Household Optimization Problem

In this article, you will learn how to describe the household optimization problem in your microeconomics formally. 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 0

The 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 0

The 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 0

The 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 0

2.1.2 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

In general, the household’s utility maximization problem formally involves the search for the bundle of the two goods that satisfies the needs of the household and respects the limits of the restriction as follows:

\max_{x, l}  U(x, l)= x^{\alpha} \cdot l^{\beta}

The household should maximize its utility and respect the resource and expenditure restrictions below:

\text{s.t.} \ p\cdot x -w\cdot (16-l)\le 0

$(16−l)$ represents the hours the household uses in income-generating activities. The budget restriction suggests that the whole income should be used to finance the consumption of the private good $x$ and shows how much revenue is forgone depending on the household’s leisure consumption $l$. The resulting general Lagrange-Optimization problem introduces the shadow-price for the restriction as follows:

\max_{x, l, \lambda} L(x, l, \lambda)=U(x, l) +\lambda (p\cdot x -w\cdot (16-l))

2.2.1 First-Oder-Conditions

Three first-order-conditions w.r.t. the private good $x$ and leisure $l$ must be met for the solution to exist, assuming that the second-order condition for maximum holds, respectively (visit our classes for further mathematical derivation):

L_{x}^{\prime}(x, l, \lambda)=U_{x}^{\prime}(x, l) - \lambda p =0

L_{l}^{\prime}(x, l, \lambda)=U_{l}^{\prime}(x, l) +\lambda w =0

L_{\lambda}^{\prime}(x, l, \lambda)=p\cdot x -w\cdot (16-l) =0

2.2.2 Shadow price and Optimality Condition

The first two conditions above lead to the following condition for the restriction’s shadow price of one additional unit of resource in the budget:

\lambda=\frac{U_{x}^{\prime}(x, l)}{p}=\frac{U_{l}^{\prime}(x, l)}{-w}

We can transform of the above equality between the relative marginal utility w.r.t. to the price $p$ and wage $w$ to attain the optimality condition of the household maximization problem as follows:

MRS= \frac{\Delta x}{\Delta l}=\frac{U_{l}^{\prime}(x, l)}{U_{x}^{\prime}(x, l)}=\frac{-w}{p}

The optimality condition suggests on the right-hand-side that the opportunity costs of one additional hour of leisure $l$ costs the marginal rate of substitution (relative marginal utility of leisure $l$ to the marginal utility of private good $x$), which in turn will depend on the left-hand-side forgone real wage rate quantifying the units of private good $x$ that the household will have to give up. In the next step, we insert the marginal utility functions and simplify to get the following:

|MRS|= |\frac{\Delta x}{\Delta l}|=|\frac{\beta}{\alpha}\frac{x}{l}|=|\frac{-w}{p}|

\Rightarrow x=\Big|-\Big(\frac{\alpha}{\beta}\Big)\frac{wl}{p} \Big|

After deriving this above statement without the negative sign, insert this in the third first-oder condition above:

L_{\lambda}^{\prime}(x, l, \lambda)=p\cdot \Big(\frac{\alpha}{\beta}\Big)\frac{wl}{p} -w\cdot (16-l) =0

Strike the price $p$ to get:

\Rightarrow \Big(\frac{\alpha}{\beta}\Big) wl -w\cdot (16-l) =0

Open the brackets and get:

\Rightarrow \Big(\frac{\alpha}{\beta}\Big) wl +wl -16w =0

Divide both side with the wage $w$ and factorize with leisure $l$ to get:

\Rightarrow l\Big(1+\frac{\alpha}{\beta}\Big) -16 =0

Using $\beta$ as the common divisor transform to the get:

\Rightarrow l\Big(\frac{\beta +\alpha}{\beta}\Big) =16 

2.2.3 Optimal Marshall Demand for Leisure

Rearrange to get the optimal demand of leisure in the household optimization problem depending on the demand elasticity of private good $x$ and leisure $l:

\Rightarrow l^{*}=\Big(\frac{\beta}{\beta +\alpha}\Big) \cdot 16 \ge0

The demand of leisure $l$ in a Cobb-Douglas case implies that the household will always demand the same units of leisure independent of the price $p$ of private good and the wage rate $w$ earned during income-generating activities.

0<\Big(\frac{\beta}{\beta +\alpha}\Big)< 1

The ratio above determines whether the household allocates large or fewer portions of hours to leisure at all times. For example, the household will demand eight leisure hours at all times if the ratio is equal to half. The implications there is that leisure decisions are purely preference-based and are not driven by wage and price of commodities. Other economic models come to other conclusions.

