A manufacturer has $600 to spend on the production of a certain product and knows that if "x" units of capital and y units of labor are allocated to production, then "P" units will be produced with P(x.y)=120x4/5y1/5, Assume that each unit of capital costs $3 and each unit of labor power costs $5. Use Lagrange multipliers to find how many units of capital and labor power should be used in order to maximize output.
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- Consider the following consumer’s problem: max u(c1 ) + βu(c2) c1c2 subject to c1 + s1 = y1 c2 = y2 + (1 + r)s1 u(c)={1/(1-1/σ}c1-1/σ a) Describe all equations using economic terminology.b) Derive the intertemporal budget constraint for this consumer.c) Set the Langrangean for this problem.d) Find the first order conditions for a maximum.e) What is the effect of an increase in the interest rate on savings? Show in equation and explain.Total Costfor producer= L(i+1)Total output = Q = 10L –L2Price of the output = $1L=Loan amountQ = Quantity of shoes produced. i= Interest ratea. Find the optimal interest rate for the moneylender who wants to maximize profit. Find the optimal loan amount that borrowers want to use to maximize their profit. Assume that if borrowers were not in this business of producing shoes, they would have worked in agriculture and earned an amount $16. [Hint: There are 2 equations and 2 unknowns] b. If moneylenders know that borrowers may default, set up the necessary equations to highlight this situation and graph it. Assume N time periods where N is not very large.QUESTION 1An individual lives for two periods and decides how much to consume in each period.- In the first period his consumption equals C1 and his income Y1 = 200- In the second period his consumption equals C2 and his income Y2 = 100He can save or borrow money in the first period to finance his consumption in the second period.The interest rate he gets in case he saves or has to pay in case he borrows money equals 7%.Determine the budget constraint of this individual. C2 = −0.935·C1 +314C2 =−1.07·C1 +314C2 =−0.8·C1 +314C2 =−1.08·C1 +314 QUESTION 2The total production of a good y is determined by the production function y = 3L2/3K1/3, where L is labour input and K capital input.The reward (factor prices) for labour and capital are, l = 27 en r = 2, respectively.The producer needs to produce 9000 units of good y.How much units of labour will he hire if he wants to miminize his total costs? 1587,4839,953000515,23 QUESTION 3A good is traded on a perfectly competitive…
- You are an economic advisor to the government. Discuss your opinion . a) How COVID-19 pandemic will affect the consumption behavior as well as the investment done by the firms and household for the next two years? b) What are the actions or policies that the government can implement to face this situation? please answers with analysis and --graph (if possible)Write down the Planner’s problem, as well as the Lagrangian. (Hint: There should be two constraints because aggregate consumption of each good cannot exceed the economy’s endowment.) Label the multipliers ϕ1 and ϕ2. What are the first order conditions of the Planner’s problem? (Hint: There should be 6.). Find the solution to the Planner’s problem and label the quantities (xpA; ypA) and (xpB; ypB). If we compare the first order conditions of the Planner’s problem to the first order conditions of a corresponding competitive equilibrium (CE), we can obtain a relationship between ϕx, ϕy, λ, px, and py so that the CE and the Planner’s problem give us the same solution. This is called decentralizing the Planner’s problem.Decentralize the Planner’s problem. (find the equilibrium prices and the value of λ so that the two problems have the same solution.) Can you relate this to the two welfare theorems?Assume an economy producing only two goods (shoes and computers) with a fixed amount ofproductive resources and technology and employing all its productive resources to the maximum.Production in this economy is subjected to the law of diminishing marginal returns and resourcesare assumed to be fully optimized. In addition, the cost of sacrificing shoes for computers andvice versa is 1. On the basis of the foregoing assumptions, answer the following questions:i. Draw the economy’s production possibility frontier on hindsight of the relevantassumption.ii. Why are points outside the frontier unattainable? iii. Identify three ways by which the economy can attain the level of production outside thePPF. iv. What happens to the PPF when technological change overwhelmingly favours theproduction of computers? v. What happens to the PPF when the economy discovers an improved technology forproducing shoes? vi. Assume now that the sacrifice ratio is greater than 1, show what will happen to the…
