# Price-health Model

## Introduction

### Question: What's the Cost of Our Diagnosis Method

In our project, we lower down the cost extremely by using engineered bacteria.
Biosensors’ main advantage is the ability to replicate, which make them really cheap. With little cost of culture medium, we can cut down the cost of production. .

### Question: What's the Price for Early Diagnosis and Screening Now

Expensive.
Question: What’s the effect of cutting the cost.
Generally speaking, cutting down the price can improve its capacity to more rural and undeveloped areas.

### Model: The Effect of Cutting Cost

We are building a model of the price of medical prices and the possibility of people to use them.

## Grossman Model

The Grossman model of health demand is a model for studying the demand for health and medical care by Michael Grossman. The model based demand for medical care on the interaction between a demand function for health and a production function for health. Andrew Jones, Nigel Rice, and Paul Contoyannis call the model the "founding father of demand for health models".
In this model, health is a durable capital good which is inherited and depreciates over time. Investment in health takes the form of medical care purchases and other inputs and depreciation is interpreted as natural deterioration of health over time. In the model, health enters the utility function directly as a good people derive pleasure from and indirectly as an investment which makes more healthy time available for market and non-market activities.
The model creates a dynamic system of equations which can be cast as an optimization problem where utility is optimized over gross investment in health in each period, consumption of medical care, and time inputs in the gross investment function in each period. In this way, the length of life of the agent is partially endogenous to the model.

$M(t)=m[w(t),p(t),A(t),E(t),\Omega(t)]$

In this function, M(t) is the medical demand in t period, which can refer to the using rate of medical facilities or medical expenses. Others variables are: w means wealth, p means price, A means age, E means education status, Ω means other environment variables. This function means that demand of health is decided by price, age, education and other environment variables.
Solutions to the problem of this model generally show that the rate of return on health capital must equal the opportunity cost of said capital. Thus, increases in the depreciation rate over time cause the optimal stock of health to decrease. If the marginal efficiency of capital curve is inelastic, gross investment grows over time. In practical terms, this model thus predicts that older people will have more sick time and time spent on increasing health and have higher medical expenditures than younger people. Another implication is that since increases in wages shift the marginal efficiency of capital curve to the right and increases the curve's slope, an increase in wage will increase the demand for health capital.

## Budget and Visitor Demand Curve

Suppose the consumer's budget for a fixed time is Y, the other commodity price is Po, and the price of the visit is Pv, then the total amount he spends is Y. As shown in the figure, the point M is the number of consumptions for other products in the absence of a visit, and the point N is the number of visits when no other items are purchased, and the budget line is MN. The equilibrium point of the consumer is at point E, which is the tangent point of the indifference curve U2 and the budget line. At point E, the slope of the indifference curve is equal to the slope of the budget line. The marginal replacement rate (MRS) is the rate at which the consumer is willing to use other commodities instead of seeing a doctor. The slope of the budget line is the opposite of the price ratio.

Fig 1. The relation between commodity and treatment frequency

After having a budget curve, consumers will be treated. As shown in the figure, the horizontal axis represents the number of visits, and the vertical axis represents the sum of other commodities. Consumers choose between visiting and other products. From V1 to V2 to V3, respectively, the increase in the number of visits indicates that the visit is cheaper than other products, and the price of the visit changes.
The graph mainly shows the income effect and the substitution effect. Among them, the income effect is from U3 to U3', indicating that the income has increased, so that the consumption of other commodities will increase, and the number of visits will increase. U3’ represents the budget constraint for increased income, and U3 represents the budget constraint for fixed income and price changes. The substitution effect is from E1 to E2, and then to E3, the consumption of other commodities has not changed, but the number of visits is increasing, indicating that the health status is reduced or the income is used in the direction of treatment. This is because the income is more used. Come to buy a doctor. In summary, the graph shows that changes in visit costs and or changes in income will change the number of visits.

Fig 2. The demand curve

Assume that at two different times, in addition to health conditions, all other aspects of the consumer's economy are the same. At the equilibrium point E, he is very healthy, and when E' indicates that he is ill, the whole health condition is relatively poor. Changes in health status will affect Allen's preference for visits and other commodities, as reflected by different indifference curves and points E in different physician health maps.
As shown in the figure, at the equilibrium point of illness and non-illness, his visit has the same marginal replacement rate as other commodities. This means that prices, income levels, hobbies, health conditions and other environmental factors can affect the consumption of medical services.

Fig 3. Relation between price and amount of need

When health is used as a consumer product, income is positively related to health; as an investment, income increases and consumption decreases.

(Author:Bozitao Zhong)