Suppose the economy has 100 units of labor (L) and 100 units of capital (K), and it chooses to allocate three times more labor and capital in the production of good X than in good Y. Suppose also that the production of goods X and Y are represented by the following functions: X = Lx0.25Kx0.25 Y = 2LY0.25KY0.25 Compute for Lx, Ly, Kx and Ky Calculate the MRTSLK of each firm Is this allocation efficient? How so?

Suppose the economy has 100 units of labor (L) and 100 units of capital (K), and it chooses to allocate three times more labor and capital in the production of good X than in good Y. Suppose also that the production of goods X and Y are represented by the following functions: X = Lx0.25Kx0.25 Y = 2LY0.25KY0.25 Compute for Lx, Ly, Kx and Ky Calculate the MRTSLK of each firm Is this allocation efficient? How so?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.4P

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Suppose the economy has 100 units of labor (L) and 100 units of capital (K), and it chooses to allocate three times more labor and capital in the production of good X than in good Y. Suppose also that the production of goods X and Y are represented by the following functions: X = LX0.25KX0.25; Y = 2LY0.25KY0.25. Compute for Lx, Ly, Kx and Ky Calculate the MRTSLK of each firm Is this allocation efficient? How so? How many unit of X and Y will be produced? Suppose the utility function of the two individuals in the economy is given by UA = 2XA0.5YA0.5 ; UB = XB0.5YB0.5 and X and Y are allocated equally between the two individuals. Compute for XA, YA ,XB ,YB Calculate the MRSXY of each individual Is the allocation efficient? How so? Determine if at the current allocation, the efficiency in optimal mix attained. Compute for MRTXY What is the MRSXY? Is optimal mix efficiency attained? How so?
Problem 1: Winter melons A farmer is using land and fertilizer to produce winter melons. The production function is q = L√F. The price of one unit of land is 3, the price of unit of fertilizer is 1. The farmer is currently using 4 units of land and 4 units of fertilizer. b) Suppose now that the farmer cannot modify the amount of land but can modify the amount of fertilizer. Write the cost function. c) The farmer can bring the winter melons to a market and sell them at price 1. Find the optimal quantity to produce. Show that the profit is negative. d) Suppose that now the farmer can buy more land. Do you think that, after buying sufficiently many units of land, the farmer can eventually make a positive profit on the market?
A country's production is described by the function 2 1 f(x, y) = 30x³y3 units, when x units of labor and y units of capital were used. a) Compute and f,- b) What is the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 100 units and 30 units, respectively? c) In order to increase the country's productivity, should the government have encouraged capital investment rather than increasing expenditure on labor?
Robinson Crusoe has exactly 12 hours per day to spend gathering coconuts ( C) or catching fish (F). Let tp denote the time (in hours) spent catching fish and tc the time (in hours) spent gathering coconuts per day. Robinson can catch 3 fish per hour or he can gather 6 coconuts per hour; his "production technology" can thus be described by: F = 3tp and C = 6tc. Robinson's utility function is U(F,C) = FC where F is his consumption of fish and C is his consumption of coconuts. 1. (a) Write down Robinson's time constraint and hence find an equation for his Production Possibility Frontier (PPF). (b) Find the number of fish Robinson will catch per day and the number of coconuts he will gather per day.
what are the two distinct input bundles (P, W) in the diagram which give q = 48?
Suppose a soap-manufacturing production process is described by the following equation: Y = a + b log K + с log L Where, Y= Output (number of soaps produced) K=Capital L=Labor a, b and c are constants Suppose 0<a<1, 0< b<1 a. Find the Marginal Product of Labor (MPL) and Marginal Product of Capital (MPK) in the production of soap b. Is MPL diminishing, increasing or constant as L increases? c. Is MPK diminishing, increasing or constant as K increases?
Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used () and the amount of capital used () by the following function: S- 20 L0 C0.7 In this formula L represents the units of labor input and Cthe units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 50,000 tons of steel at minimum cost. If your answer is zere, enter "p" Min 40 120 s.t. 20 S0000 (b) Solve the optimization problem you formulated in part a. Hint: When using Excel Solver, start with an initial >oand C0 Do not round intermediate calculations. Round your answers to the nearest whole number 2991 2318 Cost S 397448 ide Feedback
The production of a product is characterised by the function f(x, y) = 16x/4y3/4, where x is the number of units of labour and y is the number of units of capital needed to produce f(x, y) units of the product. Each unit of labour costs £50 and each unit of capital costs £100. You are given a budget of £500,000 and are asked to calculate the number of units of labour that maximise the production. Enter your answer as an integer. Answer:
A T-shirt screener can screen t-shirts (q) in two different ways. He can either use a fast screening machine (F) or a slow screening machine (S). Screen use is defined in terms of ”hours” running. His production function is f(F, S) = 10F + 6S. (a) The screener wants to be able to produce 120 shirts. List three feasible and efficient production plans (combinations of inputs) for doing this. (b) Graph the screener’s isoquant curve for q = 120. (c) The hourly cost of using the fast machine is $800 and the hourly cost of the slow machine is $200. What is the cost minimizing (optimal) combination of inputs for producing 120 shirts? (d) Suppose the screener must now produce 400 shirts. What is the cost minimizing combination of inputs?
An economy depends on two basic products, wheat and oil. To produce 1 metric ton of wheat requires 0.25 metric tons of wheat and 0.33 metric tons of oil. Production of 1 metric ton of oil consumes 0.08 metric tons of wheat and 0.11 metric tons of oil. Find the production that will satisfy a demand for 540 metric tons of wheat and 870 metric tons of oil. The input-output matrix is A= 0.25 0.08 0.33 0.11 .
Problem 1: Winter melons A farmer is using land and fertilizer to produce winter melons. The production function is q = L√F. The price of one unit of land is 3, the price of unit of fertilizer is 1. The farmer is currently using 4 units of land and 4 units of fertilizer. a) Would it be possible to produce the same number of winter melons at a lower cost? In particular, would the farmer save some money by replacing a little bit of land with the needed amount of fertilizer to keep production constant? Or by doing the other way round? b) Suppose now that the farmer cannot modify the amount of land but can modify the amount of fertilizer. Write the cost function. c) The farmer can bring the winter melons to a market and sell them at price 1. Find the optimal quantity to produce. Show that the profit is negative.
Prof. Smith and Prof. Jones are going to produce a new textbook. The production function for the book is: ?=?1/2?1/2?= is the number of pages in the finished book?= is the number of working hours spent by Smith?= is the number of working hours spent by JonesSmith's labor is valued at 3 TL per working hour and Jones's labor is valued at 12 TL per working hour. After having spent 900 hours preparing the first draft, Smith cannot contribute any more to the book. Jones will revise the Smith's draft to complete the book.a) How many hours will Jones have to spend to produce a finished book of 300 pages?b) What is the marginal cost of the 300th page of the finished book?