> Bonds are sold on the secondary market at market value but that is just between private investors right?

Let's say the market value of a government bond on the secondary market is a 4.5% yield. (If it's a bond that pays exactly 4.5% then it will cost 100. If it pays a higher or lower coupon then its market price will be adjusted so that it yields 4.5% when purchased)

If the government goes and offers to sell new bonds on the market at the par price of 100 and is only offering 4% interest, who is going to buy them? If you're an investor, it's much better for you to just buy the existing bonds on the market which give you a 4.5% return. If the government wants to sell new bonds it's going to have to offer a rate that is competitive with the bonds that are currently on the market.

The government has to match the secondary market. If they bring out bonds that perform worse than existing bonds then noone will buy them and instead grab existing ones from the secondary market instead.

>So if a government wants to borrow money it issues bonds at a fixed interest rate at a certain maturity date?

Correct. There are other ways a goverment can raise money, but issuing bonds is often seen as a good way to be able to spend more money without causing inflation. A bond is nothing more then "you pay us X now, we will pay Y back in Z years."

>How does it know what to set that rate at?

Goverments have credit ratings (similar to companies). Since a lot of investors use these ratings to decide to buy or not buy their bonds, goverments can use a similar analesys the investors use to decide there interest. Most goverments also have an institution to monitor there economy. These will probably have an inestigation ready with the effects of different interest rates.

>Bonds are sold on the secondary market at market value but that is just between private investors right?

Correct

>How does the secondary market price impact how the government services its debt given they will just offer par 100 at 4%?

The only thing that changes is who the goverment has to pay at the end of the agreed period. The secondary market doesn't change the bond, just the owner. Where it does matter is when they want to issue extra bonds. If on the secondary market your bonds are valued less then expacted new bonds will probably need a higher interest rate to be sold. While if there sold fir more they could lower the rate for New bonds.

The real trick is that governments actually auction bonds rather than selling them at a fixed price, so the market decides what they are worth at that moment. Typically though they choose a coupon rate (the percentage quoted) which matches market yields so that the bonds sell at or close to the nominal value.

> if a government wants to borrow money it issues bonds at a fixed interest rate

Government bond interest rates are not fixed. They are set at auction.

Government: Here's an IOU. It says "I pay you $100 in yearly installments over the next 20 years -- US government" and this crowd of bidders has assembled to buy it today.

Auctioneer: Do I hear $50? $70 from the man in the red shirt. Okay, I hear $75 from the woman from BigCapital Megafund. JP Morgan, with your hand in the air, $80. Last National Bank, $86. $86 from Last National, any other bids? Going once...Going twice...SOLD to Last National Bank."

Then Last National Bank pays the government $86 when the auction ends. According to the IOU, the government pays Last National Bank $5 a year for the next 20 years.

Except the auction process isn't a live auction, it's done by everyone sending their bids to a government computer system that can handle thousands of bidders. After the deadline for submitting bids, the computer calculates the auction results.

The bond contract states. We will pay you $100 in 5 years time, and we will also pay you $5 per year.

This is expressed as a 5% interest rate on the bond. And it appears as if the government are repaying you the $100 that they borrowed when they initially sold the bond, but that part is not entirely true.

The government simply has a contract stating that they will pay $100 in 5 years time, and $5 per year for each of those 5 years. Then they put that contract up for auction.

If they have set their interest rate correctly, the market will settle on exactly $100 as the price of that bond, but it rarely goes that well. Instead they might settle on a price of $99.99 or literally any other amount. If the market buys the bonds for less than the "face value" then the interest rate was set to low, if the market settles on a price more than face value, the interest rate was too high.

It's all a bit of a balancing act. The interest rate (the profit) balances against the initial investment (the price of the bond). This is true in both the primary market (when the government initially sells the bond) as well as the secondary market (when the bond holder sells it to someone else).