Question

Consider the principal-agent relationship between a patient and doctor. Suppose that the patient's utility function is given by \(U_{P}(m, x),\) where \(m\) denotes medical care (whose quantity is determined by the doctor) and \(x\) denotes other consumption goods. The patient faces budget constraint \(I_{c}=p_{m} m+x,\) where \(p_{m}\) is the relative price of medical care. The doctor's utility function is given by \(U_{d}\left(I_{d}\right)+U_{P}-\) that is, the doctor derives utility from income but, being altruistic, also derives utility from the patient's well-being. Moreover, the additive specification implies that the doctor is a perfect altruist in the sense that his or her utility increases one-for-one with the patient's. The doctor's income comes from the patient's medical expenditures: \(I_{d}=p_{m} m .\) Show that, in this situation, the doctor will generally choose a level of \(m\) that is higher than a fully informed patient would choose.

Short answer

In conclusion, the doctor's altruistic behavior leads him/her to choose a higher level of medical care than a fully informed patient would choose. This is because the doctor derives utility from both his/her income and the patient's well-being, and as a result, the doctor's utility increases one-for-one with the patient's utility. By comparing the first-order conditions for the patient's and doctor's utility maximization problems, we can observe that the weight given to the patient's utility, reflected in the marginal utilities, results in the doctor generally opting for a higher level of medical care.

Step-by-step solution

Patient's Utility Maximization Problem

Given the patient's utility function \(U_P(m, x)\), the patient will try to maximize his/her utility subject to the budget constraint \(I_c=p_m m + x\). The problem can be expressed as: $$ \max_{m, x} U_P(m, x) \text{, s.t. } I_c = p_m m + x $$

Derive the First-Order Condition for the Patient's Problem

To derive the first-order condition for the patient's problem, we can use the method of Lagrange multipliers. Define the Lagrangian as: $$ \mathcal{L}(m, x, \lambda) = U_P(m, x) + \lambda(I_c - p_m m - x) $$ Differentiate the Lagrangian with respect to \(m, x\), and \(\lambda\), and set the derivatives equal to zero: $$ \frac{\partial \mathcal{L}}{\partial m} = \frac{\partial U_P}{\partial m} - \lambda p_m = 0 $$ $$ \frac{\partial \mathcal{L}}{\partial x} = \frac{\partial U_P}{\partial x} - \lambda = 0 $$ $$ \frac{\partial \mathcal{L}}{\partial \lambda} = I_c - p_m m - x = 0 $$ We obtain the first-order condition for the patient's problem as: $$ \frac{\partial U_P}{\partial m} - \lambda p_m = 0 $$

Doctor's Utility Maximization Problem

The doctor's utility function is given by \(U_d(I_d)+U_P\). To maximize his/her utility, the doctor will choose the level of medical care \(m\) that maximizes this function, subject to the patient's budget constraint. The problem can be expressed as: $$ \max_{m} (U_d(p_m m)+U_P(m, I_c - p_m m)) $$

Derive the First-Order Condition for the Doctor's Problem

Now, we need to differentiate the doctor's utility function with respect to \(m\), and set the derivative equal to zero: $$ \frac{\mathrm{d} U_d(p_m m)}{\mathrm{d} m} + \frac{\partial U_P(m, I_c - p_m m)}{\partial m} = 0 $$ The first-order condition for the doctor's problem is: $$ p_m \frac{\mathrm{d} U_d}{\mathrm{d} I_d} + \frac{\partial U_P}{\partial m} = 0 $$

Compare the First-Order Conditions and Show that the Level of Medical Care is Higher

We have the first-order conditions for the patient's and doctor's problems as: - Patient: \(\frac{\partial U_P}{\partial m} - \lambda p_m = 0\) - Doctor: \(p_m \frac{\mathrm{d} U_d}{\mathrm{d} I_d} + \frac{\partial U_P}{\partial m} = 0\) Comparing these two first-order conditions, we have: $$ \lambda p_m = p_m \frac{\mathrm{d} U_d}{\mathrm{d} I_d} $$ Since the doctor is altruistic, he/she derives utility from both his/her income and the patient's well-being. From the equation above, we can note that the weight given to the patient's utility, \(\lambda\), is equal to the marginal utility of the doctor's income with respect to medical care, \(\frac{\mathrm{d} U_d}{\mathrm{d} I_d}\). Therefore, as the doctor's utility increases one-for-one with the patient's utility due to altruistic considerations, the doctor will generally choose a higher level of medical care than a fully informed patient would choose.

Key concepts

Utility Maximization in Microeconomics

Utility maximization is a fundamental concept in microeconomics reflecting the idea that individuals and entities make choices to achieve the highest level of satisfaction or 'utility' from their available resources. Understanding this concept involves studying how consumers allocate their limited income across various goods and services to optimize their overall well-being under given prices and a budget constraint.

In the context of healthcare, patients seek to maximize their utility, represented by a utility function, such as the one indicated as 'U_P(m, x)', where 'm' stands for medical care, and 'x' for other consumption. The goal for the patient is to find the perfect balance between spending on medical care and other goods, so they achieve maximum satisfaction without exceeding their income or budget. The concept also includes the idea of diminishing marginal utility, meaning that as one consumes more medical care, the added satisfaction from an additional unit might decline.

Budget Constraint

The budget constraint represents the combinations of goods and services a consumer can purchase with their limited income at current prices. It is the foundational underpinning of the utility maximization problem in microeconomics. In our healthcare example, the budget constraint is modeled by the equation 'I_c = p_mm + x', where 'I_c' is the consumer's income, 'p_m' is the price of medical care, 'm' is the quantity of medical care, and 'x' is the spending on other goods.

A consumer can only purchase combinations of 'm' and 'x' that lie on or below the budget constraint line. This constraint forces the patient to make trade-offs between medical care and other goods because the resources (income) available are limited. If prices change or income varies, the budget constraint shifts, affecting the consumer's optimization problem.

Lagrange Multipliers Method

The Lagrange multipliers method is a mathematical optimization technique that is particularly useful in solving problems involving constraints, such as the utility maximization problem under a budget constraint. By introducing a Lagrange multiplier (often denoted by \( \lambda \)), this method creates a new function, the Lagrangian, which combines the objective function (in this case, the utility function 'U_P(m, x)') with the constraint (the budget equation 'I_c = p_mm + x').

The essence of this method is to take partial derivatives of the Lagrangian with respect to each variable, including the multiplier \lambda, and setting them equal to zero. These steps uncover the first-order conditions that characterize the optimal choice. The multiplier itself indicates the rate at which utility changes as the budget changes, providing insight into the value or 'shadow price' of relaxing the budget constraint by one unit.