3. Consider an economy with two individuals, Eve and Adam. There are twogoods-apples(A) and figs (F). Eve has fixed-coefficient preferences{}where the superscript denotes e for Eve. Eve has an endowment of 20 apples andnothing else. Adam owns only labour, and consumes only apples, thusua (A, F) = A.u(A, F) = minNote that Adam gets no utility from leisure, so he supplies labour inelastically.Labour (L) is needed to produce figs, using the technologyF = L.Suppose initially the endowment of Adam is K units of labour.(a) Compute algebraically the competitive equilibria of this economy for bothK< 20 and K ≥ 20.(b) Depict the solutions for each case graphically.(c) Now suppose the endowment of Adam goes up to K' > K. Is it possible that hebecomes worse off?

Eva utility=min(A/2,F/2)Endowment =20Adam utilityU=AAdam production function of figF=L

3. Consider an economy with two individuals, Eve and Adam. There are two goods-apples(A) and figs (F). Eve has fixed-coefficient preferences u(A, F) = min. where the superscript denotes e for Eve. Eve has an endowment of 20 apples and nothing else. Adam owns only labour, and consumes only apples, thus

3. Consider an economy with two individuals, Eve and Adam. There are two goods-apples(A) and figs (F). Eve has fixed-coefficient preferences u(A, F) = min. where the superscript denotes e for Eve. Eve has an endowment of 20 apples and nothing else. Adam owns only labour, and consumes only apples, thus

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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2. The standard model of consumer behavior assumes that income is exogenous. Of course, a person's income usually depends upon the number of hours the individual works. Suppose that an individual has T hours each day which can be allocated toward working time, H, or leisure time, L-that is, T = H + L. The individual earns w dollars for each hour worked. Then the individual's income is M + wH where M denotes any non-labor income. The individual has preferences over leisure time, L, and a consumption good, X, which can be represented by a quasi-concave utility function, U(X, L). The consumption good, X, can be purchased at the price Px. The individual seeks to maximize the utility subject to the constraints on time and money. (1) Formulate the individual's problem as an optimization problem with two constraints a time constraint and a money constraint. Derive the first-order conditions. Give an economic interpretation of the Lagrange multipliers. (2) The two-constraints problem can be…
7. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and Ya are A's endowments, Xb and Yb are B's, and Xo = Xa + Xb, Yo = Ya + Yb), the competitive equilibrium allocation of the two commodities represents a mutual tangency of both consumers' indifference curves with each other and with a common budget line. (a) The conditions require that A and B consume at a point on their budget lines where their indifference curves have the same marginal rate of substitution, and equilibrium for the market as a whole requires that the sum of the individuals' demands for each commodity must equal the totals (Xo and Yo) available. Mathematically express the point of equilibrium, that is, express the above statement with equations in relating each endowments, equilibrium points (Xa, Xb, Ya, Yb), and totals ((Xo, Yo). (b) Suppose MRSa = Ya/Xa, MRSB = Yb/Xb, Xa = 10, Ya, = 100, Xb = 50, Yb = 20, and let good Y be the numeraire (Py =1). Verify that the competitive…
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= 5. Consider an economy with a single (representative) agent with utility function u(x, lc) = x¹/514/5 and an endowment of 0 units of x and 10 units of time, which the agent can use as leisure (c) or labor (L) (i.e., L + lc = 10). The agent owns a firm that produces good x using L as an input, with technology of production given by x(L) = 3√L. Let the price of x be p = 1 and let w denote the price of time (i.e., the price of leisure and the wage). Find the competitive price of time w and the competitive allocation.
6. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and d Ya are A's endowments, Xb and Yb are B's, and Xo = Xa + Xb, Yo = Ya + Yb), the competitive equilibrium allocation of the two commodities represents a mutual tangency of both consumers' indifference curves with each other and with a common budget line. (a) The conditions require that A and B consume at a point on their budget lines where their indifference curves have the same marginal rate of substitution. and equilibrium for the market as a whole requires that the sum of the individuals' demands for each commodity must equal the totals (Xo and Yo) available. Mathematically express the point of equilibrium, that is, express the above statement with equations in relating each endowments, equilibrium points (Xa, Xb, Ya, Yb), and totals ((Xo, Yo).
5.18 In a two-good, two-consumer economy, utility functions are u¹ (x₁, x₂) = x₁(x₂)², u² (x₁, x₂) = (x₁)²x₂. Total endowments are (10, 20). (a) A social planner wants to allocate goods to maximise consumer 1's utility while holding con- sumer 2's utility at u² = 8000/27. Find the assignment of goods to consumers that solves the planner's problem and show that the solution is Pareto efficient. (b) Suppose, instead, that the planner just divides the endowments so that e¹ = (10, 0) and e² = (0, 20) and then lets the consumers transact through perfectly competitive markets. Find the Walrasian equilibrium and show that the WEAs are the same as the solution in part (a).
6. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and d Ya are A's endowments, Xb and Yb are B's, and Xo = Xa + Xb, Yo = Ya + Yb), the competitive equilibrium allocation of the two commodities represents a mutual tangency of both consumers' indifference curves with each other and with a common budget line. (b) Suppose MRSa = Ya/Xa, MRSb = Yb/Xb, Xa = 10, Ya, = 100, Xb= 50, Yb = 20, and let good Y be the numeraire (Py = 1). Verify that the competitive equilibrium price is Px = 2.
6.
In an environment in which all education is privately provided, suppose that a family with one child has $20,000 per year to spend on private goods and education. Draw this family’s budget constraint. Suppose now that an option of free public education with spending of $4,000 per pupil is introduced to this family. Draw three different indifference curves corresponding to the following three situations: (a) a free public education would increase the amount of money that is spent on the child’s education; (b) a free public education would decrease the amount of money that is spent on the child’s education; (c) a free public education would not affect the amount of money spent on the child’s education.