According to the asset-market approach in economics, exchange rates are determined by the supply and demand for different currencies in the foreign exchange market. Changes in various economic variables, such as inflation rates, can influence these supply and demand dynamics, leading to shifts in exchange rates. Let’s examine how the exchange rate between the U.S. dollar ($) and the British pound (£) might change in both the short run and the long run when U.K. inflation rates are expected to increase, while keeping other factors constant (ceteris paribus).

Short Run: In the short run, an increase in U.K. inflation rates, ceteris paribus, can be expected to have the following effects on the exchange rate:

1. Effect on Relative Interest Rates: Higher inflation in the U.K. could lead to higher nominal interest rates in order to compensate for the loss of purchasing power. This would make U.K. assets more attractive to investors seeking higher returns. As a result, there would be an increase in demand for British assets, including the British pound, in the short run.
2. Effect on Capital Flows: Higher interest rates in the U.K. could also attract foreign capital, as investors seek to take advantage of the higher returns. This increase in capital flows could create additional demand for the pound.
3. Expected Depreciation: Despite the increased demand for British assets, the expectation of higher inflation erodes the real value of the pound over time. Investors may anticipate that the purchasing power of the pound will decrease relative to other currencies due to higher inflation. This expectation of future depreciation can lead to a decrease in the demand for the pound.

Overall, in the short run, the increase in demand due to higher interest rates and capital flows could partially offset the decrease in demand caused by the expectation of future depreciation. As a result, the pound might experience a slight appreciation against the U.S. dollar.

Long Run: In the long run, the exchange rate would likely be more affected by the expectation of higher inflation:

1. Purchasing Power Parity (PPP): According to the theory of purchasing power parity, exchange rates should adjust to reflect changes in relative price levels (inflation rates) between countries. If the U.K. has higher expected inflation rates than the U.S., this implies that the U.K. pound is losing value in terms of purchasing power. As a result, in the long run, the pound would be expected to depreciate against the U.S. dollar to compensate for the anticipated inflation differential.
2. Currency Depreciation: The expectation of future depreciation due to higher inflation can lead to a decrease in demand for the pound as investors and traders look to hold other currencies that are expected to maintain their value better.

In summary, in the long run, the asset-market approach suggests that the U.K. pound would likely depreciate against the U.S. dollar due to the expectation of higher inflation in the U.K. This depreciation would align the exchange rate with changes in relative purchasing power between the two currencies.