Advanced Microeconomics Problem Set 4

What condition must be satisfied for a Nash Equilibrium to exist in a first-price auction?

How do reaction functions illustrate bidding behavior in auctions?

Define efficiency in the context of first-price auctions.

What bidding strategy is typically employed by risk-averse bidders in first-price auctions?

How does the concept of a bid shading apply to first-price auction strategies?

What role does information asymmetry play in first-price auctions?

Explain how the expected utility theory influences bidding strategies in auctions.

What is the significance of the first-order condition in determining optimal bidding in first-price auctions?

What does the increasing nature of the bidding function b(v) imply about bidders' strategies in a first-price auction?

How is the expected revenue from a first-price auction (RFPA) affected by the number of bidders N, according to the given derivation?

What is the condition for a Nash equilibrium in the context of agents contributing in a public goods setting?

In the context of auction theory, explain the significance of the expected revenue formula for the second-price auction (RSPA).

What role does the uniform distribution of bidders' values play in the analysis of Nash Equilibrium in auction settings?

Describe the reaction function for agent i when the contribution of agent j is known.

What does it imply if $g_1^* > 0$ and $g_2^* > 0$ cannot be a Nash equilibrium?

Describe how reaction functions might differ between first-price and second-price auctions.

In a first-price auction, how does the bidding function behave as bidders' values increase?

What is the bidding function for two bidders in a first-price auction?

As the number of bidders N increases, how does the bid at a fixed valuation v change?

What is the behavior of the optimal bid as the number of bidders approaches infinity in a first-price auction?

Explain the concept of efficiency in auctions in relation to bidder behavior.

What strategy might a bidder employ in a first-price auction to maximize their chances of winning?

How do reaction functions illustrate the strategic interdependence between agents in a Nash equilibrium?

Explain the reason each agent has an incentive to drive too fast from a social perspective.

What optimal fine should be imposed on agent i in case of an accident to align incentives?

How are total costs, including fines, assessed when optimal fines are imposed on agents?

In a scenario where agent i receives utility only with no accident, what would be the appropriate fine?

Identify the Nash Equilibrium conditions in the context of agents’ driving speeds and accident probabilities.

Describe how reaction functions can be applied to the driving speeds of the agents.

Discuss the implications of efficiency in auctions related to the externality in driving speeds.

What bidding strategies might agents employ in a first-price auction, analogous to driving decisions?

Explain the relationship between individual utility and the social cost of accidents in the context of externalities.

How does the concept of externalities complicate the decision-making process for agents driving cars?

What is the condition that an agent must satisfy to reach a private optimum in contributions to a public good?

In the context of public goods, how can we express the efficient amount of the public good, $G^*$?

What are the reaction functions of the agents for contributing to the public good?

Describe the Nash equilibrium in the context of public good contributions.

How does the efficiency of public good provision relate to private contributions?

What is the implication of the first-order condition (FOC) in the optimal contribution problem?

How is utility maximization represented mathematically in the context of public goods?

Why are public goods often undersupplied in a private provision scenario?

What role do bidding strategies play in the efficiency of auctions for public goods?

How do externalities relate to the provision of public goods in auction settings?

The faster agents drive, the probability of accidents decreases.

Each agent has a social incentive to drive slower to minimize utility losses.

The optimal fine to be imposed on agent i in case of an accident is dependent on the cost of the accident to the agent.

If both agents act independently regarding their driving speeds, they are likely to achieve an optimal social outcome.

The utility function of agent i becomes negative if an accident occurs.

The total costs incurred by agents, including fines, can sometimes exceed the total cost of the accidents.

The analysis of the accident probabilities does not account for the speeds chosen by agents.

Optimal fines should be set at zero to encourage agents to drive faster.

The derivative of the bidding function b(v) with respect to v is negative.

As the number of bidders N approaches infinity, b(v) tends to 0.

The expected revenue from a first-price auction decreases as the number of bidders increases.

In a second-price auction, each bidder's value is uniformly distributed on [0, 1].

The bidding function b2(v) is equal to v^2.

The expected revenues from the first-price auction and second-price auction can be calculated using the functions derived in the content.

The derivative of b(v) with respect to N produces a negative value.

B(v) is defined to be decreasing as bidders submit higher bids.

In the auction model, f(x) remains constant regardless of the value of x.

The function F(v) is defined to be a decreasing function.

The relationship RFPA = RSPA shows that expected revenue in a first-price auction equals that in a second-price auction.

The equation $RFPA = N \int_0^1 b(v)f(v)F_{N-1}(v) dv$ indicates that RF PA depends solely on the function $f(v)$ and not on other variables.

In the provided context, the equation $rac{N(N-1)}{x} = RSPA$ represents a valid derivation of revenue.

Changing the order of integration in the equation for RFPA can yield the integral representation $RSPA = N (N-1) \int_0^1 x f(x)F_{N-2}(x) dx$.

The given equations suggest that if $b(v) = 12v$, then the functional form of RFPA would necessarily be linear.

In a Nash equilibrium, both agents have to be best-responding to each other's actions, meaning that $g_1^$ and $g_2^$ cannot both be equal to zero.

The reaction function for agent i indicates optimal contributions that depend solely on the contributions of agent j.

If $g_1^* > 0$ and $g_2^* > 0$ were possible in a Nash equilibrium, it would imply that $b_1 - b_2 = 0$.

An increase in the number of bidders in a first-price auction leads to a decrease in the equilibrium bid at a fixed valuation $v$.

The optimal contribution $g_1^$ in a public goods setting is given by the condition $g_1^ = b_1$ when $g_2^* = 0$.

The first-order condition (FOC) for maximizing utility in contributions to a public good is represented by $-1=0$.

