Developing a comprehensive financial plan to cater for phone repair services is a task that requires an intricate understanding of the variables at play. This is not to suggest that it is an insurmountable challenge, but rather a call to prudently delineate the particulars of your specific situation, and thus, mold a budget that optimally serves your needs. In this discourse, we'll walk through the labyrinth of budgeting for phone repair services, guided by economic, mathematical, and statistical principles.

Fundamentally, phone repair services present a classic case of demand and supply dynamics. Every individual phone user contributes to the aggregate demand, while repair shops generate the supply. These forces interact in the marketplace to establish the equilibrium price – the point at which the quantity demanded equals the quantity supplied. This price, in turn, forms the basis of our budgeting process.

Firstly, you must identify the repair needs of your phone. This is equivalent to the "demand" in our economic model. It is a function of the type and severity of the damage on the phone, the make and model, and the availability of repair parts and expertise. For instance, replacing a cracked screen on a flagship Samsung model would ostensibly be more expensive than on an entry-level model due to the higher cost of the parts and expertise required. Consequently, a higher budgetary allocation would be necessary.

Our second step involves scouting the marketplace for appropriate service providers to meet our repair needs – the "supply" side of our model. It is advisable to obtain quotations from several repair shops to get a sense of the prevailing market price. While doing this, it is crucial to consider other factors such as warranties offered, the reputation of the repair shop, and the turnaround time.

Now, to the number crunching. Given the quotations obtained, we can calculate the mean, median, and mode. These statistical measures will offer an accurate snapshot of the market price, thus providing a sound basis for our budgeting. If you come across a repair quote that is significantly lower than the mean, caution is recommended, as the quality of service might be compromised.

With this data, the next step involves factoring in the law of diminishing marginal utility, an economic principle that states that the satisfaction (utility) gained from an additional dollar spent decreases as more money is spent. In the context of phone repairs, this means that beyond a certain repair cost, the value derived from fixing the phone might not be worth the extra money spent. The tipping point at which this happens will be a key consideration in determining your budget limit.

Then, we move on to consider the risks involved. Repairs carry an inherent risk, quantifiable as the probability of the repair not rectifying the problem or causing additional damage. This risk, though often overlooked, should be factored into your budget. One could employ the principles of risk management, attaching a monetary value to these potential losses, and incorporating it into the budget.

Lastly, it is important to periodically review and adjust the budget to reflect the changing realities of the market. Prices for parts and services fluctuate due to factors such as inflation, exchange rates, and shifts in supply and demand. A budget set a year ago might no longer be realistic today. Hence, an ongoing appraisal of the budget is essential.

Phone repair services, like many other sectors of the economy, present a complex interplay of economic, mathematical, and statistical elements. A deep understanding of these elements enables one to craft a budget that not only caters for one's repair needs but also optimizes resource utilization, balances risks, and maintains relevance over time. Despite the complexity, the merits that stem from such a well-calibrated budget are certainly worth the intellectual investment.