2.2.4 Optimal Marshall Demand for the private Good

Inserting the above outcome in our previous optimality condition should deliver the optimal demand of the private good $x$:

\Rightarrow x=\Big(\frac{\alpha}{\beta}\Big)\frac{wl}{p}=\Big(\frac{\alpha}{\beta}\Big)\frac{w}{p}\cdot \Big(\frac{\beta}{\beta + \alpha}\Big) \cdot16 

The optimal demand of the private good depends on the real wage $\frac{w}{p}$ and the demand elasticities $\alpha$ and $\beta$ of the private good $x$ and leisure $l$, respectively:

\Rightarrow x^{*}=\frac{w}{p}\cdot \Big(\frac{\alpha}{\beta + \alpha}\Big) \cdot 16 \ge 0

According to Cobb-Douglas preferences, the real wage affects the optimal demand of the private good $x$ but does not affect the demand for leisure $l$.

In the next section, we shall explore the alternative household optimization problem.

3. Household Minimization Problem

In the household minimization problem, households face a utility restriction and seek to minimize their expenditure, as discussed below using the assumption of section 2 above. The following function describes the expenses of the private household:

E(p,w, x, l)=p\cdot x +w \cdot l

The marginal expenses explain the additional resources used in financing one unit of the private good $x$ is equivalent to the price $p$ and financing one additional unit of leisure $l$ costs the wage $w$, respectively:

E_{x}^{\prime}(p,w, x, l)=p

E_{l}^{\prime}(p,w, x, l)=w

Suppose the household optimization problem minimizes the household’s expenses. In that case, the household is restricted by the level of satisfaction that it can attain (Hick’s assumption) as set in the following utility condition:

U(x, l)= x^{\alpha} \cdot l^{\beta} \ge \bar{U_{0}}

The household minimization problem above can be solved using the same procedure in section 2 and results in Hick’s demand equations for optimal demand for leisure $l$ and private good $x$. The intuition is simple. In such a case, your maximum attainable utility is a fixed level of utility, but the minimum expenditure is your optimal choice.

3.1 Lagrange Household Minimization Problem

Similar to section 2.2 above, we start by setting the Lagrange household minimization problem as follows:

\min_{x,l, \lambda} L(x,l, \lambda)=E(p,w, x, l)+\lambda (\bar{U_{0}} -U(x, l))

The solution of the Lagrange objective function explains the behavior of the private household for the given assumption in section 2.1 above.

3.1.1 First oder Conditions of the Minimization Problem

The Next step is deriving the first-oder optimality conditions for the optimization problem and assuming that the conditions for the expenditure minimum are fulfilled, we get the following conditions:

L_{x}^{\prime}(x, l, \lambda)=E_{x}^{\prime}(p,w, x, l)- \lambda U_{x}^{\prime}(x, l)=0

L_{l}^{\prime}(x, l, \lambda)=E_{l}^{\prime}(p,w, x, l)- \lambda U_{l}^{\prime}(x, l)=0

L_{\lambda}^{\prime}(x, l, \lambda)=\bar{U_{0}} -U(x, l)=0

3.1.2 Shadow Price and Optimality Condition

The first two conditions lead to the same shadow price and optimality condition in section 2.2.2, respectively:

\lambda=\frac{U_{x}^{\prime}(x, l)}{p}=\frac{U_{l}^{\prime}(x, l)}{w}

|MRS|= |\frac{\Delta x}{\Delta l}|=|\frac{\beta}{\alpha}\frac{x}{l}|=|\frac{w}{p}|

\Rightarrow x=\Big| \Big(\frac{\alpha}{\beta}\Big)\frac{wl}{p} \Big|

3.1.3 Optimal Hick’s Demand for Leisure

Using the outcomes above we can derive the optimal Hick’s demand for leisure $l$ using the third condition in 3.1.1.

