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Consider the following scenario: The Czech Republic has been enacting fiscal policies to curb its budget deficits. The International Monetary Fund (IMF) has praised the Czech Republic for cutting its federal budget deficit from \(1.4 \%\) of its gross domestic product (GDP) to \(0.4 \%\) of GDP. The annual budget deficit in the Czech Republic from 2009 to 2015 can be expressed by the function $$F(x)=-18.5+1.2 x, \quad x=9,10, \ldots, 15$$ where \(x\) represents the number of years since 2000 and \(F(x)\) represents the annual budget deficit in billions of euros. Use this function to answer the following questions. Evaluate \(F(12) .\) Explain why the function value is negative.

Short Answer

Expert verified
The deficit in 2012 was -4.1 billion euros, indicating expenditure exceeded revenue.

Step by step solution

01

Understanding the Function

The function given is \( F(x) = -18.5 + 1.2x \) where \( x \) is the number of years since 2000. \( F(x) \) represents the annual budget deficit in billions of euros. We need to evaluate this function at \( x = 12 \) to find the deficit for the year 2012.
02

Substitute x in the Function

To find \( F(12) \), substitute 12 for \( x \) in the function: \[ F(12) = -18.5 + 1.2 \times 12 \]
03

Compute the Product

Calculate the product \( 1.2 \times 12 \):\[ 1.2 \times 12 = 14.4 \]
04

Evaluate the Expression

Now substitute the product from the previous step back into the expression to find \( F(12) \):\[ F(12) = -18.5 + 14.4 = -4.1 \]Thus, \( F(12) \) is \(-4.1\).
05

Interpret the Negative Value

A negative value of \( F(x) \) means that the budget is in deficit. Since \( F(x) \) is negative, the Czech Republic's budget was indeed in a deficit state for that year, indicating expenditure exceeded revenue by 4.1 billion euros.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Budget Deficit
A budget deficit occurs when a country spends more money than it brings in through revenue, such as taxes. When a nation like the Czech Republic faces a budget deficit, its expenditure surpasses its income, leading to a shortfall that must be managed. This can result from various factors including increased public spending or decreased tax income.
The Czech Republic's efforts to reduce its budget deficit were recognized by the IMF, showcasing how fiscal policy can positively impact a nation's economy. Fiscal policy involves government adjustments in spending and taxation to influence economic conditions.
By narrowing its deficit from 1.4% to 0.4% of GDP, the Czech Republic showed a significant improvement in maintaining economic stability. A lower budget deficit often signifies a healthier economy and can lead to positive economic outcomes such as lower national debt and more room for essential public services.
Gross Domestic Product (GDP)
Gross Domestic Product, or GDP, is a crucial indicator that represents the total value of all goods and services produced over a specific time period within a country. GDP is essential as it reflects the economic health and size of a country's economy.
Countries aim to optimize their GDP, as it directly influences living standards, employment rates, and social well-being. The Czech Republic's GDP is used as a benchmark to gauge its budget deficit in this exercise, illustrating the ratio of deficit in terms of GDP percentage.
  • An increasing GDP suggests economic growth and expansion.
  • A stable or declining GDP might indicate economic stagnation or contraction.
By reducing the budget deficit relative to GDP, the Czech Republic improves its fiscal stability, ensuring that its economic growth is not hampered by excessive borrowing or debt accumulation.
Mathematical Functions
Mathematical functions are powerful tools for modeling real-world scenarios. In this exercise, the function provided is a linear function of the form \( F(x) = -18.5 + 1.2x \). This type of function is crucial for predicting and understanding trends over time.
Within the context of fiscal policy, the function here models the annual budget deficit of the Czech Republic in billions of euros. Each year is represented by \( x \), calculated from the year 2000.
  • Substitute \( x \) to find the deficit for a given year. For example, \( x = 12 \) gives the deficit for 2012.
The result, \( F(12) = -4.1 \), indicates a deficit of 4.1 billion euros for 2012. The negative value signifies that expenses exceeded revenues. Understanding these functions enables policymakers and analysts to make informed decisions based on past data and anticipate future financial needs.

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