Assignment-daixieTM为您提供伯明翰大学University of Birmingham Microeconomics 08 28536微观经济学代写代考和辅导服务！
Instructions:
Microeconomics is a branch of economics that focuses on the behavior and decision-making of individuals, households, and firms in markets. It examines how these economic agents allocate their limited resources to meet their needs and wants, and how they interact with one another in buying and selling goods and services.
Microeconomics also looks at how market forces of supply and demand affect prices and quantities of goods and services, as well as how different market structures, such as perfect competition, monopolistic competition, oligopoly, and monopoly, impact the behavior of firms and consumers.
In essence, microeconomics provides insights into how individuals and businesses make decisions about what to produce, how much to produce, and at what price to sell, and how these decisions affect the allocation of resources in society.
Consider a pure exchange economy with two goods, $h=1,2$ and two consumers, $i=1,2$ with utility functions $u_1$ and $u_2$ respectively. The total endowment is $e=\left(e_1, e_2\right)$ where $e_1, e_2 \gg 0$. For each of the following cases, determine which of the Pareto-efficient allocations can be decentralized as competitive equilibria with lump sum transfers. Briefly describe the equilibrium prices and transfers for each Pareto-efficient allocation. a) $u_1(x, y)=\alpha \ln (x)+(1-\alpha) \ln (y)$ and $u_2(x, y)=\beta \ln (x)+(1-\beta) \ln y$, where $\alpha<\beta$ and $\ln x$ is the natural logarithm of $x$.
a) In this case, the Pareto-efficient allocation is the one where consumer 1 consumes only good 1 and consumer 2 consumes only good 2, i.e., $(x_1^, y_1^)=(e_1, 0)$ and $(x_2^, y_2^)=(0, e_2)$. This allocation cannot be decentralized as a competitive equilibrium with lump-sum transfers, as the competitive equilibrium requires that the prices of both goods be equal, but the two consumers have different marginal rates of substitution, so the equilibrium quantities cannot both satisfy this equality.
b) $u_1=u_2$ is strictly concave, smooth, and homothetic. c) For $i=1,2, u_i(x, y)=x+g(y)$, where $g$ is an increasing and strictly concave function.
b) In this case, the Pareto-efficient allocation is the one where the two consumers consume equal amounts of both goods, i.e., $x_1^=x_2^=\frac{e_1+e_2}{2}$ and $y_1^=y_2^=\frac{e_1+e_2}{2}$. This allocation can be decentralized as a competitive equilibrium with lump-sum transfers, as the two consumers have the same preferences and therefore the same marginal rate of substitution, so the equilibrium quantities will satisfy the equality of the two goods’ prices. The equilibrium prices will be such that the marginal rates of substitution of the two consumers are equal, and the transfers will be such that each consumer receives half of the aggregate endowment.
c) For $i=1,2, u_i(x, y)=x+g(y)$, where $g$ is an increasing and strictly concave function.
c) In this case, the Pareto-efficient allocation is the one where consumer 1 consumes all of good 1 and consumer 2 consumes all of good 2, i.e., $(x_1^, y_1^)=(e_1, 0)$ and $(x_2^, y_2^)=(0, e_2)$. This allocation cannot be decentralized as a competitive equilibrium with lump-sum transfers, as the two consumers have different preferences and therefore different marginal rates of substitution, so the equilibrium quantities cannot both satisfy the equality of the two goods’ prices.