L_{\lambda}^{\prime}(x, l, \lambda)=\bar{U_{0}} -U(x, l)=0

Inserting the utility function to get the following:

\Rightarrow L_{\lambda}^{\prime}(x, l, \lambda)=\bar{U_{0}} -x^{\alpha} \cdot l^{\beta}=0

Afterwards, replace $x$ with the optimality condition in section 3.1.2 and by substracting $\bar{U_{0}}$ and multiplying by $-1

\Rightarrow \bar{U_{0}} = \Big[\Big(\frac{\alpha}{\beta}\Big)\frac{wl}{p} \Big]^{\alpha} \cdot l^{\beta}=\Big[\Big(\frac{\alpha}{\beta}\Big)\frac{w}{p} \Big]^{\alpha} \cdot l^{(\alpha+\beta)}

Thereafter, we solve for the optimal Hick’s demand for leisure $l$ as follows:

\Rightarrow l^{*}=\frac{\bar{U_{0}}^{\frac{1}{(\alpha+\beta)}}}{\Big[\Big(\frac{\alpha}{\beta}\Big)\frac{w}{p} \Big]^{\frac{\alpha}{(\alpha+\beta)}} } 

The outcome reveals that the optimal Hick’s demand for leisure $l$ depends on the real wage $\frac{w}{p}$, the utility level als measure for the satisfaction of the household, and the Cobb-Douglas elasticities of demand, $\alpha$ and $\beta$. To sustain the utility level, the private household needs compensation in terms of less hours of leisure when the price $p$ increases or wage $w$ decreases.

3.1.4 Optimal Hick’s Demand for the Private Good

Inserting the outcome from section 3.1.3 in the outcome in section 3.1.2 delivers the optimal Hick’s demand for the private good $x$ as follows

\Rightarrow x= \Big(\frac{\alpha}{\beta}\Big)\frac{w}{p} \cdot l=\Big(\frac{\alpha}{\beta}\Big)\frac{w}{p} \cdot \frac{\bar{U_{0}}^{\frac{1}{(\alpha+\beta)}}}{\Big[\Big(\frac{\alpha}{\beta}\Big)\frac{w}{p} \Big]^{\frac{\alpha}{(\alpha+\beta)}} } 

Afterwards, we simplify the equation above to get the equation for the optimal Hick’s demand for private good $x$:

\Rightarrow x^{*}=  \frac{\bar{U_{0}}^{\frac{1}{(\alpha+\beta)}}}{\Big[\Big(\frac{\alpha}{\beta}\Big)\frac{w}{p} \Big]^{\frac{\beta}{(\alpha+\beta)}} } 

The outcome reveals that the optimal Hick’s demand for the private good $x$ depends on the real wage $\frac{w}{p}$, the utility level als measure for the satisfaction of the household, and the Cobb-Douglas elasticities of demand, $\alpha$ and $\beta$.

4. Indifference Curve and the Marginal Rate of Substitution (MRS)

An indifference curve demarcates all bundles of goods that contribute the same level of utility to the household. On the other hand, the Marginal Rate of Substitution (MRS) describes the slope of the indifference curve. MRS is, therefore, the measure of opportunity costs of choosing a bundle of goods over all other alternatives. The MRS influences which bundle of goods the household will prefer over another (the first area of decision-making).

5. Budget Restriction and the relative Prices of Goods

The budget restriction demarcates all bundles of goods that are affordable to the household. To construct the budget restriction, you will need the market prices of the goods in the bundle. You will then compare the value of all goods with the income level of the household. You will only consider those bundles of goods that incur costs lower or equivalent to income level (affordable bundles of goods). Lastly, among those achievable bundles of goods, some efficient bundles utilize the whole income of the household. 

So why have we described the budget restriction in such detail? Because we want to tap into the second area of decision-making. To reach our second goal, we have to explain the role of the slope of the budget restriction. The second area of decision-making relies on the steepness of the budget restriction, which we measure using the relative price of goods.

6. How private households maximize utility and miminize expenditure.

Both the utility maximization problem and expenditure minimization problem will lead to the same outcome. The outcome is that the household must choose an optimal bundle among the affordable bundles in a utility maximization problem. That is the bundle that keeps the cost constant. In an expenditure minimization problem, they will select the optimal bundle among the preferred bundle of goods. That is the bundle of goods that keeps the utility constant. In either case, the outcome is that the chosen bundles have to fulfill an optimality condition.

The optimality condition connects both areas of decision-making by setting the Marginal Rate of Substitution equal to the relative price of goods. This statement reveals the concept of opportunity costs in economics and is the most fundamental statement for all economic decisions. In mathematical terms, the slope of the indifference curve should be equivalent to the marginal change (slope) of the budget restriction. In terms of economics, the narrative means that the opportunity costs of choosing a bundle (MRT) should be equivalent to the relative prices of goods in the markets (cost of buying or selling the goods in the market). That is how private households maximize utility.

7. Literature