0.5 u(36)<u(0.5 \times 4+0.5 \times 36)=u(20)$. Thus without information offer of the corrupted manager is still accepted as it gives the same EV: $(36+4) / 2=20$ but with certainty
Maximum price of information should make this agent indifferent:
u(20)=E U^{\text {inf }}(Q)=0.5 u(20-Q)+0.5 u(36-Q)
Due to risk-aversion $0.5 u(20-Q)+0.5 u(36-Q)<u\left(\frac{20-Q}{2}+\frac{36-Q}{2}\right)=u(28-Q)$

It implies that $u(20)=E U^{\text {int }}(Q)=0.5 u(20-Q)+0.5 u(36-Q)<u(28-Q)$
As $u(w)$ is increasing then $20<28-Q, Q_{D a n}<8=Q_

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