If $g_2^* = R_2(0) = b_2$ and $g_1^* > 0$, it follows that $R_1(b_2)$ must be less than or equal to $0$ for equilibrium to hold.

In the context of first-price auctions, the bidding function can be considered increasing in $v$ for $N = 3$ bidders.

A scenario where $g_1^* = g_2^* = 0$ represents an efficient outcome in public goods provision.

The uniform distribution of bidders' values in an auction ensures that every bidder has an equal probability of winning regardless of their bid.

The optimal contribution to a public good is determined by the equation $G^* = b_1 + b_2$.

In maximizing utilities, the conditions $g_1 + x_1 = heta_1$ and $g_2 + x_2 = heta_2$ can both be negative.

The first-order conditions yield $b_1 + b_2 = g_1 + g_2$ indicating an equilibrium.

The reaction functions $R_i(g_j)$ illustrate the optimal contribution of agent $i$ based on their own contribution only.

If the costs are structured such that $u_i(x_i) - c_i = 0$, agent $i$ is at a private optimum.

The sum of individual utilities is maximized when all agents contribute $g_1$ and $g_2$ regardless of their income constraints.

An increase in $b_1$ or $b_2$ leads to a higher total efficient contribution to the public good.

Each agent's contribution is unaffected by the other agent's decision in determining the public good amount.

The expression $rac{eta_1}{g_1 + g_2} - 1 = 0$ elucidates the optimal reaction function.

In the equation $G = g_1 + g_2$, $G$ can remain positive even if both $g_1$ and $g_2$ are below zero.

The value function is represented as b(v) = 12 ______.

In the formula for RFPA, the integration runs from 0 to ______.

The result of the calculation shows that RFPA is equal to ______ PA.

The equation RSP A involves evaluating the integral of v(1 - ______) dv.

The expression involves changing the order of ______ in the integration process.

Agent i's utility is a function of their speed, represented as u_i(xi), where u_i'(xi) > 0 and u_i''(xi) < 0, indicating that the utility is ______ in speed.

The probability of an accident, p(x1, x2), is ______ in each argument, indicating that higher speeds increase the likelihood of an incident.

If agent i is fined ti in the case of an accident, the optimal fine should account for the ______ imposed on agent i.

Each agent has an incentive to drive too fast from the social point of view, which leads to a ______ in total costs from accidents.

In a situation where agent i only receives utility if there is no accident, the appropriate fine should be directly tied to their ______.

The social problem includes maximizing the sum of the utilities of the two agents, written as max u_i(xi) + u_j(xj) minus the probability of accidents times the total ______.

The optimal driving speed xi for agent i must satisfy the first-order condition, which includes the derivative of the accident probability and the ______ adjusted for the costs incurred.

When assessing total costs, including fines, it is important to compare them against the total cost of the ______ to evaluate economic efficiency.

The bidding function b(v) is calculated by integrating xf(x)F(N-2)(x)dx from 0 to v, resulting in b(v) = ______/v ∫ x · x^(N−2) dx.

As N approaches infinity, the bidding function b(v) approaches _____ according to the limit.

The expected revenue for the first-price auction (R______) is given by N times the integral of b(v)f(v)F(N−1)(v)dv from 0 to 1.

Taking the derivative of b(v) with respect to v shows that the function is _____ in nature.

For two bidders uniformly distributed on [0, 1], F(v) is equal to _____ when calculating expected revenues.

The derivative of b(v) with respect to N indicates that an increase in N results in an increase in the bid because _____ is greater than 0.

The bidding function for two bidders in a first-price auction is given by _____(v), where each bidder's value is uniformly distributed.

When calculating expected revenues, the formula for the second-price auction involves the integral of vf(v)(1 − F(v))F(N−2)(v) dv from 0 to 1, represented as _____.

The term F(v) = v indicates that the bidder's values are _____ distributed.

In the expression for b(v), xf(x)F(N−2)(x) dx is integrated from 0 to v to find the overall _____ of the bidding function.

In a Nash equilibrium, both agents have to be best-responding to each other's ______.

The reaction function for agent i is given by Ri (gj ) = max {0, bi − ______}.

If both g1∗ > 0 and g2∗ > 0 cannot be Nash equilibria, then g1∗ = g2∗ = ______ may also not be an equilibrium.

In a first-price auction, the equilibrium bidding function is derived by analyzing how bids relate to ______.

As the number of bidders N increases, the bid at a fixed valuation v may ______.

The optimal bid as N approaches infinity tends to ______.

Private provision of public goods is often not efficient, leading to ______ behavior by agents.

In order to reach a private optimum in contributions to a public good, an agent must satisfy a certain ______.

The cumulative distribution function of bidders' values in a uniform distribution is represented as ______ = v.

The first-order condition (FOC) plays a crucial role in determining optimal ______ in auction settings.

The optimal xi has to satisfy the condition that the derivative of the utility function minus the cost function is equal to ______.

The efficient amount of public good is denoted as ______.

In maximizing the sum of utilities, the expression is written as max u1(G, x1) + u2(G, x2) subject to G = g1 + g2 and g1 ______ 0.

The optimal contributions of agents to the public good can be represented as reaction functions, denoted by ______.

The first-order condition (FOC) in the optimal contribution problem states that b1/(g1 + g2) -1 = ______.

When substituting for x1 and x2, we replace them with ω1 - g1 and ______, respectively.

To find the equilibrium contributions, one must analyze the contributions g1 and g2 with respect to the parameters ______, b1 and b2.

In a Nash equilibrium, the contributions of the agents become stable when ______ is equal to zero.

The equation G* can be expressed as the sum of the parameters ______ and b2.

The social optimal level of public good provision can often be achieved through aligning individual contributions with ______.