Question

Find \(Y=\) and \(C^{*}\) from the following: \\[ \begin{array}{l} Y=C+1_{0}+C_{0} \\ C=25+6 Y^{1 / 2} \\ l_{0}=16 \\ C_{0}=14 \end{array} \\]

Short answer

\( Y = 121 \) and \( C^{*} = 91 \).

Step-by-step solution

Simplify the Equation for Y

Rearrange the equation for total output Y: given that \( Y = C + I_0 + C_0 \), we substitute the values of \( I_0 \) and \( C_0 \). This gives us: \[ Y = C + 16 + 14 \] Therefore, simplify it to: \[ Y = C + 30 \]

Substitute C into the Y Equation

Now substitute \( C = 25 + 6Y^{1/2} \) into the simplified equation from Step 1: \[ Y = (25 + 6Y^{1/2}) + 30 \] Simplify further: \[ Y = 55 + 6Y^{1/2} \]

Isolate and Simplify

Isolate both terms containing \(Y\): \[ Y - 6Y^{1/2} = 55 \]

Substitute and Solve Quadratically

Now, let \( x = Y^{1/2} \), implying \( x^2 = Y \). Substitute to obtain: \[ x^2 - 6x = 55 \] Rearrange the equation: \[ x^2 - 6x - 55 = 0 \] Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve: Plug in values: \( a = 1, b = -6, c = -55 \) \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-55)}}{2 \cdot 1} \] \[ x = \frac{6 \pm \sqrt{36 + 220}}{2} \] \[ x = \frac{6 \pm \sqrt{256}}{2} \] \[ x = \frac{6 \pm 16}{2} \], yielding solutions \( x = 11 \) and \( x = -5 \). Discard \( x = -5 \) since a square root cannot be negative.

Find Y

Review \( x = 11 \) which was \( Y^{1/2} \); find \( Y \) as \( x^2 \): \[ Y = 11^2 = 121 \] Hence, \( Y = 121 \).

Calculate C^{*}

Now, substitute \( Y = 121 \) back into the consumption equation \( C = 25 + 6Y^{1/2} \): \[ C = 25 + 6 \cdot 11 = 25 + 66 = 91 \]. Therefore, \( C^{*} = 91 \).

Key concepts

Quadratic Equations

Quadratic equations are a fundamental part of mathematics, appearing frequently in various applications, including those in economics. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \), where **a**, **b**, and **c** are constants, and \( x \) represents an unknown variable. The solution to this equation provides the value(s) of \( x \) which makes the equation true. To solve it, you can use the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is a reliable tool for finding solutions, especially when factoring is not straightforward. In our exercise, the equation was broken down to solve for \( x^2 - 6x = 55 \), leading to \( x^2 - 6x - 55 = 0 \). Upon applying the quadratic formula, we arrive at the solutions \( x = 11 \) and \( x = -5 \). The result \( x = 11 \) is used, as square roots cannot be negative. This approach highlights the importance of quadratic equations in economic models, where they often describe relationships between key economic quantities.

Economic Modeling

Economic modeling refers to the creation of abstract representations of economic processes. These models help in understanding, explaining, and predicting economic behaviors. Economic models often use mathematical equations to express relationships between different economic variables and assumptions. In the context of our exercise, the model expressed how consumption \( C \), investment \( I_0 \), and autonomous consumption \( C_0 \) relate to output \( Y \).

• Output Equation: \( Y = C + I_0 + C_0 \)
• Consumption Equation: \( C = 25 + 6 Y^{1/2} \)

These equations capture the essence of the economic model by showing how output is derived from other economic activities like consumption and investment. Models can vary in complexity, from simple equations to large systems with numerous variables, but their primary function remains consistent: to map economic relationships and help forecast future economic events.

Consumption Function

The consumption function is an economic concept that depicts the relationship between total consumption and total income in an economy. It shows how households' consumption levels change with variations in their income. The function can take many forms, but in our exercise, it is represented as:

• \( C = 25 + 6 Y^{1/2} \)

Here, the consumption function suggests that a part of consumption is autonomous (fixed at \( 25 \)) and another part is dependent on the square root of the output \( Y \), influenced by the factor \( 6 \). This implies that as output increases, so does consumption, following a square-root proportionality. Understanding this relationship is crucial for analyzing consumer behavior and formulating policy decisions, as it provides insights into how consumption might respond to changes in economic output.

Output Equation

The output equation in economic modeling links the output of an economy or a firm to various inputs. It represents the total amount of goods and services produced, combining factors such as consumption, investment, and autonomous consumption. In this exercise, the output equation is given by:

• \( Y = C + I_0 + C_0 \)
• With specific values, \( Y = C + 16 + 14 = C + 30 \)

This formulation shows that total output \( Y \) is the sum of consumption and other key components, such as investment \( I_0 \) and autonomous consumption \( C_0 \). Solving for \( Y \) involves substituting the consumption function, resulting in a quadratic equation, as previously discussed. This structure allows us to calculate how changes in consumption and the constants \( I_0 \) and \( C_0 \) affect overall economic output. By analyzing this equation, we gain a clearer understanding of the macroeconomic environment and its dynamics.