In this article, you will learn how to formally describe the household optimization problem in your microeconomics. To understand how private households make their consumption and resource allocation decisions in the real world, we shall look at the crucial elements that affect optimal decision-making, e.g., price, income, and preference changes. A private household is one of the primary agents making economic decisions relevant to other economic agents, e.g., government and firms. Here is a revisit of the household maximization problem described in many microeconomic and economics textbooks. But with a clear outline and a better understanding of how households set goals, their opportunity cost optimization process, and the interpretation of outcomes.
1. Introduction: Two Areas of Household Optimization Problem
The private household, also generally called consumers, is the smallest unit of private decision-making in an economy. The key concept you want to understand is how private households (or consumers) generally make optimal decisions. The reason is that decision-making is the process by which societal transactions concerning the satisfaction of needs occur and determine the outcomes of social interactions within an economy. In Maslow’s hierarchy of needs there are different levels of needs that individual in Society seek to satisfy to achieve the needs of each hierarchy.
There are two areas of decision-making within a private household based on the concept of opportunity costs, and these are as follows:
- Utility Maximization applying the Maximum Principle of Opportunity Cost: Which bundle of goods does the household prefer over another? In the bundle of goods are not only goods and services to be consumed but also the consumption of leisure, which means that the preferences of a household capture the dimension of goods, services, and leisure allocation. Indirectly, the need to allocate scarce time as a resource in labor activity is captured. Using utility functions and indifference curves, we capture the resulting preferences of the individuals. The utility describes the satisfaction that the goods contribute to a household on an ordinal or cardinal scale.
- Expenditure Minimization applying the Minimum Principle of Opportunity Costs: Which resources can the household spend and how? Private households are the holders of the economy’s scarce resources and determine their allocation. The budget restriction limits the choice of bundles of goods that a private individual can afford.
Any decision-making problem a household wants to resolve will rotate around the above general household optimization problems. In the next section, we shall dig deeper into the rational decision-making process based on the two concepts.
2. Household Maximization Problem
In the household maximization problem, households face a resource restriction and seek to maximize their utility, as discussed below. Let us review the neoclassical model of labor supply and demand for leisure and private goods consumption that you can find in many economic research models and textbooks.
2.1 Basic Model of a Private Good and Leisure Consumption
Let us assume an economy consisting of one infinitely lived household made of one individual with the following preferences towards the bundle of two goods: Leisure $l$ per day and some amount of private good $x$ per day (see literature below).
2.1.1 Cobb-Douglas Preferences
The following Cobb-Douglas utility function describes the household’s preferences between different bundles on a particular day:
U(x, l)= x^{\alpha} \cdot l^{\beta}Further, we assume that $0 < \alpha < 1$ and $0< \beta <1$ imply that the household has diminishing marginal utility. The marginal utility of consuming an additional unit of the private good $x$ is as follows:
U^{\prime}(x, l)= \alpha x^{(\alpha-1)} \cdot l^{\beta}=\alpha \frac{l^{\beta}}{x^{(1-\alpha)}} \ge 0The change in marginal utility is the second derivative of the utility explaining the impact of one additional unit of private good $x$:
U^{\prime \prime}(x, l)= (\alpha-1)\cdot \alpha x^{(\alpha-2)} \cdot l^{\beta}=-(1-\alpha)\cdot \alpha \frac{l^{\beta}}{x^{(2-\alpha)}} \le 0The marginal utility of consuming an additional unit of leisure $x$ is as follows:
U^{\prime}(x, l)= \beta x^{\alpha} \cdot l^{(\beta-1)} =\beta \frac{x^{\alpha}}{l^{(1-\beta)}}\ge 0The change in marginal utility is the second derivative of the utility explaining the impact of one additional unit of private good $x$:
U^{\prime \prime}(x, l)= (\beta-1)\cdot \beta x^{\alpha} \cdot l^{(\beta-2)}=-(1-\beta)\cdot \beta \frac{x^{\alpha}}{l^{(2-\beta)}}\le 02.1.2 Scarce Resources and Budget Restriction
Now, let us turn to the resource restriction of the household due to resource scarcity. First, we assume that the household has 16 hours per day at its disposal for income-generating activities to afford its consumption, or it can use all the 16 hours for leisure activities, which means that $0<l<16$. By investing time in income-generating activities, the household is assumed to earn a wage rate of $w$ per hour. But why only 16 hours and not 24 hours per day? The reason is quite simple. By natural circumstances, the household needs time to rest for at least 8 hours, as assumed in this model. Further, let us take a market-driven price $p$ of the private good that matches the marginal costs of producing an additional unit of the good using a linear production technology without fixed costs and generating zero profits. In summary, the budget restriction of the households involves the following expenditures for the bundle limited by available resources:
p\cdot x +w\cdot l\le 16w
2.2 Lagrange household maximization problem
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