Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 13: Problem 3

Consider an economy with just one technique available for the production of each good. \\[ \begin{array}{lcc} \text { Good } & \text { Food } & \text { Cloth } \\ \hline \text { Labor per unit output } & 1 & 1 \\ \text { Land per unit output } & 2 & 1 \\ \hline \end{array} \\] a. Suppose land is unlimited but labor equals 100 . Write and sketch the production possibility fronticr. b. Suppose labor is unlimited but land equals \(150 .\) Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals \(150 .\) Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility frontier? When is land fully employed? Labor? Both? Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of \(p_{F} / p_{C}=1.1\) as at \(p_{F} / p_{C}=1.9\) h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and \(0.9,\) respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case.

Short Answer

Expert verified
Answer: Introducing capital as a third factor of production changes the production possibility frontier by introducing additional constraints on the production of Food and Cloth. The PPF will still be concave, but the endpoints and shape will be determined by the availability of capital, along with labor and land. The relative price of Food as a function of its output will continue to have an increasing relationship, but the curve may change due to the capital requirements for producing each good.

Step by step solution

01

Understanding the given table

The table presents the labor and land requirements for producing one unit of each good. For example, producing one unit of Food requires one unit of labor and two units of land.
02

a) Production possibility frontier with unlimited land and limited labor

Since labor is limited to 100 units and land is unlimited, we can start by determining how much of each good can be produced using all available labor. If all labor is allocated to Food production, the economy can produce 100 units of Food (100 units of labor divided by 1 unit of labor per Food). Similarly, if all labor is allocated to Cloth production, the economy can produce 100 units of Cloth. This defines the endpoints of the line of the production possibility frontier. Sketch this line on a graph with Food on the horizontal axis and Cloth on the vertical axis.
03

b) Production possibility frontier with unlimited labor and limited land

Since land is limited to 150 units and labor is unlimited, we can start by determining how much of each good can be produced using all available land. If all land is allocated to Food production, the economy can produce 75 units of Food (150 units of land divided by 2 units of land per Food). Similarly, if all land is allocated to Cloth production, the economy can produce 150 units of Cloth. This defines the endpoints of the line of the production possibility frontier. Sketch this line on a graph with Food on the horizontal axis and Cloth on the vertical axis.
04

c) Production possibility frontier with limited labor and land

Given that both labor and land are limited (100 and 150 units, respectively), we need to consider both resources at the same time. The intercepts are the points where only one of the goods is produced. If all labor is allocated to Food production, the economy can produce 100 units of Food. If all land is allocated to Cloth production, the economy can produce 150 units of Cloth. Alternatively, if all land is allocated to Food production, the economy can produce 75 units of Food. And if all labor is allocated to Cloth production, the economy can produce 100 units of Cloth. The production possibility frontier in this case is the concave curve connecting these points. Sketch this curve on a graph with Food on the.horizontal axis and Cloth on the vertical axis.
05

d) Explaining the concavity of the production possibility frontier

The concavity of the production possibility frontier of part (c) is due to the increasing opportunity cost of producing one good as more of that good is produced. This is because the resources (land and labor) are not evenly divisible between the two goods. As more of one good is produced, both labor and land are allocated more unequally between the two goods, resulting in a lower marginal output of the good with less resource allocation.
06

e) Sketching the relative price of Food as a function of its output in part (c)

When sketching the relative price of Food as a function of its output, consider that the slope of the production possibility frontier represents the opportunity cost of producing Food in terms of Cloth. Since the PPF has a concave shape in part (c), the slope becomes steeper as more Food is produced, indicating that the relative price of Food increases as more Food is produced. Sketch a curve showing the relationship between the output of Food and its relative price, with Food output on the horizontal axis and the relative price on the vertical axis.
07

f) Relative price of Food when trading 4 units of Food for 5 units of Cloth

If consumers insist on trading 4 units of Food for 5 units of Cloth, they value one unit of Food at 1.25 units of Cloth. The relative price of Food is \(\frac{1.25}{1} = 1.25\). Consumers are expressing their preference for the trade ratio in the form of exchange rates, which directly reflects the relative price of the goods.
08

g) Production at price ratios of \(p_{F} / p_{c} = 1.1\) and \(p_{F} / p_{c} = 1.9\)

