1. Consider an economy where the production function takes the form Y = z(N + K), where N is labor and K is capital. K is exogenous and fixed at the level Ko: K = Ko. The total time endowment of the consumer in this economy is h, so N + 1 = h, where is leisure. The consumer has preferences over consumption and leisure that can be represented by convex indifference curves. The government in this economy has spending/consumption G, fixed exogenously. Assume that zKo > G. Also assume that the first welfare theorem holds for this economy. 1) On the graph draw the PPF (aggregate resource constraint) for this econ- omy. Make sure to label axes and the points of intersection of PPF with both axes. If the PPF has a kink you should clearly mark the x- and y-coordinates of the kink. (Recall, leisure should be on the horizon axis and consumption on the vertical axis). On the same graph show where is the competitive equilibrium for this economy (denote it A). 2) Assume now that the total factor productivity increases to the new level Znew. On the same graph, draw the new PPF for this economy. Make sure to label the points of the intersection of the new PPF with both axes. If the new PPF has a kink you should clearly mark the x- and y-coordinates of the kink. On the same graph, show where is the new competitive equilibrium for this economy (denote it B). What will happen to the consumption in the new equilibrium (increases/decreases)? Provide intuition. 3) Reproduce the original graph from part 1) of the problem. Make sure to place point A (competitive equilibrium) on the graph. Assume now that the government consumption G increases to the new level Gnew. Assume that zKo Gnew. On the same graph, draw the new PPF for this economy. Make sure to label the points of the intersection of the new PPF with both axes. If the new PPF has a kink you should clearly mark the x- and y-coordinates of the kink. On the same graph show where is the new competitive equilibrium for this economy (denote it D). What will happen to the consumption in the new equilibrium (increases/decreases)? Provide intuition.

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…

a) A planner in a small, open economy has a utility function U = x1X2 and production takes place according to the production functions Y1 = 10L, – 0.5L Y2 = 10L2 – L The world prices of the goods are p1 10 and p2 5. The total amount of labour %| %3D available in the economy is 5 units. (i) Formulate the Lagrange equation. (ii) Write down the Karush-Khun-Tucker conditions. (iii) Determine a possible solution.

11. A worker derives utility from consumption, C and leisure, L U = U(C, L) and initially faces the constraints p.C = N+w.H H = 24 - L where p is the nominal price of consumption, N> 0 is unearned income, w is the nominal wage and H is hours worked. (a) Derive the budget constraint in terms of C and L and show on a diagram an initial optimal choice of C and L, in which both are positive. (b) State this equilibrium algebraically, and briefly explain how it can be derived. (c) Show that, assuming normality of leisure, for some value N*, if N> N*, the individual will choose not to work. Is the initial level of N in your answers to parts a) and b) greater or less than N*?

3. Consider a parent who is altruistic towards her child, but also cares about her own consumption. The parent's utility over her own consumption and that of her child is up = log(co) +a log(ci) where c is the child's consumption, and a > 0 is the degree of parental altruism. Suppose that the parent can invest in the child's human capital by spending money (e) on her education; education generates human capital h /() and human capital is paid at rate w. The parent has a total income of (a) Write down an expression for the child's future consumption in terms of the parent's choice of e. (b) Now write down the Lagrangian for the parent's decision problem.

Consider a model in which individuals live for two periods and have utility functions of the form U = In(C1) + In(C2). They earn income of $100 in the first period and save S to finance consumption in the second period. The interest rate, r, is 10%.  1. Set up the individual's lifetime utility maximization problem. Solve for the optimal C1, C2, and S. (Hint: Rewrite C2 in terms of income, C1, and r.) Draw a graph showing the opportunity set. (5 marks) 2. The government imposes a 20% tax on labor income. Solve for the new optimal levels of C1, C2, and S. Explain any differences between the new level of savings and the level in part (a), paying attention to any income and substitution effects.  3. Instead of the labor income tax, the government imposes a 20% tax on interest income. Solve for the new optimal levels of C1, C2, and S. (Hint: What is the new after tax interest rate?) Explain any differences between the new level of savings and the level in a, paying attention to any income…

Answer please with added diagrams      b) Assume that your optimal choice of consumption and leisure is to work 8 hours per day. Illustrate this choice diagrammatically using the feasible set and indifference curves. c) Use indifference curves and the feasible set to show why, given the properties of the optimal choice in part b it is not optimal to work, say, 10, or 6 hours per day.

question ii

b. Using Fisher's Intertemporal Choice model, consider the following scenario: Suppose Milo earns $1,750 in the first period and $2,500 in the second period. If he consumes $1,200 in the first period and $1,550 in the second period, what is the i. interest rate? ii. Now if Milo's consumption changes to $1,800 in the first period and $2,000 in the second period, what is the new interest rate?

u (#1, #2) = x}x; If the price of good 1 is $4/unit, the price of good 2 is $9/unit, and income = $164... What is this person's optimal consumption level for good 2?