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Chapter 4: Problem 3

a. On a given evening, J. P. enjoys the consumption of cigars ( \(c\) ) and brandy ( \(b\) ) according to the function $$U(c, b)=20 c-c^{2}+18 b-3 b^{2}$$ How many cigars and glasses of brandy does he consume during an evening? (cost is no object to J. P.) b. Lately, however, J. P. has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to \(5 .\) How many glasses of brandy and cigars will he consume under these circumstances?

Short Answer

Expert verified
Answer: Without any constraint on cost, J.P. will consume 10 cigars and 3 glasses of brandy. However, under the constraint that the sum of his consumption is limited to 5, he will consume \(\frac{7}{5}\) cigars and \(\frac{18}{5}\) glasses of brandy.

Step by step solution

01

Find partial derivatives of U(c, b)

To find the critical points of the function, we need to find the first partial derivatives of U(c, b) with respect to both c and b: $$\frac{\partial U}{\partial c} = 20 - 2c$$ $$\frac{\partial U}{\partial b} = 18 - 6b$$
02

Set the partial derivatives equal to 0 and solve

To find the critical points, we set the partial derivatives equal to 0 and solve for c and b: $$20 - 2c = 0$$ $$18 - 6b = 0$$ Solving for c and b yields: $$c = 10$$ $$b = 3$$ So, without any constraint, J.P. will consume 10 cigars and 3 glasses of brandy to maximize his utility. #Part b: Limited consumption#
03

Introduce the constraint function and find its gradient

We introduce the constraint function g(c, b) for the sum of cigars and brandy equal to 5: $$g(c, b) = c + b - 5$$ Now we find the gradient, ∇g(c, b): $$\nabla g(c, b) = \left(\frac{\partial g}{\partial c}, \frac{\partial g}{\partial b}\right) = (1, 1)$$
04

Set the gradients of U(c, b) and the constraint equal using Lagrange multipliers

We introduce λ, the Lagrange multiplier, and set the gradients of U(c, b) and the constraint equal: $$\nabla U(c, b) = \lambda \nabla g(c, b)$$ From our previous calculations, we know that: $$\nabla U(c, b) = \left(20 - 2c, 18 - 6b\right) = \lambda (1, 1)$$ Now we can set up a system of equations: $$20 - 2c = \lambda$$ $$18 - 6b = \lambda$$ $$c + b = 5$$
05

Solve the system of equations

We can solve the system of equations using substitution or elimination: Subtract the first from the second equation to eliminate λ: $$6b - 2c = -2$$ Combine with the constraint equation: $$5c = 7$$, Solving for c yields: $$c = \frac{7}{5}$$ Substitute this value back into constraint equation to find b: $$b = 5 - \frac{7}{5} = \frac{18}{5}$$ Under the given constraint, J.P. will consume \(\frac{7}{5}\) cigars and \(\frac{18}{5}\) glasses of brandy to maximize his utility.

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Most popular questions from this chapter

Two of the simplest utility functions are: 1\. Fixed proportions: \(U(x, y)=\min [x, y]\) 2\. Perfect substitutes: \(U(x, y)=x+y\) a. For each of these utility functions, compute the following: \(\bullet\) Demand functions for \(x\) and \(y\) \(\bullet\) Indirect utility function \(\bullet\) Expenditure function b. Discuss the particular forms of these functions you calculated-why do they take the specific forms they do?

Mr. A derives utility from martinis \((m)\) in proportion to the number he drinks: \\[U(m)=m.\\] \(\mathrm{Mr}, \mathrm{A}\) is particular about his martinis, however: He only enjoys them made in the exact proportion of two parts gin \((g)\) to one part vermouth ( \(v\) ). Hence we can rewrite Mr. A's utility function as \\[U(m)=U(g, v)=\min \left(\frac{g}{2}, v\right).\\] a. Graph Mr. A's indifference curve in terms of \(g\) and \(v\) for various levels of utility. Show that, regardless of the prices of the two ingredients, Mr. A will never alter the way he mixes martinis. b. Calculate the demand functions for \(g\) and \(v\). c. Using the results from part (b), what is Mr. A's indirect utility function? d. Calculate Mr. A's expenditure function; for each level of utility, show spending as a function of \(p_{8}\) and \(p_{v}\). Hint: Because this problem involves a fixed-proportions utility function, you cannot solve for utility-maximizing decisions by using calculus.

In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by $$U(x, y)=\left(x-x_{0}\right)^{\alpha} y^{\beta, }$$[in this function the elasticity of substitution \(\sigma=1 /(1-\delta)]\) a. Show that the indirect utility function for the utility function just given is $$V=I\left(p_{x}^{r}+p_{y}^{r}\right)^{-1 / r}$$ where \(r=\delta /(\delta-1)=1-\sigma\) b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by $$E=V\left(p_{x}^{r}+p_{y}^{r}\right)^{1 / r}$$ f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.

The CES utility function we have used in this chapter is given by $$U(x, y)=\frac{x^{0}}{\delta}+\frac{y^{0}}{\delta}$$ a. Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion $$\frac{x}{y}=\left(\frac{p_{x}}{p_{y}}\right)^{1 /(\delta-1)}$$ b. Show that the result in part (a) implies that individuals will allocate their funds equally between \(x\) and \(y\) for the CobbDouglas case \((\delta=0),\) as we have shown before in several problems. c. How does the ratio \(p_{x} x / p_{y} y\) depend on the value of \(\delta\) ? Explain your results intuitively. (For further details on this function, see Extension E4.3.) d. Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.

Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies ( \(t\) ) and soda ( \(s\) ), and these provide him a utility of utility \(=U(t, s)=\sqrt{t s}\) a. If Twinkies cost \(\$ 0.10\) each and soda costs \(\$ 0.25\) per cup, how should Paul spend the \(\$ 1\) his mother gives him to maximize his utility? b. If the school tries to discourage Twinkie consumption by increasing the price to \(\$ 0.40,\) by how much will Paul's mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)?
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