The production levels in both cases would be exactly the same because the concave shape of the PPF in part (c) causes the opportunity cost and relative prices to rise as production increases. For both ratios, producers would choose to produce on the PPF where the opportunity cost equals the relative price. Therefore, at both relative prices, producers would be indifferent between producing the same combination of Food and Cloth, leading to the same level of production for both goods.
09

h) Production possibility frontier with capital requirements and limited capital

Now, we have 100 units of capital available and capital requirements per unit of Food and Cloth are 0.8 and 0.9, respectively. We must determine the endpoints of the production possibility frontier by allocating all available capital to one of the two goods while still meeting labor and land constraints. Calculate the maximum amount of each good that can be produced given the capital, labor, and land constraints. This will give you the endpoints of the production possibility frontier, which will also be concave due to the unequal requirements of the two goods. Sketch this curve on a graph with Food on the horizontal axis and Cloth on the vertical axis. Finally, analyze the relative price of Food as a function of its output as calculated in step 6 and sketch the new curve representing this relationship.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Opportunity Cost
Opportunity cost is a fundamental concept in economics that describes the value of the next best alternative that is foregone when a choice is made. It's the cost associated with not choosing the next best option. For instance, in the context of the production possibility frontier (PPF), when an economy decides to produce more food, it must use resources that could have been used to produce cloth. As a result, the opportunity cost of producing more food is the amount of cloth that could have been produced with those same resources.

Understanding opportunity cost helps in making informed decisions about resource allocation. It is crucial for businesses and individuals to consider opportunity costs to optimize their outcomes. For students grappling with the idea, a simple way to think about it is by considering everyday decisions, like choosing to study an extra hour for an exam instead of watching a movie. The opportunity cost here is the enjoyment and relaxation that the movie would have provided.
Resource Allocation
Resource allocation involves the distribution of resources among competing uses to maximize efficiency and output. In our exercise, we're looking at how an economy allocates its limited resources—labor, land, and capital—to produce two goods, food and cloth. The PPF demonstrates this allocation visually, with each point on the frontier representing an efficient allocation of resources.

In practice, determining the best resource allocation can be complex, as it involves trade-offs. For example, if an economy fully employs its labor to produce food, it cannot produce any cloth, and vice versa. This is illustrated in the exercise when we discuss limited resources of labor and land and calculate the maximum outputs of food and cloth. Equally important for students to consider is how changes in the availability of resources, like the introduction of capital in our exercise, affect this allocation and subsequently the PPF.
Economic Trade-Offs
Economic trade-offs are crucial to understanding how choices affect an economy's resource utilization and output. The trade-offs are vividly represented by the PPF, which shows the combinations of goods that can be produced with a fixed set of resources. As we move along the PPF, to increase the production of one good, we must decrease the production of another due to limited resources, which is the trade-off.

In the provided exercise, sketching different PPFs based on various combinations of labor and land constraints exemplifies how shifting resources from the production of one good to another results in different trade-offs. A key takeaway for students should be that these trade-offs are guided by opportunity costs. For instance, if the economy is at a point along the PPF where the trade-off between food and cloth is favorable, moving away from that point will increase the opportunity cost. As a learning tip, students can reinforce their understanding of economic trade-offs by applying these concepts to real-life scenarios, such as budgeting time or money.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are \(m\) individuals in the economy and that each individual is characterized by a skill level, \(a_{p}\), which indicates his or her ability to earn income. Without loss of generality suppose also that individuals are ordered by increasing ability. Pretax income itself is determined by skill level and effort, \(c_{b}\) which may or may not be sensitive to taxation. That is, \(I_{i}=I\left(a_{b} c_{i}\right) .\) Suppose also that the utility cost of effort is given by \(\psi(c), \psi^{\prime}>0, \psi^{\prime \prime}<0, \psi(0)=0 .\) Finally, the government wishes to choose a schedule of income taxes and transfers, \(T(I),\) which maximizes social welfare subject to a government budget constraint satisfying \(\sum_{i=1}^{m} T\left(I_{i}\right)=R\) (where \(R\) is the amount needed to finance public goods). a Suppose that each individual's income is unaffected by effort and that each person's utility is given by \(u_{i}=u_{i}\left[I_{i}-T\left(I_{i}\right)-\right.\) \(\psi(c)]\). Show that maximization of a CES social welfare function requires perfect equality of income no matter what the precise form of that function. (Note: for some individuals \(T\left(I_{i}\right)\) may be negative.) b. Suppose now that individuals' incomes are affected by effort. Show that the results of part (a) still hold if the government based income taxation on \(a_{i}\) rather than on \(I_{i}\) c. In general show that if income taxation is based on observed income, this will affect the level of effort individuals undertake d. Characterization of the optimal tax structure when income is affected by effort is difficult and often counterintuitive. Diamond \(^{25}\) shows that the optimal marginal rate schedule may be U-shaped, with the highest rates for both low- and highincome people. He shows that the optimal top rate marginal rate is given by \\[ T^{\prime}\left(I_{\max }\right)=\frac{\left(1+e_{L, w}\right)\left(1-k_{i}\right)}{2 c_{L, w}+\left(1+e_{L, w}\right)\left(1-k_{i}\right)} \\] where \(k,\left(0 \leq k_{l} \leq 1\right)\) is the top income person's relative weight in the social welfare function and \(e_{L w}\) is the elasticity of labor supply with respect to the after-tax wage rate. Try a few simulations of possible values for these two parameters, and describe what the top marginal rate should be. Give an intuitive discussion of these results.

