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Chapter 16: Problem 1

Suppose the production possibility frontier for cheeseburgers (C) and milkshakes \((M)\) is given by \\[ C+2 M=600 \\] a. Graph this function. b. Assuming that people prefer to eat two cheeseburgers with every milkshake, how much of each product will be produced? Indicate this point on your graph. c. Given that this fast-food economy is operating efficiently, what price ratio \(\left(P / P_{M}\right)\) must prevail?

Short Answer

Expert verified
Answer: The price ratio for an efficient economy is \(P_C / P_M = -\frac{1}{2}\).

Step by step solution

01

Understand the production possibility frontier equation

The given equation, \(C+2M=600\), shows the maximum combinations of cheeseburgers (C) and milkshakes (M) that can be produced with the available resources.
02

Graph the PPF equation

First, we need to rewrite the equation in the form of a function: \(M = \frac{1}{2} (600 - C)\). Now we can graph this equation on the coordinate plane. The x-axis will represent cheeseburgers (C), and the y-axis will represent milkshakes (M). Sketch the graph of the line, and label the PPF curve.
03

Find the preferred combination of cheeseburgers and milkshakes

The problem states that people prefer to eat two cheeseburgers with every milkshake. We can represent this as a ratio: \(C = 2M\). Now we need to find the amounts of C and M that satisfy both the preference condition and the PPF equation. 1. Substitute the preference condition \(C = 2M\) into the PPF equation: \((2M) + 2M = 600\). 2. Solve for M: \(4M = 600\), so \(M = 150\). 3. Plug the value of M back into the preference condition to find C: \(C = 2 \cdot 150 = 300\). So, the preferred combination is 300 cheeseburgers and 150 milkshakes. 4. Locate this point on the graph and label it as the preferred point (P).
04

Determine the price ratio for an efficient economy

Finally, we need to determine the price ratio \(P_C / P_M\) for an efficient economy. To do this, we can use the concept of marginal rate of transformation (MRT), which represents the opportunity cost of producing one more cheeseburger in terms of milkshakes. The MRT is the slope of the tangent line to the PPF at the preferred point (P). In our case, the PPF is a straight downward-sloping line, so its slope is constant along the entire PPF. The slope of the PPF can be found by calculating the negative of the derivative of the PPF equation with respect to C: \(-\frac{dM}{dC} = -\frac{1}{2}\). Therefore, the price ratio for an efficient economy is \(P_C / P_M = -\frac{1}{2}\).

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Most popular questions from this chapter

The purpose of this problem is to examine the relationship among returns to scale, factor intensity, and the shape of the production possibility frontier. Suppose there are fixed supplies of capital and labor to be allocated between the production of good \(X\) and good \(Y\). The production function for \(X\) is given by \\[ X=K^{\alpha} L^{\beta} \\] and for \(Y\) by \\[ Y=K^{\gamma} L^{s} \\] where the parameters \(\alpha, \beta, \gamma, \delta\) will take on different values throughout this problem. Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for \(X\) and \(Y\) in the following cases: a. \(\quad \alpha=\beta=\gamma=\delta=\frac{1}{2}\) b. \(\quad \alpha=\beta=\frac{1}{2}, \gamma=\frac{1}{3}, \delta=\frac{2}{3}\) \(c_{*} \quad \alpha=\beta=\frac{1}{2}, \gamma=\delta=\frac{2}{3}\) d. \(\alpha=\beta=\gamma=\delta=\frac{2}{3}\) e. \(\quad \alpha=\beta=.6, \gamma=.2, \delta=1.0\) f. \(\quad \alpha=\beta=.7, \gamma=.6, \delta=.8\) Do increasing returns to scale always lead to a convex production possibility frontier? Explain.

The country of Podunk produces only wheat and cloth, using as inputs land and labor. Both are produced by constant retums-to-scale production functions. Wheat is the relatively landintensive commodity. a. Explain, in words or with diagrams, how the price of wheat relative to cloth (p) deter mines the land-labor ratio in each of the two industries. b. Suppose that \(p\) is given by external forces (this would be the case if Podunk were a "small" country trading freely with a "large" world). Show, using the Edgeworth box, that if the supply of labor increases in Podunk, the output of cloth will rise and the output of wheat will fall.

Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (X) or chicken soup (I'). Smith's utility function is given by \\[ U_{r}=X_{-}^{3} Y \\] whereas Jones' is given by \\[ U_{j}=X^{s} Y^{s} \\] The individuals do not care whether they produce \(X\) or \(Y\), and the production function for cach good is given by \\[ \begin{array}{l} X=2 L \\ Y=3 L \end{array} \\] where \(L\) is the total labor devoted to production of each good. Using this information, a. What must the price ratio, \(P_{x} / P_{Y},\) be? b. Given this price ratio, how much Xand Fwill Smith and Jones demand? (Hint: Set the wage equal to 1 here. c. How should labor be allocated between \(X\) and \(Y\) to satisfy the demand calculated in part (b)?

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

Suppose an economy produces only two goods, \(X\) and \(Y\). Production of good \(X\) is given by where \(K_{x}\) and \(L_{x}\) are the inputs of capital and labor devoted to \(X\) production. The production function for good Fis given by \\[ =\sin ^{3} y^{-4} \lg ^{2} x^{2} \\] where \(K_{\text {; }}\) a.nd \(\mathrm{L}_{\text {, are the inputs of capital and labor devoted to } \mathrm{F} \text { production. The supply of }}\) capital is fixed at 100 units and the supply of labor is fixed at 200 units. Hence, if both units are fully employed, \\[ \begin{array}{l} K_{x}+K_{Y}=K_{T}=100 \\ L_{x}+L_{Y}=L_{T}=200 \end{array} \\] Using this information, complete the following questions. a Show how the capital-labor ratio in \(X\) production \(\left(K / L_{x}=k_{x}\right)\) must be related to the capital-labor ratio in \(\mathrm{F}\) production \(\left(K_{y} / L_{Y}=k_{y}\right)\) if production is to be efficient. b. Show that the capital-labor ratios for the two goods are constrained by \\[ a_{x} k_{x}+\left(l-a_{x}\right) k_{y}=\underline{K}_{T}-\underline{100}_{-} \\] where \(a_{x}\) is the share of total labor devoted to \(X\) production [that is, \(a_{x}=L, / L_{r}=L_{s} /\) \((L x+L y) J\) c. Use the information from parts (a) and (b) to compute the efficient capital-labor ratio for good Xfor any value of \(a_{x}\) between 0 and 1 d. Graph the Edgeworth production box for this economy and use the information from part (c) to develop a rough sketch of the production contract curve. e. Which good, \(X\) or \(Y\), is capital intensive in this economy? Explain why the production possibility curve for the economy is concave. \(f\) Calculate the mathematical form of the production possibility frontier for this economy (this calculation may be rather tedious!). Show that, as expected, this is a concave function.
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