Question

Mr. A derives utility from martinis (M) in proportion to the number he drinks: $$U(M)=\mathbf{M}$$ Mr. A is very 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 $$U(M)=U(G, V)=\min I j, v |$$ 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 \(P_{G}\) and \(P_{v}\) Hint: Because this problem involves a fixed proportions utility function you cannot solve for utility- maximizing decisions by using calculus.

Short answer

Also, provide the demand functions for G and V, the indirect utility function, and the expenditure function for Mr. A. Answer: Mr. A consumes gin (G) and vermouth (V) in a 2-to-1 ratio. The demand functions for G and V are as follows: $$G = \frac{2I}{3(P_{G}+P_{V})}$$ $$V = \frac{I}{3(P_{G}+P_{V})}$$ The indirect utility function is: $$U = \frac{I}{3(P_{G}+P_{V})}$$ And the expenditure function is: $$E(U, P_{G}, P_{V}) = U(2P_{G} + P_{V})$$

Step-by-step solution

a. Graph the indifference curve and show that Mr. A will never alter the way he mixes martinis

We are given the utility function: $$U(G, V) = \min(2G, V)$$ To graph the indifference curves, set the utility to a constant (i.e., U = k), and then solve for the relationship between G and V: $$k = \min(2G, V)$$ For the optimal consumption, where G and V are in a 2-to-1 ratio, k = 2G = V. As a result, the indifference curves will be straight lines following the equation: $$V = 2G$$ Regardless of the prices of the two ingredients, Mr. A will always strive to consume gin and vermouth in the 2-to-1 ratio, as deviating from it would decrease his utility. The slope of the indifference curve represents the trade-off for Mr. A, and as such, it is a constant, implying that his consumption pattern will not change due to prices.

b. Calculate the demand functions for G and V

Let \(P_{G}\) and \(P_{V}\) be the prices of gin and vermouth, respectively, and I be the income. Since the consumption pattern remains the same, the demand functions for gin and vermouth follow the given ratio: $$G = \frac{2}{3}M$$ $$V = \frac{1}{3}M$$ We can also express the demand in terms of income and prices: $$G = \frac{2}{3}\frac{I}{P_{G}+P_{V}} = \frac{2I}{3(P_{G}+P_{V})}$$ $$V = \frac{1}{3}\frac{I}{P_{G}+P_{V}} = \frac{I}{3(P_{G}+P_{V})}$$

c. Determine Mr. A's indirect utility function

Replace the derived demand functions (in terms of income and prices) in the utility function: $$U = \min(2\frac{2I}{3(P_{G}+P_{V})}, \frac{I}{3(P_{G}+P_{V})})$$ Since we're consuming in the optimal 2-to-1 ratio, we can simplify the expression: $$U = \frac{I}{3(P_{G}+P_{V})}$$

d. Calculate Mr. A's expenditure function

To find the expenditure function, we need to express the income in terms of the utility, \(P_{G}\) and \(P_{V}\). We can do so using our indirect utility function: $$I = 3U(P_{G}+P_{V})$$ Now, replace the derived expression for I in the demand functions: $$E(U, P_{G}, P_{V}) = G * P_{G} + V * P_{V} = ( \frac{2}{3} * 3U(P_{G}+P_{V})) * P_{G} + (\frac{1}{3} * 3U(P_{G}+P_{V})) * P_{V}$$ Simplifying the expression gives us the expenditure function: $$E(U, P_{G}, P_{V}) = U(2P_{G} + P_{V})$$ This function shows Mr. A's expenditure on gin and vermouth for each level of utility, given the prices of these ingredients.

Key concepts

Utility Function

Imagine a consumer trying to maximize happiness with their choices, just like Mr. A in our exercise. The theory of utility is about quantifying this happiness or satisfaction. In economics, a utility function represents a consumer's preference for various bundles of goods and services, which reflects their level of satisfaction. For Mr. A, the utility function is exceptionally straightforward, as it is based solely on the quantity of martinis, which in turn is dependent on two ingredients - gin and vermouth.

Utility functions can come in many forms and complexities, but Mr. A's is a fixed proportions utility function, meaning that the satisfaction achieved from the goods is based on consuming them in specific proportions. Mr. A's utility function, defined as the minimum amount of either twice the gin or vermouth consumed, illustrates his fixed and unchanging preference for the 2:1 mix ratio in his martinis.

Indifference Curves

An indifference curve is a visual tool that shows a combination of two goods between which a consumer is indifferent because they provide the same level of utility or satisfaction. Typically, indifference curves are bowed inward, reflecting the principle of diminishing marginal rates of substitution between goods. However, Mr. A's preference for the strict 2:1 gin-to-vermouth mix leads to straight-line indifference curves.

The slope of Mr. A's indifference curves is rigid due to his unyielding preference for the proportion in which he mixes his martinis. No matter the prices of gin and vermouth, Mr. A's satisfaction only remains constant when he consumes the ingredients in the exact ratio, which is graphically represented by the linear nature of his indifference curves.

Demand Functions

A demand function expresses the quantity of a good a consumer will purchase at various prices, holding other factors constant. In our exercise, the demand functions for gin (G) and vermouth (V) were determined by Mr. A's income and the prices of G and V, while maintaining the fixed proportion consumption pattern.

The demand functions quantify the relationship between Mr. A's consumption of gin and vermouth, and the prices of these goods. By responding exclusively to price changes and income while adhering to the 2-to-1 ratio, we outlined the clear equations defining how much of each good Mr. A will demand. These functions are pillars in microeconomics for understanding how consumers adjust their consumption in response to economic changes.

Indirect Utility Function

The indirect utility function reflects the maximum utility a consumer can reach based on their income and the prices of goods, rather than the actual quantities consumed. It's the indirect counterpart of the utility function and comes in handy when considering the effect of prices and income on overall satisfaction.

In the case of Mr. A, once we substituted the demand functions into the utility function, we could express his utility in terms of income and the prices of gin and vermouth. This approach captures the essence of how external factors such as budget constraints and market prices impact the consumption decisions of individuals. From a learning perspective, understanding the conversion from utility to indirect utility equips students with an appreciation for the broader economic context influencing consumer behavior.

Expenditure Function

The expenditure function measures the minimum amount of money needed to achieve a certain level of utility, given the prices of goods. It's almost the inverse of the indirect utility function, representing the consumer's necessary spending for varying levels of satisfaction.

In our scenario, solving for Mr. A's expenditure function illustrated how he allocates his budget to gin and vermouth in order to maintain his preferred level of satisfaction. By expressing expenditure as a function of utility and the prices of gin and vermouth, we demonstrate the practical application of these theoretical concepts. The expenditure function is crucial for economic analysis because it provides insights into consumer budgeting and the cost associated with achieving desired levels of utility.