Suppose there are only three goods \(\left(x_{1}, x_{2}, x_{3}\right)\) in an economy and that the excess demand functions for \(x_{2}\) and \(x_{3}\) are given by \\[ \begin{array}{l} E D_{2}=-\frac{3 p_{2}}{p_{1}}+\frac{2 p_{3}}{p_{1}}-1 \\ E D_{3}=-\frac{4 p_{2}}{p_{1}}-\frac{2 p_{3}}{p_{1}}-2 \end{array} \\] a Show that these functions are homogencous of degree 0 in \(p_{1}, p_{2},\) and \(p_{3}\) b. Use Walras' law to show that, if \(E D_{2}=E D_{3}=0,\) then \(E D_{1}\) must also be \(0 .\) Can you also use Walras' law to calculate \(E D_{1} ?\) c. Solve this system of equations for the equilibrium relative prices \(p_{2} / p_{1}\) and \(p_{3} / p_{1}\). What is the equilibrium value for \(p_{3} / p_{2} ?\)

Suppose the production possibility frontier for guns \((x)\) and butter \((y)\) is given by \\[ x^{2}+2 y^{2}=900 \\] a. Graph this frontier. b. If individuals always prefer consumption bundles in which \(y=2 x\), how much \(x\) and \(y\) will be produced? c. At the point described in part (b), what will be the \(R P T\) and hence what price ratio will cause production to take place at that point? (This slope should be approximated by considering small changes in \(x\) and \(y\) around the optimal point.) d. Show your solution on the figure from part (a).

In the country of Ruritania there are two regions, \(A\) and \(B\). Two goods \((x \text { and } y)\) are produced in both regions. Production functions for region \(A\) are given by \\[ \begin{array}{l} x_{A}=\sqrt{l_{x}} \\ y_{A}=\sqrt{l_{y}} \end{array} \\] here \(l_{x}\) and \(l_{y}\) are the quantities of labor devoted to \(x\) and \(y\) production, respectively. Total labor available in region \(A\) is 100 units; that is, \\[ l_{x}+l_{y}=100 \\] Using a similar notation for region \(B\), production functions are given by \\[ \begin{array}{l} x_{B}=\frac{1}{2} \sqrt{l_{x}} \\ y_{B}=\frac{1}{2} \sqrt{l_{y}} \end{array} \\] There are also 100 units of labor available in region \(B\) ? \\[ l_{x}+l_{y}=100 \\] a. Calculate the production possibility curves for regions \(A\) and \(B\). b. What condition must hold if production in Ruritania is to be allocated efficiently between regions \(A\) and \(B\) (assuming labor cannot move from one region to the other \() ?\) c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions). How much total \(y\) can Ruritania produce if total \(x\) output is \(12 ?\) Hint: \(\mathrm{A}\) graphical analysis may be of some help here.

In Example 13.3 we showed how a Pareto efficiency exchange equilibrium can be described as the solution to a constrained maximum problem, In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one person in a two-good economy and that his or her utility function is given by \(U(x, y)\). Suppose also that this economy's production possibility frontier can be written in implicit form as \(T(x, y)=0\) a. What is the constrained optimization problem that this economy will seek to solve if it wishes to make the best use of its available resources? b. What are the first-order conditions for a maximum in this situation? c. How would the efficient situation described in part (b) be brought about by a perfectly competitive system in which this individual maximizes utility and the firms underlying the production possibility frontier maximize profits. d. Under what situations might the first-order conditions described in part (b) not yield a utility maximum?
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free