- Suppose that there are only 10 individuals in the economy each with the following utility function over present and future consumption: U (c1, c2) = c1 +C2, where ci is consumption today, and c2 is consumption tomorrow. Consumption tomorrow is less valued because people are impatient and prefer consuming now rather than later. Buying 1 unit of consumption today costs $1 today and buying 1 unit of consumption tomorrow costs $1 tomorrow. All individuals have income of $10 dollars today and no income tomorrow (because they will be retired) but they can save at the market interest rater> 0. How much of his or her income will an individual consume today given that the interest rate is 0.3? O. Less than half of it O. Exactly half of it O. The individual is indifferent between consuming today and saving O. More than half of it O. All of it O. None of it How much of his or her income will an individual consume today given that the interest rate is 0.5? O. Less than half of it…Consider an economy with one consumption good, 100 identical consumers and 100identical firms. Each consumer is endowed with one unit of time and one unit ofcapital. The agent can spend time either working or enjoying leisure. A represen-tative consumerís utility function is u (x; l) = ln x + 2 ln l, where x is consumptionof goods and l is leisure. Each firm hires consumers to work and rents capital fromconsumers to produce goods: y = f (K; L) = K^1/2L^1/2, where L is the amount oflabor hired and K is the amount of capital rented. Both consumers and firms takegoods price p, wage rate w and rental rate r as given. Normalize p = 1. 1. Set up a representative consumerís utility-maximization problem.Derive the Marshallian demands for consumption and leisure as functions ofwage w and rental rate r 2. Set up a representative firm's profit-maximization problem. Writedown the first-order conditions regarding the choices of K and L 3. Set up all the market-clearing conditions. 4. Use the…Assume an economy produces only two goods ( shoes and computers ) with a fixed amount of productive resources and technology and employing all its productive resources to the maximum. Production in this economy is subjected to the laws of diminishing marginal returns and resources are assumed to be fully optimized. In addition, the cost of sacrificing shoes for computers and vice versa is 1. On the basis of the forgoing assumptions , drawthe economy's production possibility frontier on hindsight of the relevant assumption and why the points outside the frontier unattainable?.
- Assume an economy producing only two goods (shoes and computers) with a fixed amount of productive resources and technology and employing all its productive resources to the maximum.Production in this economy is subjected to the law of diminishing marginal returns and resourcesare assumed to be fully optimized. In addition, the cost of sacrificing shoes for computers andvice versa is 1. On the basis of the foregoing assumptions, answer the following questions: i. Draw the economy’s production possibility frontier on hindsight of the relevantassumption.please solve it completely. Bob has preferences over consumption in period 0 and 1 of the form U(x, y) = xy, where x and y are Bob's consumption in period 0 and 1 respectively. He has $15,000 in the bank now and is trying to decide between two different investment opportunities, A and B. A: invest $10,000 in period 0 and receive $20,000 in period 1. B: invest $2,000 in period 0 and receive $6,000 in period 1. If Bob can borrow and lend at a rate of interest of 50 percent, which investment opportunity will he choose? Given your answer in (a), how much will he consume in each period if the price of the good is $1 in both periods? Given your answer in (a), how much will he consume in each period if the price in period 0 is $1 and the inflation is 20%? Assuming that the price of consumption is $1 in both periods and the borrowing rate is 50% and the lending rate is 100%. Given your answer in (a), how much will he consume in each period?Assume that someone has inherited 2,000 bottles of wine from a rich uncle. He or she intends to drink these bottles over the next 40 years. Suppose that this person’s utility function for wine is given by u(c(t)) = (c(t))0.5, where c(t) is each instant t consumption of bottles. Assume also this person discounts future consumption at the rate δ = 0.05. Hence this person’s goal is to maximize 0ʃ40 e–0.05tu(c(t))dt = 0ʃ40 e–0.05t(c(t))0.5dt. Let x(t) represent the number of bottle of wine remaining at time t, constrained by x(0) = 2,000, x(40) = 0 and dx(t)/dt = – c(t): the stock of remaining bottles at each instant t is decreased by the consumption of bottles at instant t. The current value Hamiltonian expression yields: H = e–0.05t(c(t))0.5 + λ(– c(t)) + x(t)(dλ/dt). This person’s wine consumption decreases at a continuous rate of ??? percent per year. The number of bottles being consumed in the 30th year is approximately ???